the scientific method of problem solving is

What is the Scientific Method: How does it work and why is it important?

The scientific method is a systematic process involving steps like defining questions, forming hypotheses, conducting experiments, and analyzing data. It minimizes biases and enables replicable research, leading to groundbreaking discoveries like Einstein's theory of relativity, penicillin, and the structure of DNA. This ongoing approach promotes reason, evidence, and the pursuit of truth in science.

Updated on November 18, 2023

Beginning in elementary school, we are exposed to the scientific method and taught how to put it into practice. As a tool for learning, it prepares children to think logically and use reasoning when seeking answers to questions.

Rather than jumping to conclusions, the scientific method gives us a recipe for exploring the world through observation and trial and error. We use it regularly, sometimes knowingly in academics or research, and sometimes subconsciously in our daily lives.

In this article we will refresh our memories on the particulars of the scientific method, discussing where it comes from, which elements comprise it, and how it is put into practice. Then, we will consider the importance of the scientific method, who uses it and under what circumstances.

What is the scientific method?

The scientific method is a dynamic process that involves objectively investigating questions through observation and experimentation . Applicable to all scientific disciplines, this systematic approach to answering questions is more accurately described as a flexible set of principles than as a fixed series of steps.

The following representations of the scientific method illustrate how it can be both condensed into broad categories and also expanded to reveal more and more details of the process. These graphics capture the adaptability that makes this concept universally valuable as it is relevant and accessible not only across age groups and educational levels but also within various contexts.

Steps in the scientific method

While the scientific method is versatile in form and function, it encompasses a collection of principles that create a logical progression to the process of problem solving:

• Define a question : Constructing a clear and precise problem statement that identifies the main question or goal of the investigation is the first step. The wording must lend itself to experimentation by posing a question that is both testable and measurable.
• Gather information and resources : Researching the topic in question to find out what is already known and what types of related questions others are asking is the next step in this process. This background information is vital to gaining a full understanding of the subject and in determining the best design for experiments. 
• Form a hypothesis : Composing a concise statement that identifies specific variables and potential results, which can then be tested, is a crucial step that must be completed before any experimentation. An imperfection in the composition of a hypothesis can result in weaknesses to the entire design of an experiment.
• Perform the experiments : Testing the hypothesis by performing replicable experiments and collecting resultant data is another fundamental step of the scientific method. By controlling some elements of an experiment while purposely manipulating others, cause and effect relationships are established.
• Analyze the data : Interpreting the experimental process and results by recognizing trends in the data is a necessary step for comprehending its meaning and supporting the conclusions. Drawing inferences through this systematic process lends substantive evidence for either supporting or rejecting the hypothesis.
• Report the results : Sharing the outcomes of an experiment, through an essay, presentation, graphic, or journal article, is often regarded as a final step in this process. Detailing the project's design, methods, and results not only promotes transparency and replicability but also adds to the body of knowledge for future research.
• Retest the hypothesis : Repeating experiments to see if a hypothesis holds up in all cases is a step that is manifested through varying scenarios. Sometimes a researcher immediately checks their own work or replicates it at a future time, or another researcher will repeat the experiments to further test the hypothesis.

Where did the scientific method come from?

Oftentimes, ancient peoples attempted to answer questions about the unknown by:

• Making simple observations
• Discussing the possibilities with others deemed worthy of a debate
• Drawing conclusions based on dominant opinions and preexisting beliefs

For example, take Greek and Roman mythology. Myths were used to explain everything from the seasons and stars to the sun and death itself.

However, as societies began to grow through advancements in agriculture and language, ancient civilizations like Egypt and Babylonia shifted to a more rational analysis for understanding the natural world. They increasingly employed empirical methods of observation and experimentation that would one day evolve into the scientific method . 

In the 4th century, Aristotle, considered the Father of Science by many, suggested these elements , which closely resemble the contemporary scientific method, as part of his approach for conducting science:

• Study what others have written about the subject.
• Look for the general consensus about the subject.
• Perform a systematic study of everything even partially related to the topic.

By continuing to emphasize systematic observation and controlled experiments, scholars such as Al-Kindi and Ibn al-Haytham helped expand this concept throughout the Islamic Golden Age . 

In his 1620 treatise, Novum Organum , Sir Francis Bacon codified the scientific method, arguing not only that hypotheses must be tested through experiments but also that the results must be replicated to establish a truth. Coming at the height of the Scientific Revolution, this text made the scientific method accessible to European thinkers like Galileo and Isaac Newton who then put the method into practice.

As science modernized in the 19th century, the scientific method became more formalized, leading to significant breakthroughs in fields such as evolution and germ theory. Today, it continues to evolve, underpinning scientific progress in diverse areas like quantum mechanics, genetics, and artificial intelligence.

Why is the scientific method important?

The history of the scientific method illustrates how the concept developed out of a need to find objective answers to scientific questions by overcoming biases based on fear, religion, power, and cultural norms. This still holds true today.

By implementing this standardized approach to conducting experiments, the impacts of researchers’ personal opinions and preconceived notions are minimized. The organized manner of the scientific method prevents these and other mistakes while promoting the replicability and transparency necessary for solid scientific research.

The importance of the scientific method is best observed through its successes, for example: 

• “ Albert Einstein stands out among modern physicists as the scientist who not only formulated a theory of revolutionary significance but also had the genius to reflect in a conscious and technical way on the scientific method he was using.” Devising a hypothesis based on the prevailing understanding of Newtonian physics eventually led Einstein to devise the theory of general relativity .
• Howard Florey “Perhaps the most useful lesson which has come out of the work on penicillin has been the demonstration that success in this field depends on the development and coordinated use of technical methods.” After discovering a mold that prevented the growth of Staphylococcus bacteria, Dr. Alexander Flemimg designed experiments to identify and reproduce it in the lab, thus leading to the development of penicillin .
• James D. Watson “Every time you understand something, religion becomes less likely. Only with the discovery of the double helix and the ensuing genetic revolution have we had grounds for thinking that the powers held traditionally to be the exclusive property of the gods might one day be ours. . . .” By using wire models to conceive a structure for DNA, Watson and Crick crafted a hypothesis for testing combinations of amino acids, X-ray diffraction images, and the current research in atomic physics, resulting in the discovery of DNA’s double helix structure .

Final thoughts

As the cases exemplify, the scientific method is never truly completed, but rather started and restarted. It gave these researchers a structured process that was easily replicated, modified, and built upon. 

While the scientific method may “end” in one context, it never literally ends. When a hypothesis, design, methods, and experiments are revisited, the scientific method simply picks up where it left off. Each time a researcher builds upon previous knowledge, the scientific method is restored with the pieces of past efforts.

By guiding researchers towards objective results based on transparency and reproducibility, the scientific method acts as a defense against bias, superstition, and preconceived notions. As we embrace the scientific method's enduring principles, we ensure that our quest for knowledge remains firmly rooted in reason, evidence, and the pursuit of truth.

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What Is the Scientific Method?

The scientific method is a systematic way of conducting experiments or studies so that you can explore the things you observe in the world and answer questions about them. The scientific method, also known as the hypothetico-deductive method, is a series of steps that can help you accurately describe the things you observe or improve your understanding of them.

Ultimately, your goal when you use the scientific method is to:

• Find a cause-and-effect relationship by asking a question about something you observed
• Collect as much evidence as you can about what you observed, as this can help you explore the connection between your evidence and what you observed
• Determine if all your evidence can be combined to answer your question in a way that makes sense

Francis Bacon and René Descartes are usually credited with formalizing the process in the 16th and 17th centuries. The two philosophers argued that research shouldn’t be guided by preset metaphysical ideas of how reality works. They supported the use of inductive reasoning to come up with hypotheses and understand new things about reality.

Scientific Method Steps

The scientific method is a step-by-step problem-solving process. These steps include:

Observe the world around you. This will help you come up with a topic you are interested in and want to learn more about. In many cases, you already have a topic in mind because you have a related question for which you couldn't find an immediate answer.

Either way, you'll start the process by finding out what people before you already know about the topic, as well as any questions that people are still asking about. You may need to look up and read books and articles from academic journals or talk to other people so that you understand as much as you possibly can about your topic. This will help you with your next step.

Ask questions. Asking questions about what you observed and learned from reading and talking to others can help you figure out what the "problem" is. Scientists try to ask questions that are both interesting and specific and can be answered with the help of a fairly easy experiment or series of experiments. Your question should have one part (called a variable) that you can change in your experiment and another variable that you can measure. Your goal is to design an experiment that is a "fair test," which is when all the conditions in the experiment are kept the same except for the one you change (called the experimental or independent variable).

Form a hypothesis and make predictions based on it.  A hypothesis is an educated guess about the relationship between two or more variables in your question. A good hypothesis lets you predict what will happen when you test it in an experiment. Another important feature of a good hypothesis is that, if the hypothesis is wrong, you should be able to show that it's wrong. This is called falsifiability. If your experiment shows that your prediction is true, then your hypothesis is supported by your data.

Test your prediction by doing an experiment or making more observations.  The way you test your prediction depends on what you are studying. The best support comes from an experiment, but in some cases, it's too hard or impossible to change the variables in an experiment. Sometimes, you may need to do descriptive research where you gather more observations instead of doing an experiment. You will carefully gather notes and measurements during your experiments or studies, and you can share them with other people interested in the same question as you. Ideally, you will also repeat your experiment a couple more times because it's possible to get a result by chance, but it's less possible to get the same result more than once by chance.

Draw a conclusion. You will analyze what you already know about your topic from your literature research and the data gathered during your experiment. This will help you decide if the conclusion you draw from your data supports or contradicts your hypothesis. If your results contradict your hypothesis, you can use this observation to form a new hypothesis and make a new prediction. This is why scientific research is ongoing and scientific knowledge is changing all the time. It's very common for scientists to get results that don't support their hypotheses. In fact, you sometimes learn more about the world when your experiments don't support your hypotheses because it leads you to ask more questions. And this time around, you already know that one possible explanation is likely wrong.

Use your results to guide your next steps (iterate). For instance, if your hypothesis is supported, you may do more experiments to confirm it. Or you could come up with a hypothesis about why it works this way and design an experiment to test that. If your hypothesis is not supported, you can come up with another hypothesis and do experiments to test it. You'll rarely get the right hypothesis in one go. Most of the time, you'll have to go back to the hypothesis stage and try again. Every attempt offers you important information that helps you improve your next round of questions, hypotheses, and predictions.

Share your results. Scientific research isn't something you can do on your own; you must work with other people to do it.   You may be able to do an experiment or a series of experiments on your own, but you can't come up with all the ideas or do all the experiments by yourself .

Scientists and researchers usually share information by publishing it in a scientific journal or by presenting it to their colleagues during meetings and scientific conferences. These journals are read and the conferences are attended by other researchers who are interested in the same questions. If there's anything wrong with your hypothesis, prediction, experiment design, or conclusion, other researchers will likely find it and point it out to you.

It can be scary, but it's a critical part of doing scientific research. You must let your research be examined by other researchers who are as interested and knowledgeable about your question as you. This process helps other researchers by pointing out hypotheses that have been proved wrong and why they are wrong. It helps you by identifying flaws in your thinking or experiment design. And if you don't share what you've learned and let other people ask questions about it, it's not helpful to your or anyone else's understanding of what happens in the world.

Scientific Method Example

Here's an everyday example of how you can apply the scientific method to understand more about your world so you can solve your problems in a helpful way.

Let's say you put slices of bread in your toaster and press the button, but nothing happens. Your toaster isn't working, but you can't afford to buy a new one right now. You might be able to rescue it from the trash can if you can figure out what's wrong with it. So, let's figure out what's wrong with your toaster.

Observation. Your toaster isn't working to toast your bread.

Ask a question. In this case, you're asking, "Why isn't my toaster working?" You could even do a bit of preliminary research by looking in the owner's manual for your toaster. The manufacturer has likely tested your toaster model under many conditions, and they may have some ideas for where to start with your hypothesis.

Form a hypothesis and make predictions based on it. Your hypothesis should be a potential explanation or answer to the question that you can test to see if it's correct. One possible explanation that we could test is that the power outlet is broken. Our prediction is that if the outlet is broken, then plugging it into a different outlet should make the toaster work again.

Test your prediction by doing an experiment or making more observations. You plug the toaster into a different outlet and try to toast your bread.

If that works, then your hypothesis is supported by your experimental data. Results that support your hypothesis don't prove it right; they simply suggest that it's a likely explanation. This uncertainty arises because, in the real world, we can't rule out the possibility of mistakes, wrong assumptions, or weird coincidences affecting the results. If the toaster doesn’t work even after plugging it into a different outlet, then your hypothesis is not supported and it's likely the wrong explanation.

Use your results to guide your next steps (iteration). If your toaster worked, you may decide to do further tests to confirm it or revise it. For example, you could plug something else that you know is working into the first outlet to see if that stops working too. That would be further confirmation that your hypothesis is correct.

If your toaster failed to toast when plugged into the second outlet, you need a new hypothesis. For example, your next hypothesis might be that the toaster has a shorted wire. You could test this hypothesis directly if you have the right equipment and training, or you could take it to a repair shop where they could test that hypothesis for you.

Share your results. For this everyday example, you probably wouldn't want to write a paper, but you could share your problem-solving efforts with your housemates or anyone you hire to repair your outlet or help you test if the toaster has a short circuit.

What the Scientific Method Is Used For

The scientific method is useful whenever you need to reason logically about your questions and gather evidence to support your problem-solving efforts. So, you can use it in everyday life to answer many of your questions; however, when most people think of the scientific method, they likely think of using it to:

Describe how nature works . It can be hard to accurately describe how nature works because it's almost impossible to account for every variable that's involved in a natural process. Researchers may not even know about many of the variables that are involved. In some cases, all you can do is make assumptions. But you can use the scientific method to logically disprove wrong assumptions by identifying flaws in the reasoning.

Do scientific research in a laboratory to develop things such as new medicines.

Develop critical thinking skills.  Using the scientific method may help you develop critical thinking in your daily life because you learn to systematically ask questions and gather evidence to find answers. Without logical reasoning, you might be more likely to have a distorted perspective or bias. Bias is the inclination we all have to favor one perspective (usually our own) over another.

The scientific method doesn't perfectly solve the problem of bias, but it does make it harder for an entire field to be biased in the same direction. That's because it's unlikely that all the people working in a field have the same biases. It also helps make the biases of individuals more obvious because if you repeatedly misinterpret information in the same way in multiple experiments or over a period, the other people working on the same question will notice. If you don't correct your bias when others point it out to you, you'll lose your credibility. Other people might then stop believing what you have to say.

Why Is the Scientific Method Important?

When you use the scientific method, your goal is to do research in a fair, unbiased, and repeatable way. The scientific method helps meet these goals because:

It's a systematic approach to problem-solving. It can help you figure out where you're going wrong in your thinking and research if you're not getting helpful answers to your questions. Helpful answers solve problems and keep you moving forward. So, a systematic approach helps you improve your problem-solving abilities if you get stuck.

It can help you solve your problems.  The scientific method helps you isolate problems by focusing on what's important. In addition, it can help you make your solutions better every time you go through the process.

It helps you eliminate (or become aware of) your personal biases.  It can help you limit the influence of your own personal, preconceived notions . A big part of the process is considering what other people already know and think about your question. It also involves sharing what you've learned and letting other people ask about your methods and conclusions. At the end of the process, even if you still think your answer is best, you have considered what other people know and think about the question.

The scientific method is a systematic way of conducting experiments or studies so that you can explore the world around you and answer questions using reason and evidence. It's a step-by-step problem-solving process that involves: (1) observation, (2) asking questions, (3) forming hypotheses and making predictions, (4) testing your hypotheses through experiments or more observations, (5) using what you learned through experiment or observation to guide further investigation, and (6) sharing your results.

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The 6 Scientific Method Steps and How to Use Them

General Education

When you’re faced with a scientific problem, solving it can seem like an impossible prospect. There are so many possible explanations for everything we see and experience—how can you possibly make sense of them all? Science has a simple answer: the scientific method.

The scientific method is a method of asking and answering questions about the world. These guiding principles give scientists a model to work through when trying to understand the world, but where did that model come from, and how does it work?

In this article, we’ll define the scientific method, discuss its long history, and cover each of the scientific method steps in detail.

What Is the Scientific Method?

At its most basic, the scientific method is a procedure for conducting scientific experiments. It’s a set model that scientists in a variety of fields can follow, going from initial observation to conclusion in a loose but concrete format.

The number of steps varies, but the process begins with an observation, progresses through an experiment, and concludes with analysis and sharing data. One of the most important pieces to the scientific method is skepticism —the goal is to find truth, not to confirm a particular thought. That requires reevaluation and repeated experimentation, as well as examining your thinking through rigorous study.

There are in fact multiple scientific methods, as the basic structure can be easily modified.  The one we typically learn about in school is the basic method, based in logic and problem solving, typically used in “hard” science fields like biology, chemistry, and physics. It may vary in other fields, such as psychology, but the basic premise of making observations, testing, and continuing to improve a theory from the results remain the same.

The History of the Scientific Method

The scientific method as we know it today is based on thousands of years of scientific study. Its development goes all the way back to ancient Mesopotamia, Greece, and India.

The Ancient World

In ancient Greece, Aristotle devised an inductive-deductive process , which weighs broad generalizations from data against conclusions reached by narrowing down possibilities from a general statement. However, he favored deductive reasoning, as it identifies causes, which he saw as more important.

Aristotle wrote a great deal about logic and many of his ideas about reasoning echo those found in the modern scientific method, such as ignoring circular evidence and limiting the number of middle terms between the beginning of an experiment and the end. Though his model isn’t the one that we use today, the reliance on logic and thorough testing are still key parts of science today.

The Middle Ages

The next big step toward the development of the modern scientific method came in the Middle Ages, particularly in the Islamic world. Ibn al-Haytham, a physicist from what we now know as Iraq, developed a method of testing, observing, and deducing for his research on vision. al-Haytham was critical of Aristotle’s lack of inductive reasoning, which played an important role in his own research.

Other scientists, including Abū Rayhān al-Bīrūnī, Ibn Sina, and Robert Grosseteste also developed models of scientific reasoning to test their own theories. Though they frequently disagreed with one another and Aristotle, those disagreements and refinements of their methods led to the scientific method we have today.

Following those major developments, particularly Grosseteste’s work, Roger Bacon developed his own cycle of observation (seeing that something occurs), hypothesis (making a guess about why that thing occurs), experimentation (testing that the thing occurs), and verification (an outside person ensuring that the result of the experiment is consistent).

After joining the Franciscan Order, Bacon was granted a special commission to write about science; typically, Friars were not allowed to write books or pamphlets. With this commission, Bacon outlined important tenets of the scientific method, including causes of error, methods of knowledge, and the differences between speculative and experimental science. He also used his own principles to investigate the causes of a rainbow, demonstrating the method’s effectiveness.

Scientific Revolution

Throughout the Renaissance, more great thinkers became involved in devising a thorough, rigorous method of scientific study. Francis Bacon brought inductive reasoning further into the method, whereas Descartes argued that the laws of the universe meant that deductive reasoning was sufficient. Galileo’s research was also inductive reasoning-heavy, as he believed that researchers could not account for every possible variable; therefore, repetition was necessary to eliminate faulty hypotheses and experiments.

All of this led to the birth of the Scientific Revolution , which took place during the sixteenth and seventeenth centuries. In 1660, a group of philosophers and physicians joined together to work on scientific advancement. After approval from England’s crown , the group became known as the Royal Society, which helped create a thriving scientific community and an early academic journal to help introduce rigorous study and peer review.

Previous generations of scientists had touched on the importance of induction and deduction, but Sir Isaac Newton proposed that both were equally important. This contribution helped establish the importance of multiple kinds of reasoning, leading to more rigorous study.

As science began to splinter into separate areas of study, it became necessary to define different methods for different fields. Karl Popper was a leader in this area—he established that science could be subject to error, sometimes intentionally. This was particularly tricky for “soft” sciences like psychology and social sciences, which require different methods. Popper’s theories furthered the divide between sciences like psychology and “hard” sciences like chemistry or physics.

Paul Feyerabend argued that Popper’s methods were too restrictive for certain fields, and followed a less restrictive method hinged on “anything goes,” as great scientists had made discoveries without the Scientific Method. Feyerabend suggested that throughout history scientists had adapted their methods as necessary, and that sometimes it would be necessary to break the rules. This approach suited social and behavioral scientists particularly well, leading to a more diverse range of models for scientists in multiple fields to use.

The Scientific Method Steps

Though different fields may have variations on the model, the basic scientific method is as follows:

#1: Make Observations 

Notice something, such as the air temperature during the winter, what happens when ice cream melts, or how your plants behave when you forget to water them.

#2: Ask a Question

Turn your observation into a question. Why is the temperature lower during the winter? Why does my ice cream melt? Why does my toast always fall butter-side down?

This step can also include doing some research. You may be able to find answers to these questions already, but you can still test them!

#3: Make a Hypothesis

A hypothesis is an educated guess of the answer to your question. Why does your toast always fall butter-side down? Maybe it’s because the butter makes that side of the bread heavier.

A good hypothesis leads to a prediction that you can test, phrased as an if/then statement. In this case, we can pick something like, “If toast is buttered, then it will hit the ground butter-first.”

#4: Experiment

Your experiment is designed to test whether your predication about what will happen is true. A good experiment will test one variable at a time —for example, we’re trying to test whether butter weighs down one side of toast, making it more likely to hit the ground first.

The unbuttered toast is our control variable. If we determine the chance that a slice of unbuttered toast, marked with a dot, will hit the ground on a particular side, we can compare those results to our buttered toast to see if there’s a correlation between the presence of butter and which way the toast falls.

If we decided not to toast the bread, that would be introducing a new question—whether or not toasting the bread has any impact on how it falls. Since that’s not part of our test, we’ll stick with determining whether the presence of butter has any impact on which side hits the ground first.

#5: Analyze Data

After our experiment, we discover that both buttered toast and unbuttered toast have a 50/50 chance of hitting the ground on the buttered or marked side when dropped from a consistent height, straight down. It looks like our hypothesis was incorrect—it’s not the butter that makes the toast hit the ground in a particular way, so it must be something else.

Since we didn’t get the desired result, it’s back to the drawing board. Our hypothesis wasn’t correct, so we’ll need to start fresh. Now that you think about it, your toast seems to hit the ground butter-first when it slides off your plate, not when you drop it from a consistent height. That can be the basis for your new experiment.

#6: Communicate Your Results

Good science needs verification. Your experiment should be replicable by other people, so you can put together a report about how you ran your experiment to see if other peoples’ findings are consistent with yours.

This may be useful for class or a science fair. Professional scientists may publish their findings in scientific journals, where other scientists can read and attempt their own versions of the same experiments. Being part of a scientific community helps your experiments be stronger because other people can see if there are flaws in your approach—such as if you tested with different kinds of bread, or sometimes used peanut butter instead of butter—that can lead you closer to a good answer.

A Scientific Method Example: Falling Toast

We’ve run through a quick recap of the scientific method steps, but let’s look a little deeper by trying again to figure out why toast so often falls butter side down.

#1: Make Observations

At the end of our last experiment, where we learned that butter doesn’t actually make toast more likely to hit the ground on that side, we remembered that the times when our toast hits the ground butter side first are usually when it’s falling off a plate.

The easiest question we can ask is, “Why is that?”

We can actually search this online and find a pretty detailed answer as to why this is true. But we’re budding scientists—we want to see it in action and verify it for ourselves! After all, good science should be replicable, and we have all the tools we need to test out what’s really going on.

Why do we think that buttered toast hits the ground butter-first? We know it’s not because it’s heavier, so we can strike that out. Maybe it’s because of the shape of our plate?

That’s something we can test. We’ll phrase our hypothesis as, “If my toast slides off my plate, then it will fall butter-side down.”

Just seeing that toast falls off a plate butter-side down isn’t enough for us. We want to know why, so we’re going to take things a step further—we’ll set up a slow-motion camera to capture what happens as the toast slides off the plate.

We’ll run the test ten times, each time tilting the same plate until the toast slides off. We’ll make note of each time the butter side lands first and see what’s happening on the video so we can see what’s going on.

When we review the footage, we’ll likely notice that the bread starts to flip when it slides off the edge, changing how it falls in a way that didn’t happen when we dropped it ourselves.

That answers our question, but it’s not the complete picture —how do other plates affect how often toast hits the ground butter-first? What if the toast is already butter-side down when it falls? These are things we can test in further experiments with new hypotheses!

Now that we have results, we can share them with others who can verify our results. As mentioned above, being part of the scientific community can lead to better results. If your results were wildly different from the established thinking about buttered toast, that might be cause for reevaluation. If they’re the same, they might lead others to make new discoveries about buttered toast. At the very least, you have a cool experiment you can share with your friends!

Key Scientific Method Tips

Though science can be complex, the benefit of the scientific method is that it gives you an easy-to-follow means of thinking about why and how things happen. To use it effectively, keep these things in mind!

Don’t Worry About Proving Your Hypothesis

One of the important things to remember about the scientific method is that it’s not necessarily meant to prove your hypothesis right. It’s great if you do manage to guess the reason for something right the first time, but the ultimate goal of an experiment is to find the true reason for your observation to occur, not to prove your hypothesis right.

Good science sometimes means that you’re wrong. That’s not a bad thing—a well-designed experiment with an unanticipated result can be just as revealing, if not more, than an experiment that confirms your hypothesis.

Be Prepared to Try Again

If the data from your experiment doesn’t match your hypothesis, that’s not a bad thing. You’ve eliminated one possible explanation, which brings you one step closer to discovering the truth.

The scientific method isn’t something you’re meant to do exactly once to prove a point. It’s meant to be repeated and adapted to bring you closer to a solution. Even if you can demonstrate truth in your hypothesis, a good scientist will run an experiment again to be sure that the results are replicable. You can even tweak a successful hypothesis to test another factor, such as if we redid our buttered toast experiment to find out whether different kinds of plates affect whether or not the toast falls butter-first. The more we test our hypothesis, the stronger it becomes!

What’s Next?

Want to learn more about the scientific method? These important high school science classes will no doubt cover it in a variety of different contexts.

Test your ability to follow the scientific method using these at-home science experiments for kids !

Need some proof that science is fun? Try making slime

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Melissa Brinks graduated from the University of Washington in 2014 with a Bachelor's in English with a creative writing emphasis. She has spent several years tutoring K-12 students in many subjects, including in SAT prep, to help them prepare for their college education.

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Scientific Method

Science is an enormously successful human enterprise. The study of scientific method is the attempt to discern the activities by which that success is achieved. Among the activities often identified as characteristic of science are systematic observation and experimentation, inductive and deductive reasoning, and the formation and testing of hypotheses and theories. How these are carried out in detail can vary greatly, but characteristics like these have been looked to as a way of demarcating scientific activity from non-science, where only enterprises which employ some canonical form of scientific method or methods should be considered science (see also the entry on science and pseudo-science ). Others have questioned whether there is anything like a fixed toolkit of methods which is common across science and only science. Some reject privileging one view of method as part of rejecting broader views about the nature of science, such as naturalism (Dupré 2004); some reject any restriction in principle (pluralism).

Scientific method should be distinguished from the aims and products of science, such as knowledge, predictions, or control. Methods are the means by which those goals are achieved. Scientific method should also be distinguished from meta-methodology, which includes the values and justifications behind a particular characterization of scientific method (i.e., a methodology) — values such as objectivity, reproducibility, simplicity, or past successes. Methodological rules are proposed to govern method and it is a meta-methodological question whether methods obeying those rules satisfy given values. Finally, method is distinct, to some degree, from the detailed and contextual practices through which methods are implemented. The latter might range over: specific laboratory techniques; mathematical formalisms or other specialized languages used in descriptions and reasoning; technological or other material means; ways of communicating and sharing results, whether with other scientists or with the public at large; or the conventions, habits, enforced customs, and institutional controls over how and what science is carried out.

While it is important to recognize these distinctions, their boundaries are fuzzy. Hence, accounts of method cannot be entirely divorced from their methodological and meta-methodological motivations or justifications, Moreover, each aspect plays a crucial role in identifying methods. Disputes about method have therefore played out at the detail, rule, and meta-rule levels. Changes in beliefs about the certainty or fallibility of scientific knowledge, for instance (which is a meta-methodological consideration of what we can hope for methods to deliver), have meant different emphases on deductive and inductive reasoning, or on the relative importance attached to reasoning over observation (i.e., differences over particular methods.) Beliefs about the role of science in society will affect the place one gives to values in scientific method.

The issue which has shaped debates over scientific method the most in the last half century is the question of how pluralist do we need to be about method? Unificationists continue to hold out for one method essential to science; nihilism is a form of radical pluralism, which considers the effectiveness of any methodological prescription to be so context sensitive as to render it not explanatory on its own. Some middle degree of pluralism regarding the methods embodied in scientific practice seems appropriate. But the details of scientific practice vary with time and place, from institution to institution, across scientists and their subjects of investigation. How significant are the variations for understanding science and its success? How much can method be abstracted from practice? This entry describes some of the attempts to characterize scientific method or methods, as well as arguments for a more context-sensitive approach to methods embedded in actual scientific practices.

1. Overview and organizing themes

2. historical review: aristotle to mill, 3.1 logical constructionism and operationalism, 3.2. h-d as a logic of confirmation, 3.3. popper and falsificationism, 3.4 meta-methodology and the end of method, 4. statistical methods for hypothesis testing, 5.1 creative and exploratory practices.

• 5.2 Computer methods and the ‘new ways’ of doing science

6.1 “The scientific method” in science education and as seen by scientists

6.2 privileged methods and ‘gold standards’, 6.3 scientific method in the court room, 6.4 deviating practices, 7. conclusion, other internet resources, related entries.

This entry could have been given the title Scientific Methods and gone on to fill volumes, or it could have been extremely short, consisting of a brief summary rejection of the idea that there is any such thing as a unique Scientific Method at all. Both unhappy prospects are due to the fact that scientific activity varies so much across disciplines, times, places, and scientists that any account which manages to unify it all will either consist of overwhelming descriptive detail, or trivial generalizations.

The choice of scope for the present entry is more optimistic, taking a cue from the recent movement in philosophy of science toward a greater attention to practice: to what scientists actually do. This “turn to practice” can be seen as the latest form of studies of methods in science, insofar as it represents an attempt at understanding scientific activity, but through accounts that are neither meant to be universal and unified, nor singular and narrowly descriptive. To some extent, different scientists at different times and places can be said to be using the same method even though, in practice, the details are different.

Whether the context in which methods are carried out is relevant, or to what extent, will depend largely on what one takes the aims of science to be and what one’s own aims are. For most of the history of scientific methodology the assumption has been that the most important output of science is knowledge and so the aim of methodology should be to discover those methods by which scientific knowledge is generated.

Science was seen to embody the most successful form of reasoning (but which form?) to the most certain knowledge claims (but how certain?) on the basis of systematically collected evidence (but what counts as evidence, and should the evidence of the senses take precedence, or rational insight?) Section 2 surveys some of the history, pointing to two major themes. One theme is seeking the right balance between observation and reasoning (and the attendant forms of reasoning which employ them); the other is how certain scientific knowledge is or can be.

Section 3 turns to 20 th century debates on scientific method. In the second half of the 20 th century the epistemic privilege of science faced several challenges and many philosophers of science abandoned the reconstruction of the logic of scientific method. Views changed significantly regarding which functions of science ought to be captured and why. For some, the success of science was better identified with social or cultural features. Historical and sociological turns in the philosophy of science were made, with a demand that greater attention be paid to the non-epistemic aspects of science, such as sociological, institutional, material, and political factors. Even outside of those movements there was an increased specialization in the philosophy of science, with more and more focus on specific fields within science. The combined upshot was very few philosophers arguing any longer for a grand unified methodology of science. Sections 3 and 4 surveys the main positions on scientific method in 20 th century philosophy of science, focusing on where they differ in their preference for confirmation or falsification or for waiving the idea of a special scientific method altogether.

In recent decades, attention has primarily been paid to scientific activities traditionally falling under the rubric of method, such as experimental design and general laboratory practice, the use of statistics, the construction and use of models and diagrams, interdisciplinary collaboration, and science communication. Sections 4–6 attempt to construct a map of the current domains of the study of methods in science.

As these sections illustrate, the question of method is still central to the discourse about science. Scientific method remains a topic for education, for science policy, and for scientists. It arises in the public domain where the demarcation or status of science is at issue. Some philosophers have recently returned, therefore, to the question of what it is that makes science a unique cultural product. This entry will close with some of these recent attempts at discerning and encapsulating the activities by which scientific knowledge is achieved.

Attempting a history of scientific method compounds the vast scope of the topic. This section briefly surveys the background to modern methodological debates. What can be called the classical view goes back to antiquity, and represents a point of departure for later divergences. [ 1 ]

We begin with a point made by Laudan (1968) in his historical survey of scientific method:

Perhaps the most serious inhibition to the emergence of the history of theories of scientific method as a respectable area of study has been the tendency to conflate it with the general history of epistemology, thereby assuming that the narrative categories and classificatory pigeon-holes applied to the latter are also basic to the former. (1968: 5)

To see knowledge about the natural world as falling under knowledge more generally is an understandable conflation. Histories of theories of method would naturally employ the same narrative categories and classificatory pigeon holes. An important theme of the history of epistemology, for example, is the unification of knowledge, a theme reflected in the question of the unification of method in science. Those who have identified differences in kinds of knowledge have often likewise identified different methods for achieving that kind of knowledge (see the entry on the unity of science ).

Different views on what is known, how it is known, and what can be known are connected. Plato distinguished the realms of things into the visible and the intelligible ( The Republic , 510a, in Cooper 1997). Only the latter, the Forms, could be objects of knowledge. The intelligible truths could be known with the certainty of geometry and deductive reasoning. What could be observed of the material world, however, was by definition imperfect and deceptive, not ideal. The Platonic way of knowledge therefore emphasized reasoning as a method, downplaying the importance of observation. Aristotle disagreed, locating the Forms in the natural world as the fundamental principles to be discovered through the inquiry into nature ( Metaphysics Z , in Barnes 1984).

Aristotle is recognized as giving the earliest systematic treatise on the nature of scientific inquiry in the western tradition, one which embraced observation and reasoning about the natural world. In the Prior and Posterior Analytics , Aristotle reflects first on the aims and then the methods of inquiry into nature. A number of features can be found which are still considered by most to be essential to science. For Aristotle, empiricism, careful observation (but passive observation, not controlled experiment), is the starting point. The aim is not merely recording of facts, though. For Aristotle, science ( epistêmê ) is a body of properly arranged knowledge or learning—the empirical facts, but also their ordering and display are of crucial importance. The aims of discovery, ordering, and display of facts partly determine the methods required of successful scientific inquiry. Also determinant is the nature of the knowledge being sought, and the explanatory causes proper to that kind of knowledge (see the discussion of the four causes in the entry on Aristotle on causality ).

In addition to careful observation, then, scientific method requires a logic as a system of reasoning for properly arranging, but also inferring beyond, what is known by observation. Methods of reasoning may include induction, prediction, or analogy, among others. Aristotle’s system (along with his catalogue of fallacious reasoning) was collected under the title the Organon . This title would be echoed in later works on scientific reasoning, such as Novum Organon by Francis Bacon, and Novum Organon Restorum by William Whewell (see below). In Aristotle’s Organon reasoning is divided primarily into two forms, a rough division which persists into modern times. The division, known most commonly today as deductive versus inductive method, appears in other eras and methodologies as analysis/​synthesis, non-ampliative/​ampliative, or even confirmation/​verification. The basic idea is there are two “directions” to proceed in our methods of inquiry: one away from what is observed, to the more fundamental, general, and encompassing principles; the other, from the fundamental and general to instances or implications of principles.

The basic aim and method of inquiry identified here can be seen as a theme running throughout the next two millennia of reflection on the correct way to seek after knowledge: carefully observe nature and then seek rules or principles which explain or predict its operation. The Aristotelian corpus provided the framework for a commentary tradition on scientific method independent of science itself (cosmos versus physics.) During the medieval period, figures such as Albertus Magnus (1206–1280), Thomas Aquinas (1225–1274), Robert Grosseteste (1175–1253), Roger Bacon (1214/1220–1292), William of Ockham (1287–1347), Andreas Vesalius (1514–1546), Giacomo Zabarella (1533–1589) all worked to clarify the kind of knowledge obtainable by observation and induction, the source of justification of induction, and best rules for its application. [ 2 ] Many of their contributions we now think of as essential to science (see also Laudan 1968). As Aristotle and Plato had employed a framework of reasoning either “to the forms” or “away from the forms”, medieval thinkers employed directions away from the phenomena or back to the phenomena. In analysis, a phenomena was examined to discover its basic explanatory principles; in synthesis, explanations of a phenomena were constructed from first principles.

During the Scientific Revolution these various strands of argument, experiment, and reason were forged into a dominant epistemic authority. The 16 th –18 th centuries were a period of not only dramatic advance in knowledge about the operation of the natural world—advances in mechanical, medical, biological, political, economic explanations—but also of self-awareness of the revolutionary changes taking place, and intense reflection on the source and legitimation of the method by which the advances were made. The struggle to establish the new authority included methodological moves. The Book of Nature, according to the metaphor of Galileo Galilei (1564–1642) or Francis Bacon (1561–1626), was written in the language of mathematics, of geometry and number. This motivated an emphasis on mathematical description and mechanical explanation as important aspects of scientific method. Through figures such as Henry More and Ralph Cudworth, a neo-Platonic emphasis on the importance of metaphysical reflection on nature behind appearances, particularly regarding the spiritual as a complement to the purely mechanical, remained an important methodological thread of the Scientific Revolution (see the entries on Cambridge platonists ; Boyle ; Henry More ; Galileo ).

In Novum Organum (1620), Bacon was critical of the Aristotelian method for leaping from particulars to universals too quickly. The syllogistic form of reasoning readily mixed those two types of propositions. Bacon aimed at the invention of new arts, principles, and directions. His method would be grounded in methodical collection of observations, coupled with correction of our senses (and particularly, directions for the avoidance of the Idols, as he called them, kinds of systematic errors to which naïve observers are prone.) The community of scientists could then climb, by a careful, gradual and unbroken ascent, to reliable general claims.

Bacon’s method has been criticized as impractical and too inflexible for the practicing scientist. Whewell would later criticize Bacon in his System of Logic for paying too little attention to the practices of scientists. It is hard to find convincing examples of Bacon’s method being put in to practice in the history of science, but there are a few who have been held up as real examples of 16 th century scientific, inductive method, even if not in the rigid Baconian mold: figures such as Robert Boyle (1627–1691) and William Harvey (1578–1657) (see the entry on Bacon ).

It is to Isaac Newton (1642–1727), however, that historians of science and methodologists have paid greatest attention. Given the enormous success of his Principia Mathematica and Opticks , this is understandable. The study of Newton’s method has had two main thrusts: the implicit method of the experiments and reasoning presented in the Opticks, and the explicit methodological rules given as the Rules for Philosophising (the Regulae) in Book III of the Principia . [ 3 ] Newton’s law of gravitation, the linchpin of his new cosmology, broke with explanatory conventions of natural philosophy, first for apparently proposing action at a distance, but more generally for not providing “true”, physical causes. The argument for his System of the World ( Principia , Book III) was based on phenomena, not reasoned first principles. This was viewed (mainly on the continent) as insufficient for proper natural philosophy. The Regulae counter this objection, re-defining the aims of natural philosophy by re-defining the method natural philosophers should follow. (See the entry on Newton’s philosophy .)

To his list of methodological prescriptions should be added Newton’s famous phrase “ hypotheses non fingo ” (commonly translated as “I frame no hypotheses”.) The scientist was not to invent systems but infer explanations from observations, as Bacon had advocated. This would come to be known as inductivism. In the century after Newton, significant clarifications of the Newtonian method were made. Colin Maclaurin (1698–1746), for instance, reconstructed the essential structure of the method as having complementary analysis and synthesis phases, one proceeding away from the phenomena in generalization, the other from the general propositions to derive explanations of new phenomena. Denis Diderot (1713–1784) and editors of the Encyclopédie did much to consolidate and popularize Newtonianism, as did Francesco Algarotti (1721–1764). The emphasis was often the same, as much on the character of the scientist as on their process, a character which is still commonly assumed. The scientist is humble in the face of nature, not beholden to dogma, obeys only his eyes, and follows the truth wherever it leads. It was certainly Voltaire (1694–1778) and du Chatelet (1706–1749) who were most influential in propagating the latter vision of the scientist and their craft, with Newton as hero. Scientific method became a revolutionary force of the Enlightenment. (See also the entries on Newton , Leibniz , Descartes , Boyle , Hume , enlightenment , as well as Shank 2008 for a historical overview.)

Not all 18 th century reflections on scientific method were so celebratory. Famous also are George Berkeley’s (1685–1753) attack on the mathematics of the new science, as well as the over-emphasis of Newtonians on observation; and David Hume’s (1711–1776) undermining of the warrant offered for scientific claims by inductive justification (see the entries on: George Berkeley ; David Hume ; Hume’s Newtonianism and Anti-Newtonianism ). Hume’s problem of induction motivated Immanuel Kant (1724–1804) to seek new foundations for empirical method, though as an epistemic reconstruction, not as any set of practical guidelines for scientists. Both Hume and Kant influenced the methodological reflections of the next century, such as the debate between Mill and Whewell over the certainty of inductive inferences in science.

The debate between John Stuart Mill (1806–1873) and William Whewell (1794–1866) has become the canonical methodological debate of the 19 th century. Although often characterized as a debate between inductivism and hypothetico-deductivism, the role of the two methods on each side is actually more complex. On the hypothetico-deductive account, scientists work to come up with hypotheses from which true observational consequences can be deduced—hence, hypothetico-deductive. Because Whewell emphasizes both hypotheses and deduction in his account of method, he can be seen as a convenient foil to the inductivism of Mill. However, equally if not more important to Whewell’s portrayal of scientific method is what he calls the “fundamental antithesis”. Knowledge is a product of the objective (what we see in the world around us) and subjective (the contributions of our mind to how we perceive and understand what we experience, which he called the Fundamental Ideas). Both elements are essential according to Whewell, and he was therefore critical of Kant for too much focus on the subjective, and John Locke (1632–1704) and Mill for too much focus on the senses. Whewell’s fundamental ideas can be discipline relative. An idea can be fundamental even if it is necessary for knowledge only within a given scientific discipline (e.g., chemical affinity for chemistry). This distinguishes fundamental ideas from the forms and categories of intuition of Kant. (See the entry on Whewell .)

Clarifying fundamental ideas would therefore be an essential part of scientific method and scientific progress. Whewell called this process “Discoverer’s Induction”. It was induction, following Bacon or Newton, but Whewell sought to revive Bacon’s account by emphasising the role of ideas in the clear and careful formulation of inductive hypotheses. Whewell’s induction is not merely the collecting of objective facts. The subjective plays a role through what Whewell calls the Colligation of Facts, a creative act of the scientist, the invention of a theory. A theory is then confirmed by testing, where more facts are brought under the theory, called the Consilience of Inductions. Whewell felt that this was the method by which the true laws of nature could be discovered: clarification of fundamental concepts, clever invention of explanations, and careful testing. Mill, in his critique of Whewell, and others who have cast Whewell as a fore-runner of the hypothetico-deductivist view, seem to have under-estimated the importance of this discovery phase in Whewell’s understanding of method (Snyder 1997a,b, 1999). Down-playing the discovery phase would come to characterize methodology of the early 20 th century (see section 3 ).

Mill, in his System of Logic , put forward a narrower view of induction as the essence of scientific method. For Mill, induction is the search first for regularities among events. Among those regularities, some will continue to hold for further observations, eventually gaining the status of laws. One can also look for regularities among the laws discovered in a domain, i.e., for a law of laws. Which “law law” will hold is time and discipline dependent and open to revision. One example is the Law of Universal Causation, and Mill put forward specific methods for identifying causes—now commonly known as Mill’s methods. These five methods look for circumstances which are common among the phenomena of interest, those which are absent when the phenomena are, or those for which both vary together. Mill’s methods are still seen as capturing basic intuitions about experimental methods for finding the relevant explanatory factors ( System of Logic (1843), see Mill entry). The methods advocated by Whewell and Mill, in the end, look similar. Both involve inductive generalization to covering laws. They differ dramatically, however, with respect to the necessity of the knowledge arrived at; that is, at the meta-methodological level (see the entries on Whewell and Mill entries).

3. Logic of method and critical responses

The quantum and relativistic revolutions in physics in the early 20 th century had a profound effect on methodology. Conceptual foundations of both theories were taken to show the defeasibility of even the most seemingly secure intuitions about space, time and bodies. Certainty of knowledge about the natural world was therefore recognized as unattainable. Instead a renewed empiricism was sought which rendered science fallible but still rationally justifiable.

Analyses of the reasoning of scientists emerged, according to which the aspects of scientific method which were of primary importance were the means of testing and confirming of theories. A distinction in methodology was made between the contexts of discovery and justification. The distinction could be used as a wedge between the particularities of where and how theories or hypotheses are arrived at, on the one hand, and the underlying reasoning scientists use (whether or not they are aware of it) when assessing theories and judging their adequacy on the basis of the available evidence. By and large, for most of the 20 th century, philosophy of science focused on the second context, although philosophers differed on whether to focus on confirmation or refutation as well as on the many details of how confirmation or refutation could or could not be brought about. By the mid-20 th century these attempts at defining the method of justification and the context distinction itself came under pressure. During the same period, philosophy of science developed rapidly, and from section 4 this entry will therefore shift from a primarily historical treatment of the scientific method towards a primarily thematic one.

Advances in logic and probability held out promise of the possibility of elaborate reconstructions of scientific theories and empirical method, the best example being Rudolf Carnap’s The Logical Structure of the World (1928). Carnap attempted to show that a scientific theory could be reconstructed as a formal axiomatic system—that is, a logic. That system could refer to the world because some of its basic sentences could be interpreted as observations or operations which one could perform to test them. The rest of the theoretical system, including sentences using theoretical or unobservable terms (like electron or force) would then either be meaningful because they could be reduced to observations, or they had purely logical meanings (called analytic, like mathematical identities). This has been referred to as the verifiability criterion of meaning. According to the criterion, any statement not either analytic or verifiable was strictly meaningless. Although the view was endorsed by Carnap in 1928, he would later come to see it as too restrictive (Carnap 1956). Another familiar version of this idea is operationalism of Percy William Bridgman. In The Logic of Modern Physics (1927) Bridgman asserted that every physical concept could be defined in terms of the operations one would perform to verify the application of that concept. Making good on the operationalisation of a concept even as simple as length, however, can easily become enormously complex (for measuring very small lengths, for instance) or impractical (measuring large distances like light years.)

Carl Hempel’s (1950, 1951) criticisms of the verifiability criterion of meaning had enormous influence. He pointed out that universal generalizations, such as most scientific laws, were not strictly meaningful on the criterion. Verifiability and operationalism both seemed too restrictive to capture standard scientific aims and practice. The tenuous connection between these reconstructions and actual scientific practice was criticized in another way. In both approaches, scientific methods are instead recast in methodological roles. Measurements, for example, were looked to as ways of giving meanings to terms. The aim of the philosopher of science was not to understand the methods per se , but to use them to reconstruct theories, their meanings, and their relation to the world. When scientists perform these operations, however, they will not report that they are doing them to give meaning to terms in a formal axiomatic system. This disconnect between methodology and the details of actual scientific practice would seem to violate the empiricism the Logical Positivists and Bridgman were committed to. The view that methodology should correspond to practice (to some extent) has been called historicism, or intuitionism. We turn to these criticisms and responses in section 3.4 . [ 4 ]

Positivism also had to contend with the recognition that a purely inductivist approach, along the lines of Bacon-Newton-Mill, was untenable. There was no pure observation, for starters. All observation was theory laden. Theory is required to make any observation, therefore not all theory can be derived from observation alone. (See the entry on theory and observation in science .) Even granting an observational basis, Hume had already pointed out that one could not deductively justify inductive conclusions without begging the question by presuming the success of the inductive method. Likewise, positivist attempts at analyzing how a generalization can be confirmed by observations of its instances were subject to a number of criticisms. Goodman (1965) and Hempel (1965) both point to paradoxes inherent in standard accounts of confirmation. Recent attempts at explaining how observations can serve to confirm a scientific theory are discussed in section 4 below.

The standard starting point for a non-inductive analysis of the logic of confirmation is known as the Hypothetico-Deductive (H-D) method. In its simplest form, a sentence of a theory which expresses some hypothesis is confirmed by its true consequences. As noted in section 2 , this method had been advanced by Whewell in the 19 th century, as well as Nicod (1924) and others in the 20 th century. Often, Hempel’s (1966) description of the H-D method, illustrated by the case of Semmelweiss’ inferential procedures in establishing the cause of childbed fever, has been presented as a key account of H-D as well as a foil for criticism of the H-D account of confirmation (see, for example, Lipton’s (2004) discussion of inference to the best explanation; also the entry on confirmation ). Hempel described Semmelsweiss’ procedure as examining various hypotheses explaining the cause of childbed fever. Some hypotheses conflicted with observable facts and could be rejected as false immediately. Others needed to be tested experimentally by deducing which observable events should follow if the hypothesis were true (what Hempel called the test implications of the hypothesis), then conducting an experiment and observing whether or not the test implications occurred. If the experiment showed the test implication to be false, the hypothesis could be rejected. If the experiment showed the test implications to be true, however, this did not prove the hypothesis true. The confirmation of a test implication does not verify a hypothesis, though Hempel did allow that “it provides at least some support, some corroboration or confirmation for it” (Hempel 1966: 8). The degree of this support then depends on the quantity, variety and precision of the supporting evidence.

Another approach that took off from the difficulties with inductive inference was Karl Popper’s critical rationalism or falsificationism (Popper 1959, 1963). Falsification is deductive and similar to H-D in that it involves scientists deducing observational consequences from the hypothesis under test. For Popper, however, the important point was not the degree of confirmation that successful prediction offered to a hypothesis. The crucial thing was the logical asymmetry between confirmation, based on inductive inference, and falsification, which can be based on a deductive inference. (This simple opposition was later questioned, by Lakatos, among others. See the entry on historicist theories of scientific rationality. )

Popper stressed that, regardless of the amount of confirming evidence, we can never be certain that a hypothesis is true without committing the fallacy of affirming the consequent. Instead, Popper introduced the notion of corroboration as a measure for how well a theory or hypothesis has survived previous testing—but without implying that this is also a measure for the probability that it is true.

Popper was also motivated by his doubts about the scientific status of theories like the Marxist theory of history or psycho-analysis, and so wanted to demarcate between science and pseudo-science. Popper saw this as an importantly different distinction than demarcating science from metaphysics. The latter demarcation was the primary concern of many logical empiricists. Popper used the idea of falsification to draw a line instead between pseudo and proper science. Science was science because its method involved subjecting theories to rigorous tests which offered a high probability of failing and thus refuting the theory.

A commitment to the risk of failure was important. Avoiding falsification could be done all too easily. If a consequence of a theory is inconsistent with observations, an exception can be added by introducing auxiliary hypotheses designed explicitly to save the theory, so-called ad hoc modifications. This Popper saw done in pseudo-science where ad hoc theories appeared capable of explaining anything in their field of application. In contrast, science is risky. If observations showed the predictions from a theory to be wrong, the theory would be refuted. Hence, scientific hypotheses must be falsifiable. Not only must there exist some possible observation statement which could falsify the hypothesis or theory, were it observed, (Popper called these the hypothesis’ potential falsifiers) it is crucial to the Popperian scientific method that such falsifications be sincerely attempted on a regular basis.

The more potential falsifiers of a hypothesis, the more falsifiable it would be, and the more the hypothesis claimed. Conversely, hypotheses without falsifiers claimed very little or nothing at all. Originally, Popper thought that this meant the introduction of ad hoc hypotheses only to save a theory should not be countenanced as good scientific method. These would undermine the falsifiabililty of a theory. However, Popper later came to recognize that the introduction of modifications (immunizations, he called them) was often an important part of scientific development. Responding to surprising or apparently falsifying observations often generated important new scientific insights. Popper’s own example was the observed motion of Uranus which originally did not agree with Newtonian predictions. The ad hoc hypothesis of an outer planet explained the disagreement and led to further falsifiable predictions. Popper sought to reconcile the view by blurring the distinction between falsifiable and not falsifiable, and speaking instead of degrees of testability (Popper 1985: 41f.).

From the 1960s on, sustained meta-methodological criticism emerged that drove philosophical focus away from scientific method. A brief look at those criticisms follows, with recommendations for further reading at the end of the entry.

Thomas Kuhn’s The Structure of Scientific Revolutions (1962) begins with a well-known shot across the bow for philosophers of science:

History, if viewed as a repository for more than anecdote or chronology, could produce a decisive transformation in the image of science by which we are now possessed. (1962: 1)

The image Kuhn thought needed transforming was the a-historical, rational reconstruction sought by many of the Logical Positivists, though Carnap and other positivists were actually quite sympathetic to Kuhn’s views. (See the entry on the Vienna Circle .) Kuhn shares with other of his contemporaries, such as Feyerabend and Lakatos, a commitment to a more empirical approach to philosophy of science. Namely, the history of science provides important data, and necessary checks, for philosophy of science, including any theory of scientific method.

The history of science reveals, according to Kuhn, that scientific development occurs in alternating phases. During normal science, the members of the scientific community adhere to the paradigm in place. Their commitment to the paradigm means a commitment to the puzzles to be solved and the acceptable ways of solving them. Confidence in the paradigm remains so long as steady progress is made in solving the shared puzzles. Method in this normal phase operates within a disciplinary matrix (Kuhn’s later concept of a paradigm) which includes standards for problem solving, and defines the range of problems to which the method should be applied. An important part of a disciplinary matrix is the set of values which provide the norms and aims for scientific method. The main values that Kuhn identifies are prediction, problem solving, simplicity, consistency, and plausibility.

An important by-product of normal science is the accumulation of puzzles which cannot be solved with resources of the current paradigm. Once accumulation of these anomalies has reached some critical mass, it can trigger a communal shift to a new paradigm and a new phase of normal science. Importantly, the values that provide the norms and aims for scientific method may have transformed in the meantime. Method may therefore be relative to discipline, time or place

Feyerabend also identified the aims of science as progress, but argued that any methodological prescription would only stifle that progress (Feyerabend 1988). His arguments are grounded in re-examining accepted “myths” about the history of science. Heroes of science, like Galileo, are shown to be just as reliant on rhetoric and persuasion as they are on reason and demonstration. Others, like Aristotle, are shown to be far more reasonable and far-reaching in their outlooks then they are given credit for. As a consequence, the only rule that could provide what he took to be sufficient freedom was the vacuous “anything goes”. More generally, even the methodological restriction that science is the best way to pursue knowledge, and to increase knowledge, is too restrictive. Feyerabend suggested instead that science might, in fact, be a threat to a free society, because it and its myth had become so dominant (Feyerabend 1978).

An even more fundamental kind of criticism was offered by several sociologists of science from the 1970s onwards who rejected the methodology of providing philosophical accounts for the rational development of science and sociological accounts of the irrational mistakes. Instead, they adhered to a symmetry thesis on which any causal explanation of how scientific knowledge is established needs to be symmetrical in explaining truth and falsity, rationality and irrationality, success and mistakes, by the same causal factors (see, e.g., Barnes and Bloor 1982, Bloor 1991). Movements in the Sociology of Science, like the Strong Programme, or in the social dimensions and causes of knowledge more generally led to extended and close examination of detailed case studies in contemporary science and its history. (See the entries on the social dimensions of scientific knowledge and social epistemology .) Well-known examinations by Latour and Woolgar (1979/1986), Knorr-Cetina (1981), Pickering (1984), Shapin and Schaffer (1985) seem to bear out that it was social ideologies (on a macro-scale) or individual interactions and circumstances (on a micro-scale) which were the primary causal factors in determining which beliefs gained the status of scientific knowledge. As they saw it therefore, explanatory appeals to scientific method were not empirically grounded.

A late, and largely unexpected, criticism of scientific method came from within science itself. Beginning in the early 2000s, a number of scientists attempting to replicate the results of published experiments could not do so. There may be close conceptual connection between reproducibility and method. For example, if reproducibility means that the same scientific methods ought to produce the same result, and all scientific results ought to be reproducible, then whatever it takes to reproduce a scientific result ought to be called scientific method. Space limits us to the observation that, insofar as reproducibility is a desired outcome of proper scientific method, it is not strictly a part of scientific method. (See the entry on reproducibility of scientific results .)

By the close of the 20 th century the search for the scientific method was flagging. Nola and Sankey (2000b) could introduce their volume on method by remarking that “For some, the whole idea of a theory of scientific method is yester-year’s debate …”.

Despite the many difficulties that philosophers encountered in trying to providing a clear methodology of conformation (or refutation), still important progress has been made on understanding how observation can provide evidence for a given theory. Work in statistics has been crucial for understanding how theories can be tested empirically, and in recent decades a huge literature has developed that attempts to recast confirmation in Bayesian terms. Here these developments can be covered only briefly, and we refer to the entry on confirmation for further details and references.

Statistics has come to play an increasingly important role in the methodology of the experimental sciences from the 19 th century onwards. At that time, statistics and probability theory took on a methodological role as an analysis of inductive inference, and attempts to ground the rationality of induction in the axioms of probability theory have continued throughout the 20 th century and in to the present. Developments in the theory of statistics itself, meanwhile, have had a direct and immense influence on the experimental method, including methods for measuring the uncertainty of observations such as the Method of Least Squares developed by Legendre and Gauss in the early 19 th century, criteria for the rejection of outliers proposed by Peirce by the mid-19 th century, and the significance tests developed by Gosset (a.k.a. “Student”), Fisher, Neyman & Pearson and others in the 1920s and 1930s (see, e.g., Swijtink 1987 for a brief historical overview; and also the entry on C.S. Peirce ).

These developments within statistics then in turn led to a reflective discussion among both statisticians and philosophers of science on how to perceive the process of hypothesis testing: whether it was a rigorous statistical inference that could provide a numerical expression of the degree of confidence in the tested hypothesis, or if it should be seen as a decision between different courses of actions that also involved a value component. This led to a major controversy among Fisher on the one side and Neyman and Pearson on the other (see especially Fisher 1955, Neyman 1956 and Pearson 1955, and for analyses of the controversy, e.g., Howie 2002, Marks 2000, Lenhard 2006). On Fisher’s view, hypothesis testing was a methodology for when to accept or reject a statistical hypothesis, namely that a hypothesis should be rejected by evidence if this evidence would be unlikely relative to other possible outcomes, given the hypothesis were true. In contrast, on Neyman and Pearson’s view, the consequence of error also had to play a role when deciding between hypotheses. Introducing the distinction between the error of rejecting a true hypothesis (type I error) and accepting a false hypothesis (type II error), they argued that it depends on the consequences of the error to decide whether it is more important to avoid rejecting a true hypothesis or accepting a false one. Hence, Fisher aimed for a theory of inductive inference that enabled a numerical expression of confidence in a hypothesis. To him, the important point was the search for truth, not utility. In contrast, the Neyman-Pearson approach provided a strategy of inductive behaviour for deciding between different courses of action. Here, the important point was not whether a hypothesis was true, but whether one should act as if it was.

Similar discussions are found in the philosophical literature. On the one side, Churchman (1948) and Rudner (1953) argued that because scientific hypotheses can never be completely verified, a complete analysis of the methods of scientific inference includes ethical judgments in which the scientists must decide whether the evidence is sufficiently strong or that the probability is sufficiently high to warrant the acceptance of the hypothesis, which again will depend on the importance of making a mistake in accepting or rejecting the hypothesis. Others, such as Jeffrey (1956) and Levi (1960) disagreed and instead defended a value-neutral view of science on which scientists should bracket their attitudes, preferences, temperament, and values when assessing the correctness of their inferences. For more details on this value-free ideal in the philosophy of science and its historical development, see Douglas (2009) and Howard (2003). For a broad set of case studies examining the role of values in science, see e.g. Elliott & Richards 2017.

In recent decades, philosophical discussions of the evaluation of probabilistic hypotheses by statistical inference have largely focused on Bayesianism that understands probability as a measure of a person’s degree of belief in an event, given the available information, and frequentism that instead understands probability as a long-run frequency of a repeatable event. Hence, for Bayesians probabilities refer to a state of knowledge, whereas for frequentists probabilities refer to frequencies of events (see, e.g., Sober 2008, chapter 1 for a detailed introduction to Bayesianism and frequentism as well as to likelihoodism). Bayesianism aims at providing a quantifiable, algorithmic representation of belief revision, where belief revision is a function of prior beliefs (i.e., background knowledge) and incoming evidence. Bayesianism employs a rule based on Bayes’ theorem, a theorem of the probability calculus which relates conditional probabilities. The probability that a particular hypothesis is true is interpreted as a degree of belief, or credence, of the scientist. There will also be a probability and a degree of belief that a hypothesis will be true conditional on a piece of evidence (an observation, say) being true. Bayesianism proscribes that it is rational for the scientist to update their belief in the hypothesis to that conditional probability should it turn out that the evidence is, in fact, observed (see, e.g., Sprenger & Hartmann 2019 for a comprehensive treatment of Bayesian philosophy of science). Originating in the work of Neyman and Person, frequentism aims at providing the tools for reducing long-run error rates, such as the error-statistical approach developed by Mayo (1996) that focuses on how experimenters can avoid both type I and type II errors by building up a repertoire of procedures that detect errors if and only if they are present. Both Bayesianism and frequentism have developed over time, they are interpreted in different ways by its various proponents, and their relations to previous criticism to attempts at defining scientific method are seen differently by proponents and critics. The literature, surveys, reviews and criticism in this area are vast and the reader is referred to the entries on Bayesian epistemology and confirmation .

5. Method in Practice

Attention to scientific practice, as we have seen, is not itself new. However, the turn to practice in the philosophy of science of late can be seen as a correction to the pessimism with respect to method in philosophy of science in later parts of the 20 th century, and as an attempted reconciliation between sociological and rationalist explanations of scientific knowledge. Much of this work sees method as detailed and context specific problem-solving procedures, and methodological analyses to be at the same time descriptive, critical and advisory (see Nickles 1987 for an exposition of this view). The following section contains a survey of some of the practice focuses. In this section we turn fully to topics rather than chronology.

A problem with the distinction between the contexts of discovery and justification that figured so prominently in philosophy of science in the first half of the 20 th century (see section 2 ) is that no such distinction can be clearly seen in scientific activity (see Arabatzis 2006). Thus, in recent decades, it has been recognized that study of conceptual innovation and change should not be confined to psychology and sociology of science, but are also important aspects of scientific practice which philosophy of science should address (see also the entry on scientific discovery ). Looking for the practices that drive conceptual innovation has led philosophers to examine both the reasoning practices of scientists and the wide realm of experimental practices that are not directed narrowly at testing hypotheses, that is, exploratory experimentation.

Examining the reasoning practices of historical and contemporary scientists, Nersessian (2008) has argued that new scientific concepts are constructed as solutions to specific problems by systematic reasoning, and that of analogy, visual representation and thought-experimentation are among the important reasoning practices employed. These ubiquitous forms of reasoning are reliable—but also fallible—methods of conceptual development and change. On her account, model-based reasoning consists of cycles of construction, simulation, evaluation and adaption of models that serve as interim interpretations of the target problem to be solved. Often, this process will lead to modifications or extensions, and a new cycle of simulation and evaluation. However, Nersessian also emphasizes that

creative model-based reasoning cannot be applied as a simple recipe, is not always productive of solutions, and even its most exemplary usages can lead to incorrect solutions. (Nersessian 2008: 11)

Thus, while on the one hand she agrees with many previous philosophers that there is no logic of discovery, discoveries can derive from reasoned processes, such that a large and integral part of scientific practice is

the creation of concepts through which to comprehend, structure, and communicate about physical phenomena …. (Nersessian 1987: 11)

Similarly, work on heuristics for discovery and theory construction by scholars such as Darden (1991) and Bechtel & Richardson (1993) present science as problem solving and investigate scientific problem solving as a special case of problem-solving in general. Drawing largely on cases from the biological sciences, much of their focus has been on reasoning strategies for the generation, evaluation, and revision of mechanistic explanations of complex systems.

Addressing another aspect of the context distinction, namely the traditional view that the primary role of experiments is to test theoretical hypotheses according to the H-D model, other philosophers of science have argued for additional roles that experiments can play. The notion of exploratory experimentation was introduced to describe experiments driven by the desire to obtain empirical regularities and to develop concepts and classifications in which these regularities can be described (Steinle 1997, 2002; Burian 1997; Waters 2007)). However the difference between theory driven experimentation and exploratory experimentation should not be seen as a sharp distinction. Theory driven experiments are not always directed at testing hypothesis, but may also be directed at various kinds of fact-gathering, such as determining numerical parameters. Vice versa , exploratory experiments are usually informed by theory in various ways and are therefore not theory-free. Instead, in exploratory experiments phenomena are investigated without first limiting the possible outcomes of the experiment on the basis of extant theory about the phenomena.

The development of high throughput instrumentation in molecular biology and neighbouring fields has given rise to a special type of exploratory experimentation that collects and analyses very large amounts of data, and these new ‘omics’ disciplines are often said to represent a break with the ideal of hypothesis-driven science (Burian 2007; Elliott 2007; Waters 2007; O’Malley 2007) and instead described as data-driven research (Leonelli 2012; Strasser 2012) or as a special kind of “convenience experimentation” in which many experiments are done simply because they are extraordinarily convenient to perform (Krohs 2012).

5.2 Computer methods and ‘new ways’ of doing science

The field of omics just described is possible because of the ability of computers to process, in a reasonable amount of time, the huge quantities of data required. Computers allow for more elaborate experimentation (higher speed, better filtering, more variables, sophisticated coordination and control), but also, through modelling and simulations, might constitute a form of experimentation themselves. Here, too, we can pose a version of the general question of method versus practice: does the practice of using computers fundamentally change scientific method, or merely provide a more efficient means of implementing standard methods?

Because computers can be used to automate measurements, quantifications, calculations, and statistical analyses where, for practical reasons, these operations cannot be otherwise carried out, many of the steps involved in reaching a conclusion on the basis of an experiment are now made inside a “black box”, without the direct involvement or awareness of a human. This has epistemological implications, regarding what we can know, and how we can know it. To have confidence in the results, computer methods are therefore subjected to tests of verification and validation.

The distinction between verification and validation is easiest to characterize in the case of computer simulations. In a typical computer simulation scenario computers are used to numerically integrate differential equations for which no analytic solution is available. The equations are part of the model the scientist uses to represent a phenomenon or system under investigation. Verifying a computer simulation means checking that the equations of the model are being correctly approximated. Validating a simulation means checking that the equations of the model are adequate for the inferences one wants to make on the basis of that model.

A number of issues related to computer simulations have been raised. The identification of validity and verification as the testing methods has been criticized. Oreskes et al. (1994) raise concerns that “validiation”, because it suggests deductive inference, might lead to over-confidence in the results of simulations. The distinction itself is probably too clean, since actual practice in the testing of simulations mixes and moves back and forth between the two (Weissart 1997; Parker 2008a; Winsberg 2010). Computer simulations do seem to have a non-inductive character, given that the principles by which they operate are built in by the programmers, and any results of the simulation follow from those in-built principles in such a way that those results could, in principle, be deduced from the program code and its inputs. The status of simulations as experiments has therefore been examined (Kaufmann and Smarr 1993; Humphreys 1995; Hughes 1999; Norton and Suppe 2001). This literature considers the epistemology of these experiments: what we can learn by simulation, and also the kinds of justifications which can be given in applying that knowledge to the “real” world. (Mayo 1996; Parker 2008b). As pointed out, part of the advantage of computer simulation derives from the fact that huge numbers of calculations can be carried out without requiring direct observation by the experimenter/​simulator. At the same time, many of these calculations are approximations to the calculations which would be performed first-hand in an ideal situation. Both factors introduce uncertainties into the inferences drawn from what is observed in the simulation.

For many of the reasons described above, computer simulations do not seem to belong clearly to either the experimental or theoretical domain. Rather, they seem to crucially involve aspects of both. This has led some authors, such as Fox Keller (2003: 200) to argue that we ought to consider computer simulation a “qualitatively different way of doing science”. The literature in general tends to follow Kaufmann and Smarr (1993) in referring to computer simulation as a “third way” for scientific methodology (theoretical reasoning and experimental practice are the first two ways.). It should also be noted that the debates around these issues have tended to focus on the form of computer simulation typical in the physical sciences, where models are based on dynamical equations. Other forms of simulation might not have the same problems, or have problems of their own (see the entry on computer simulations in science ).

In recent years, the rapid development of machine learning techniques has prompted some scholars to suggest that the scientific method has become “obsolete” (Anderson 2008, Carrol and Goodstein 2009). This has resulted in an intense debate on the relative merit of data-driven and hypothesis-driven research (for samples, see e.g. Mazzocchi 2015 or Succi and Coveney 2018). For a detailed treatment of this topic, we refer to the entry scientific research and big data .

6. Discourse on scientific method

Despite philosophical disagreements, the idea of the scientific method still figures prominently in contemporary discourse on many different topics, both within science and in society at large. Often, reference to scientific method is used in ways that convey either the legend of a single, universal method characteristic of all science, or grants to a particular method or set of methods privilege as a special ‘gold standard’, often with reference to particular philosophers to vindicate the claims. Discourse on scientific method also typically arises when there is a need to distinguish between science and other activities, or for justifying the special status conveyed to science. In these areas, the philosophical attempts at identifying a set of methods characteristic for scientific endeavors are closely related to the philosophy of science’s classical problem of demarcation (see the entry on science and pseudo-science ) and to the philosophical analysis of the social dimension of scientific knowledge and the role of science in democratic society.

One of the settings in which the legend of a single, universal scientific method has been particularly strong is science education (see, e.g., Bauer 1992; McComas 1996; Wivagg & Allchin 2002). [ 5 ] Often, ‘the scientific method’ is presented in textbooks and educational web pages as a fixed four or five step procedure starting from observations and description of a phenomenon and progressing over formulation of a hypothesis which explains the phenomenon, designing and conducting experiments to test the hypothesis, analyzing the results, and ending with drawing a conclusion. Such references to a universal scientific method can be found in educational material at all levels of science education (Blachowicz 2009), and numerous studies have shown that the idea of a general and universal scientific method often form part of both students’ and teachers’ conception of science (see, e.g., Aikenhead 1987; Osborne et al. 2003). In response, it has been argued that science education need to focus more on teaching about the nature of science, although views have differed on whether this is best done through student-led investigations, contemporary cases, or historical cases (Allchin, Andersen & Nielsen 2014)

Although occasionally phrased with reference to the H-D method, important historical roots of the legend in science education of a single, universal scientific method are the American philosopher and psychologist Dewey’s account of inquiry in How We Think (1910) and the British mathematician Karl Pearson’s account of science in Grammar of Science (1892). On Dewey’s account, inquiry is divided into the five steps of

(i) a felt difficulty, (ii) its location and definition, (iii) suggestion of a possible solution, (iv) development by reasoning of the bearing of the suggestions, (v) further observation and experiment leading to its acceptance or rejection. (Dewey 1910: 72)

Similarly, on Pearson’s account, scientific investigations start with measurement of data and observation of their correction and sequence from which scientific laws can be discovered with the aid of creative imagination. These laws have to be subject to criticism, and their final acceptance will have equal validity for “all normally constituted minds”. Both Dewey’s and Pearson’s accounts should be seen as generalized abstractions of inquiry and not restricted to the realm of science—although both Dewey and Pearson referred to their respective accounts as ‘the scientific method’.

Occasionally, scientists make sweeping statements about a simple and distinct scientific method, as exemplified by Feynman’s simplified version of a conjectures and refutations method presented, for example, in the last of his 1964 Cornell Messenger lectures. [ 6 ] However, just as often scientists have come to the same conclusion as recent philosophy of science that there is not any unique, easily described scientific method. For example, the physicist and Nobel Laureate Weinberg described in the paper “The Methods of Science … And Those By Which We Live” (1995) how

The fact that the standards of scientific success shift with time does not only make the philosophy of science difficult; it also raises problems for the public understanding of science. We do not have a fixed scientific method to rally around and defend. (1995: 8)

Interview studies with scientists on their conception of method shows that scientists often find it hard to figure out whether available evidence confirms their hypothesis, and that there are no direct translations between general ideas about method and specific strategies to guide how research is conducted (Schickore & Hangel 2019, Hangel & Schickore 2017)

Reference to the scientific method has also often been used to argue for the scientific nature or special status of a particular activity. Philosophical positions that argue for a simple and unique scientific method as a criterion of demarcation, such as Popperian falsification, have often attracted practitioners who felt that they had a need to defend their domain of practice. For example, references to conjectures and refutation as the scientific method are abundant in much of the literature on complementary and alternative medicine (CAM)—alongside the competing position that CAM, as an alternative to conventional biomedicine, needs to develop its own methodology different from that of science.

Also within mainstream science, reference to the scientific method is used in arguments regarding the internal hierarchy of disciplines and domains. A frequently seen argument is that research based on the H-D method is superior to research based on induction from observations because in deductive inferences the conclusion follows necessarily from the premises. (See, e.g., Parascandola 1998 for an analysis of how this argument has been made to downgrade epidemiology compared to the laboratory sciences.) Similarly, based on an examination of the practices of major funding institutions such as the National Institutes of Health (NIH), the National Science Foundation (NSF) and the Biomedical Sciences Research Practices (BBSRC) in the UK, O’Malley et al. (2009) have argued that funding agencies seem to have a tendency to adhere to the view that the primary activity of science is to test hypotheses, while descriptive and exploratory research is seen as merely preparatory activities that are valuable only insofar as they fuel hypothesis-driven research.

In some areas of science, scholarly publications are structured in a way that may convey the impression of a neat and linear process of inquiry from stating a question, devising the methods by which to answer it, collecting the data, to drawing a conclusion from the analysis of data. For example, the codified format of publications in most biomedical journals known as the IMRAD format (Introduction, Method, Results, Analysis, Discussion) is explicitly described by the journal editors as “not an arbitrary publication format but rather a direct reflection of the process of scientific discovery” (see the so-called “Vancouver Recommendations”, ICMJE 2013: 11). However, scientific publications do not in general reflect the process by which the reported scientific results were produced. For example, under the provocative title “Is the scientific paper a fraud?”, Medawar argued that scientific papers generally misrepresent how the results have been produced (Medawar 1963/1996). Similar views have been advanced by philosophers, historians and sociologists of science (Gilbert 1976; Holmes 1987; Knorr-Cetina 1981; Schickore 2008; Suppe 1998) who have argued that scientists’ experimental practices are messy and often do not follow any recognizable pattern. Publications of research results, they argue, are retrospective reconstructions of these activities that often do not preserve the temporal order or the logic of these activities, but are instead often constructed in order to screen off potential criticism (see Schickore 2008 for a review of this work).

Philosophical positions on the scientific method have also made it into the court room, especially in the US where judges have drawn on philosophy of science in deciding when to confer special status to scientific expert testimony. A key case is Daubert vs Merrell Dow Pharmaceuticals (92–102, 509 U.S. 579, 1993). In this case, the Supreme Court argued in its 1993 ruling that trial judges must ensure that expert testimony is reliable, and that in doing this the court must look at the expert’s methodology to determine whether the proffered evidence is actually scientific knowledge. Further, referring to works of Popper and Hempel the court stated that

ordinarily, a key question to be answered in determining whether a theory or technique is scientific knowledge … is whether it can be (and has been) tested. (Justice Blackmun, Daubert v. Merrell Dow Pharmaceuticals; see Other Internet Resources for a link to the opinion)

But as argued by Haack (2005a,b, 2010) and by Foster & Hubner (1999), by equating the question of whether a piece of testimony is reliable with the question whether it is scientific as indicated by a special methodology, the court was producing an inconsistent mixture of Popper’s and Hempel’s philosophies, and this has later led to considerable confusion in subsequent case rulings that drew on the Daubert case (see Haack 2010 for a detailed exposition).

The difficulties around identifying the methods of science are also reflected in the difficulties of identifying scientific misconduct in the form of improper application of the method or methods of science. One of the first and most influential attempts at defining misconduct in science was the US definition from 1989 that defined misconduct as

fabrication, falsification, plagiarism, or other practices that seriously deviate from those that are commonly accepted within the scientific community . (Code of Federal Regulations, part 50, subpart A., August 8, 1989, italics added)

However, the “other practices that seriously deviate” clause was heavily criticized because it could be used to suppress creative or novel science. For example, the National Academy of Science stated in their report Responsible Science (1992) that it

wishes to discourage the possibility that a misconduct complaint could be lodged against scientists based solely on their use of novel or unorthodox research methods. (NAS: 27)

This clause was therefore later removed from the definition. For an entry into the key philosophical literature on conduct in science, see Shamoo & Resnick (2009).

The question of the source of the success of science has been at the core of philosophy since the beginning of modern science. If viewed as a matter of epistemology more generally, scientific method is a part of the entire history of philosophy. Over that time, science and whatever methods its practitioners may employ have changed dramatically. Today, many philosophers have taken up the banners of pluralism or of practice to focus on what are, in effect, fine-grained and contextually limited examinations of scientific method. Others hope to shift perspectives in order to provide a renewed general account of what characterizes the activity we call science.

One such perspective has been offered recently by Hoyningen-Huene (2008, 2013), who argues from the history of philosophy of science that after three lengthy phases of characterizing science by its method, we are now in a phase where the belief in the existence of a positive scientific method has eroded and what has been left to characterize science is only its fallibility. First was a phase from Plato and Aristotle up until the 17 th century where the specificity of scientific knowledge was seen in its absolute certainty established by proof from evident axioms; next was a phase up to the mid-19 th century in which the means to establish the certainty of scientific knowledge had been generalized to include inductive procedures as well. In the third phase, which lasted until the last decades of the 20 th century, it was recognized that empirical knowledge was fallible, but it was still granted a special status due to its distinctive mode of production. But now in the fourth phase, according to Hoyningen-Huene, historical and philosophical studies have shown how “scientific methods with the characteristics as posited in the second and third phase do not exist” (2008: 168) and there is no longer any consensus among philosophers and historians of science about the nature of science. For Hoyningen-Huene, this is too negative a stance, and he therefore urges the question about the nature of science anew. His own answer to this question is that “scientific knowledge differs from other kinds of knowledge, especially everyday knowledge, primarily by being more systematic” (Hoyningen-Huene 2013: 14). Systematicity can have several different dimensions: among them are more systematic descriptions, explanations, predictions, defense of knowledge claims, epistemic connectedness, ideal of completeness, knowledge generation, representation of knowledge and critical discourse. Hence, what characterizes science is the greater care in excluding possible alternative explanations, the more detailed elaboration with respect to data on which predictions are based, the greater care in detecting and eliminating sources of error, the more articulate connections to other pieces of knowledge, etc. On this position, what characterizes science is not that the methods employed are unique to science, but that the methods are more carefully employed.

Another, similar approach has been offered by Haack (2003). She sets off, similar to Hoyningen-Huene, from a dissatisfaction with the recent clash between what she calls Old Deferentialism and New Cynicism. The Old Deferentialist position is that science progressed inductively by accumulating true theories confirmed by empirical evidence or deductively by testing conjectures against basic statements; while the New Cynics position is that science has no epistemic authority and no uniquely rational method and is merely just politics. Haack insists that contrary to the views of the New Cynics, there are objective epistemic standards, and there is something epistemologically special about science, even though the Old Deferentialists pictured this in a wrong way. Instead, she offers a new Critical Commonsensist account on which standards of good, strong, supportive evidence and well-conducted, honest, thorough and imaginative inquiry are not exclusive to the sciences, but the standards by which we judge all inquirers. In this sense, science does not differ in kind from other kinds of inquiry, but it may differ in the degree to which it requires broad and detailed background knowledge and a familiarity with a technical vocabulary that only specialists may possess.

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scientific method

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scientific method , mathematical and experimental technique employed in the sciences . More specifically, it is the technique used in the construction and testing of a scientific hypothesis .

The process of observing, asking questions, and seeking answers through tests and experiments is not unique to any one field of science. In fact, the scientific method is applied broadly in science, across many different fields. Many empirical sciences, especially the social sciences , use mathematical tools borrowed from probability theory and statistics , together with outgrowths of these, such as decision theory , game theory , utility theory, and operations research . Philosophers of science have addressed general methodological problems, such as the nature of scientific explanation and the justification of induction .

The scientific method is critical to the development of scientific theories , which explain empirical (experiential) laws in a scientifically rational manner. In a typical application of the scientific method, a researcher develops a hypothesis , tests it through various means, and then modifies the hypothesis on the basis of the outcome of the tests and experiments. The modified hypothesis is then retested, further modified, and tested again, until it becomes consistent with observed phenomena and testing outcomes. In this way, hypotheses serve as tools by which scientists gather data. From that data and the many different scientific investigations undertaken to explore hypotheses, scientists are able to develop broad general explanations, or scientific theories.

See also Mill’s methods ; hypothetico-deductive method .

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show/hide words to know

Biased: when someone presents only one viewpoint. Biased articles do not give all the facts and often mislead the reader.

Conclusion: what a person decides based on information they get through research including experiments.

Method: following a certain set of steps to make something, or find an answer to a question. Like baking a pie or fixing the tire on a bicycle.

Research: looking for answers to questions using tools like the scientific method.

What is the Scientific Method?

If you have ever seen something going on and wondered why or how it happened, you have started down the road to discovery. If you continue your journey, you are likely to guess at some of your own answers for your question. Even further along the road you might think of ways to find out if your answers are correct. At this point, whether you know it or not, you are following a path that scientists call the scientific method. If you do some experiments to see if your answer is correct and write down what you learn in a report, you have pretty much completed everything a scientist might do in a laboratory or out in the field when doing research. In fact, the scientific method works well for many things that don’t usually seem so scientific.

The Flashlight Mystery...

Like a crime detective, you can use the elements of the scientific method to find the answer to everyday problems. For example you pick up a flashlight and turn it on, but the light does not work. You have observed that the light does not work. You ask the question, Why doesn't it work? With what you already know about flashlights, you might guess (hypothesize) that the batteries are dead. You say to yourself, if I buy new batteries and replace the old ones in the flashlight, the light should work. To test this prediction you replace the old batteries with new ones from the store. You click the switch on. Does the flashlight work? No?

What else could be the answer? You go back and hypothesize that it might be a broken light bulb. Your new prediction is if you replace the broken light bulb the flashlight will work. It’s time to go back to the store and buy a new light bulb. Now you test this new hypothesis and prediction by replacing the bulb in the flashlight. You flip the switch again. The flashlight lights up. Success!

If this were a scientific project, you would also have written down the results of your tests and a conclusion of your experiments. The results of only the light bulb hypothesis stood up to the test, and we had to reject the battery hypothesis. You would also communicate what you learned to others with a published report, article, or scientific paper.

More to the Mystery...

Not all questions can be answered with only two experiments. It can often take a lot more work and tests to find an answer. Even when you find an answer it may not always be the only answer to the question. This is one reason that different scientists will work on the same question and do their own experiments.

In our flashlight example, you might never get the light to turn on. This probably means you haven’t made enough different guesses (hypotheses) to test the problem. Were the new batteries in the right way? Was the switch rusty, or maybe a wire is broken. Think of all the possible guesses you could test.

No matter what the question, you can use the scientific method to guide you towards an answer. Even those questions that do not seem to be scientific can be solved using this process. Like with the flashlight, you might need to repeat several of the elements of the scientific method to find an answer. No matter how complex the diagram, the scientific method will include the following pieces in order to be complete.

The elements of the scientific method can be used by anyone to help answer questions. Even though these elements can be used in an ordered manner, they do not have to follow the same order. It is better to think of the scientific method as fluid process that can take different paths depending on the situation. Just be sure to incorporate all of the elements when seeking unbiased answers. You may also need to go back a few steps (or a few times) to test several different hypotheses before you come to a conclusion. Click on the image to see other versions of the scientific method. 

• Observation – seeing, hearing, touching…
• Asking a question – why or how?
• Hypothesis – a fancy name for an educated guess about what causes something to happen.
• Prediction – what you think will happen if…
• Testing – this is where you get to experiment and be creative.
• Conclusion – decide how your test results relate to your predictions.
• Communicate – share your results so others can learn from your work.

Other Parts of the Scientific Method…

Now that you have an idea of how the scientific method works there are a few other things to learn so that you will be able test out your new skills and test your hypotheses.

• Control - A group that is similar to other groups but is left alone so that it can be compared to see what happened to the other groups that are tested.
• Data - the numbers and measurements you get from the test in a scientific experiment.
• Independent variable - a variable that you change as part of your experiment. It is important to only change one independent variable for each experiment. 
• Dependent variable - a variable that changes when the independent variable is changed.
• Controlled Variable - these are variables that you never change in your experiment.

Practicing Observations and Wondering How and Why...

It is really hard not to notice things around us and wonder about them. This is how the scientific method begins, by observing and wondering why and how. Why do leaves on trees in many parts of the world turn from green to red, orange, or yellow and fall to the ground when winter comes? How does a spider move around their web without getting stuck like its victims? Both of these questions start with observing something and asking questions. The next time you see something and ask yourself, “I wonder why that does that, or how can it do that?” try out your new detective skills, and see what answer you can find. 

Try Out Your Detective Skills

Now that you have the basics of the scientific method, why not test your skills? The Science Detectives Training Room will test your problem solving ability. Step inside and see if you can escape the room. While you are there, look around and see what other interesting things might be waiting. We think you find this game a great way to learn the scientific method. In fact, we bet you will discover that you already use the scientific method and didn't even know it.

After you've learned the basics of being a detective, practice those skills in The Case of the Mystery Images . While you are there, pay attention to what's around you as you figure out just what is happening in the mystery photos that surround you.

Ready for your next challenge? Try Science Detectives: Case of the Mystery Images for even more mysteries to solve. Take your scientific abilities one step further by making observations and formulating hypothesis about the mysterious images you find within.

Acknowledgements:  

We thank John Alcock for his feedback and suggestions on this article.

Science Detectives - Mystery Room Escape was produced in partnership with the Arizona Science Education Collaborative (ASEC) and funded by ASU Women & Philanthropy.

Flashlight image via Wikimedia Commons - The Oxygen Team

Read more about: Using the Scientific Method to Solve Mysteries

View citation, bibliographic details:.

• Article: Using the Scientific Method to Solve Mysteries
• Author(s): CJ Kazilek and David Pearson
• Publisher: Arizona State University School of Life Sciences Ask A Biologist
• Site name: ASU - Ask A Biologist
• Date published: October 8, 2009
• Date accessed: August 8, 2024
• Link: https://askabiologist.asu.edu/explore/scientific-method

CJ Kazilek and David Pearson. (2009, October 08). Using the Scientific Method to Solve Mysteries . ASU - Ask A Biologist. Retrieved August 8, 2024 from https://askabiologist.asu.edu/explore/scientific-method

Chicago Manual of Style

CJ Kazilek and David Pearson. "Using the Scientific Method to Solve Mysteries ". ASU - Ask A Biologist. 08 October, 2009. https://askabiologist.asu.edu/explore/scientific-method

MLA 2017 Style

CJ Kazilek and David Pearson. "Using the Scientific Method to Solve Mysteries ". ASU - Ask A Biologist. 08 Oct 2009. ASU - Ask A Biologist, Web. 8 Aug 2024. https://askabiologist.asu.edu/explore/scientific-method

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Chapter 6: Scientific Problem Solving

If you prefer a video, click this button:

Scientific Problem Solving Video

Science is a method to discover empirical truths and patterns. Roughly speaking, the scientific method consists of

1) Observing

2) Forming a hypothesis

3) Testing the hypothesis and

4) Interpreting the data to confirm or disconfirm the hypothesis.

The beauty of science is that any scientific claim can be tested if you have the proper knowledge and equipment.

You can also use the scientific method to solve everyday problems: 1) Observe and clearly define the problem, 2) Form a hypothesis, 3) Test it, and 4) Confirm the hypothesis... or disconfirm it and start over.

So, the next time you are cursing in traffic or emotionally reacting to a problem, take a few deep breaths and then use this rational and scientific approach. Slow down, observe, hypothesize, and test.

Explain how you would solve these problems using the four steps of the scientific process.

Example: The fire alarm is not working.

1) Observe/Define the problem: it does not beep when I push the button.

2) Hypothesis: it is caused by a dead battery.

3) Test: try a new battery.

4) Confirm/Disconfirm: the alarm now works. If it does not work, start over by testing another hypothesis like “it has a loose wire.”  

• My car will not start.
• My child is having problems reading.
• I owe $20,000, but only make $10 an hour.
• My boss is mean. I want him/her to stop using rude language towards me.
• My significant other is lazy. I want him/her to help out more.

6-8. Identify three problems where you can apply the scientific method.

*Answers will vary.

Application and Value

Science is more of a process than a body of knowledge. In our daily lives, we often emotionally react and jump to quick solutions when faced with problems, but following the four steps of the scientific process can help us slow down and discover more intelligent solutions.

In your study of philosophy, you will explore deeper questions about science. For example, are there any forms of knowledge that are nonscientific? Can science tell us what we ought to do? Can logical and mathematical truths be proven in a scientific way? Does introspection give knowledge even though I cannot scientifically observe your introspective thoughts? Is science truly objective?  These are challenging questions that should help you discover the scope of science without diminishing its awesome power.

But the first step in answering these questions is knowing what science is, and this chapter clarifies its essence. Again, Science is not so much a body of knowledge as it is a method of observing, hypothesizing, and testing. This method is what all the sciences have in common.

Perhaps too science should involve falsifiability, which is a concept explored in the next chapter.

Return to Logic Home                            Next (Chapter 7, Falsifiability)

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Biology archive

Course: biology archive   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation., 2. ask a question., 3. propose a hypothesis., 4. make predictions., 5. test the predictions..

• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

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Solving Everyday Problems with the Scientific Method

• By (author): 
• Don K Mak , 
• Angela T Mak , and 
• Anthony B Mak
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• Description
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This book describes how one can use The Scientific Method to solve everyday problems including medical ailments, health issues, money management, traveling, shopping, cooking, household chores, etc. It illustrates how to exploit the information collected from our five senses, how to solve problems when no information is available for the present problem situation, how to increase our chances of success by redefining a problem, and how to extrapolate our capabilities by seeing a relationship among heretofore unrelated concepts.

One should formulate a hypothesis as early as possible in order to have a sense of direction regarding which path to follow. Occasionally, by making wild conjectures, creative solutions can transpire. However, hypotheses need to be well-tested. Through this way, The Scientific Method can help readers solve problems in both familiar and unfamiliar situations. Containing real-life examples of how various problems are solved — for instance, how some observant patients cure their own illnesses when medical experts have failed — this book will train readers to observe what others may have missed and conceive what others may not have contemplated. With practice, they will be able to solve more problems than they could previously imagine.

In this second edition, the authors have added some more theories which they hope can help in solving everyday problems. At the same time, they have updated the book by including quite a few examples which they think are interesting.

Sample Chapter(s) Chapter 1: Prelude (63 KB)

• Preface to the Second Edition
• Preface to the First Edition
• The Scientific Method
• Observation
• Recognition
• Problem Situation and Problem Definition
• Induction and Deduction
• Alternative Solutions
• Mathematics
• Probable Value
• Bibliography

FRONT MATTER

• Pages: i–xvi

https://doi.org/10.1142/9789813145313_fmatter

• Claimers and Disclaimers

Chapter 1: Prelude

https://doi.org/10.1142/9789813145313_0001

The father put down the newspaper. It had been raining for the last two hours. The rain finally stopped, and the sky looked clear. After all this raining, the negative ions in the atmosphere would have increased, and the air would feel fresh. The father suggested the family of four should go for a stroll. There was a park just about fifteen minutes walk from their house.

Chapter 2: The Scientific Method

• Pages: 3–18

https://doi.org/10.1142/9789813145313_0002

In the history of philosophical ideation, scientific discoveries, and engineering inventions, it has almost never happened that a single person (or a single group of people) has come up with an idea or a similar idea that no one has ever dreamed of earlier, or at the same time. This person may not be aware of the previous findings, nor someone else in another part of the world has comparable ideas, and thus – his idea may be very original, as far as he is concerned. However, history tells us that it is highly unlikely that no one has already come up with some related concepts.

Chapter 3: Observation

• Pages: 19–56

https://doi.org/10.1142/9789813145313_0003

Observation is the first step of the Scientific Method. However, it can infiltrate the whole scientific process – from the initial perception of a phenomenon, to proposing a solution, and right down to experimentation, where observation of the results is significant.

Chapter 4: Hypothesis

• Pages: 57–95

https://doi.org/10.1142/9789813145313_0004

In scientific discipline, a hypothesis is a set of propositions set forth to explain the occurrence of certain phenomena. In daily language, a hypothesis can be interpreted as an assumption or guess. In this book, we employ both these definitions. Within the context of the first definition, we search for an explanation of why the problem occurs to begin with. Within the context of the second definition, we look for a plausible solution to the problem.

Chapter 5: Experiment

• Pages: 96–121

https://doi.org/10.1142/9789813145313_0005

In scientific discipline, an experiment is a test under controlled conditions to investigate the validity of a hypothesis. In everyday language, experiment can be interpreted as a testing of an idea. In this book, we employ both these definitions. Within the context of the first definition, we attempt to confirm whether an explanation of an observation is correct. Within the context of the second definition, we check whether a proposed idea for a solution is valid.

Chapter 6: Recognition

• Pages: 122–144

https://doi.org/10.1142/9789813145313_0006

Before we can solve any problem, we need to recognize that a problem exists in the first place. That may seem obvious, but while some problems stick out like thorns in a bush, others are hidden like plants in a forest. As such, not only do we need to tune up our observational skills to see that a problem does exist; we should also sharpen our thinking to anticipate that a problem may arise. Thus, recognition can be considered to be a combination of observing and hypothesizing.

Chapter 7: Problem Situation and Problem Definition

• Pages: 145–152

https://doi.org/10.1142/9789813145313_0007

For just about any situation, we can look at it from different perspectives. Take the example of a piece of rock, it will look different from the eyes of a landscaper, an architect, a geologist and an artist.

Chapter 8: Induction and Deduction

• Pages: 153–164

https://doi.org/10.1142/9789813145313_0008

Once a problem has been defined, we need to find a solution. To determine which route we can take, we will have to take a look at the knowledge that we already have in hand, and we may want to search for more information when necessary. It is therefore, much more convenient if we already have an arsenal of tools that have been stored neatly and categorized in our mind. That simply means, that we should have been observing our surroundings, and preferably have come up with some general principles that can guide us in the present problem.

Chapter 9: Alternative Solutions

• Pages: 165–193

https://doi.org/10.1142/9789813145313_0009

While there are various ways to view a problem situation, and thus define a problem differently, there are also different ways to solve a problem once it is defined. Some of the solutions may be better than others. If we have the option of not requiring to make a snap judgement, we should wait till we have come up with several plausible solutions, and then decide which one would be the best. How do we know which solution is the best? We will discuss that in the chapter on Probable Value. Generally speaking, we should train ourselves to come up with a few suggestions, and weigh the pros and cons of each resolution. This would be equivalent to coming up with different hypotheses, and judging which one would provide an optimal result.

Chapter 10: Relation

• Pages: 194–225

https://doi.org/10.1142/9789813145313_0010

Relation is the connection and association among different objects, events, and ideas. Problem solving, quite often, is connected with the ability to see the various relations among diversified concepts. Understanding the affiliation of a mixture of notions can be considered as hypothesizing the existence of certain correlation.

Chapter 11: Mathematics

• Pages: 226–306

https://doi.org/10.1142/9789813145313_0011

Mathematics, even some simple arithmetic, is so important in solving some of the everyday problems, that we think a whole chapter should be written on it.

Chapter 12: Probable Value

• Pages: 307–318

https://doi.org/10.1142/9789813145313_0012

For a certain problem, we may come up with several plausible solutions. Which path should we take? Each path would only have certain chance or probability of success in resolving the problem. If each path or solution has a different reward, we can define the probable value of each path to be the multiplication of the reward by the probability. We should, most likely, choose the path that has the highest probable value. (The term “probable value” is coined by us. The idea is appropriated from the term “expected value” in Statistics. In this sense, expected value can be considered as the sum of all probable values.).

Chapter 13: Epilogue

• Pages: 319–322

https://doi.org/10.1142/9789813145313_0013

We run into problems every day. Even when we do not encounter any problems, it does not mean that they do not exist. Sometimes, we wish we could be able to recognize them earlier. The scientific method of observation, hypothesis, and experiment can help us recognize, define, and solve our problems.

BACK MATTER

• Pages: 323–332

https://doi.org/10.1142/9789813145313_bmatter

Praise for the First Edition:

“The book was fun: a clever and entertaining introduction to basic logical thinking and maths.”

“This ingenious and entertaining volume should be useful to anyone in the general public interested in self-help books; undergraduate students majoring in education or behavioral psychology; and graduates and researchers interested in problem-solving, creativity, and scientific research methodology.”

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Optimization techniques in the localization problem: a survey on recent advances.

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Stefanoni, M.; Sarcevic, P.; Sárosi, J.; Odry, A. Optimization Techniques in the Localization Problem: A Survey on Recent Advances. Machines 2024 , 12 , 569. https://doi.org/10.3390/machines12080569

Stefanoni M, Sarcevic P, Sárosi J, Odry A. Optimization Techniques in the Localization Problem: A Survey on Recent Advances. Machines . 2024; 12(8):569. https://doi.org/10.3390/machines12080569

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Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation

• Open access
• Published: 20 August 2024
• Volume 101 , article number  11 , ( 2024 )

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• Vít Dolejší   ORCID: orcid.org/0000-0001-6356-934X 1 ,
• Hyun-Geun Shin 1 &
• Miloslav Vlasák 2  

We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the hp -adaptive method and demonstrate its efficiency and usefulness on a practically relevant example.

Avoid common mistakes on your manuscript.

1 Introduction

Fluid flows in variably saturated porous media are usually described by the Richards equation [ 33 ], which is expressed in the form

where \(\partial _t\) denotes the derivative with respect to time, \(\nabla \cdot \) and \(\nabla \) are the divergence and gradient operators, respectively, \(\psi \) is the sought pressure head (= normalized pressure), z is the vertical coordinate, \(\theta \) is the water content function, \({\textbf{K}}\) is the hydraulic conductivity tensor and g is the source term. In addition, the active pore volume \(\vartheta \) is related to \(\theta \) by the following relation

where \(S_s,\theta _s\ge 0\) are material parameters. The hydraulic conductivity satisfies \({\textbf{K}}(\psi )= {\textbf{K}}_s {{{\mathcal {K}}}}_r(\psi )\) , where \({\textbf{K}}_s\) is the saturated conductivity tensor, and \({{{\mathcal {K}}}}_r\in [0,1]\) is the relative saturation. The functions \(\theta \) and \({{{\mathcal {K}}}}_r\) are given by constitutive relations, e.g., by van Genuchten’s law [ 27 ] and by Mualem’s law [ 31 ], respectively.

The Richards equation belongs to the nonlinear parabolic problems, and it can degenerate, in particular \({\textbf{K}}\rightarrow 0\) or \(\tfrac{{\mathrm d}{\vartheta }}{{\mathrm d}\psi }\rightarrow 0\) . Due to the degeneracy, the numerical solution is challenging, and various techniques have been developed for its solution in the last decades, see [ 25 ] for a survey.

In [ 14 ], we presented the adaptive space-time discontinuous Galerkin (STDG) method for the numerical solution of ( 1 ). This technique is based on a piecewise polynomial discontinuous approximation with respect to both the spatial and temporal coordinates. The resulting scheme is sufficiently stable, provides high accuracy, and is suitable for the hp -mesh adaptation. This is an important property, since the weak solution of the Richards equation is (only) piecewise regular and exhibits singularities along the material interfaces and the unsaturated/saturated zone (when \(\psi \approx 0\) ). Therefore, an adaptive method that allows different meshes at different time levels, can achieve an accurate approximation with a relatively small number of degrees of freedom.

The numerical experiments presented in [ 14 ] showed the potential of the adaptive STDG method. However, the mesh adaptation used is based on interpolation error estimates that do not guarantee an upper error bound. The aim of this work is to overcome this bottleneck, derive a posteriori error estimates, and use them in the hp -mesh adaptation framework.

A posteriori error estimates for the numerical solution of the Richards equation have been treated in many papers for different numerical methods. We mention the finite volume framework with multistep time discretization in [ 5 ], the mixed finite element method in [ 6 ], the two-point finite volume discretization in [ 8 ], the lowest-order discretization on polytopal meshes in [ 38 ], finite element techniques in [ 30 ] and the references cited therein.

Guaranteed error estimates without unknown constants are usually obtained by measuring the error in a dual norm of the residual. Introducing reconstructed fluxes from the space \(H^1(\textrm{div},{\varOmega })\) , the upper bound can then be obtained directly. In [ 18 ], we developed this approach to the higher-order STDG method for nonlinear parabolic problems, where the temporal discontinuities were treated by temporal flux reconstructions considering the time jumps.

In this paper, we extend the approach [ 18 ] to the Richards equation ( 1 ). Although the definition of the temporal and spatial flux reconstructions as well as the derivation of the upper bounds is straightforward, the proof of the lower bound (efficiency) is rather tricky since the term \(\theta (\psi )\) in the time derivative is not a polynomial function for a polynomial \(\psi \) . In contrary to [ 18 ], the proof of efficiency requires the additional oscillatory data terms. We construct spatial fluxes by solving local Neumann problems defined on space-time patches that generalize the approach from [ 22 ]. Moreover, we provide numerical experiments verifying derived error estimates. Compared to [ 18 ], the resulting effectivity indices are much closer to one. This is the first novelty of this paper.

Secondly, we deal with the errors arising due to iterative solution of nonlinear algebraic systems. We introduce a cheap stopping criterion for iterative solvers and justify it by numerical experiments. Thirdly, we introduce a space-time adaptive algorithm that employs the anisotropic hp -mesh adaptation technique [ 15 ]. The algorithm admits local adaptation of size and shape of mesh elements and the local adaptation of degrees of polynomial approximation with respect to space. However, the size of the time step can vary globally, and the degree of polynomial approximation with respect to time is fixed. Using the equidistribution principle, the algorithm gives an approximate solution with the error estimate under the given tolerance. The performance of the adaptive algorithm is demonstrated numerically, including a practically relevant example.

The rest of the paper is organized as follows. In Sect.  2 , we introduce the problem considered, its STDG discretization is briefly described in Sect.  3 . The main theoretical results are derived in Sect.  4 , where the upper and lower bounds are proved. Two possible spatial reconstructions are discussed in Sect.  5 together with the stopping criteria of iterative solvers. The numerical verification of the error estimates is given in Sect.  6 . Furthermore, we present the resulting hp -mesh adaptation algorithm in Sect.  7 and demonstrate its performance by numerical examples. Finally, we conclude with some remarks in Sect.  8 .

2 Problem Formulation

Let \({\varOmega }\subset {\mathbb {R}}^d\) ( \(d=2,3\) ) be the domain occupied by a porous medium and \(T>0\) the physical time to be reached. For simplicity, we assume that \({\varOmega }\) is polygonal. By \({\varGamma }:=\partial {\varOmega }\) , we denote the boundary of \({\varOmega }\) which consists of two disjoint parts: the Dirichlet boundary \({\varGamma _\textrm{D}}\) and the Neumann boundary \({\varGamma _\textrm{N}}\) . We write the Richards equation ( 1 ) in a different form, which is more suitable for the analysis. We seek a function \(u= u(x,t):{\varOmega }\times (0,T)\rightarrow {\mathbb {R}}\) , which represents a hydraulic head (with the physical unit \(\textrm{L}\) ). The quantity \(u\) is related to the pressure head \(\psi \) by \(u= \psi + z\) . The Richards equation ( 1 ) reads

where \(g:{\varOmega }\times (0,T)\rightarrow {\mathbb {R}}\) represents a source term if g is positive or a sink term if g is negative, \({\vartheta }:{\mathbb {R}}\rightarrow {\mathbb {R}}\) denotes the dimensionless active pore volume, and \({\textbf{K}}:{\mathbb {R}}\rightarrow {\mathbb {R}}^{d\times d}\) is the hydraulic conductivity with the physical unit \(\textrm{L}\cdot \textrm{T}^{-1}\) (L = length, T = time). Moreover, \(u_D\) is a trace of a function \(u^*\in L^2(0,T;H^1({\varOmega }))\) on \({\varGamma _\textrm{D}}\times (0,T)\) , \(g_N\in L^2(0,T; L^2({\varGamma _\textrm{N}}))\) and \(u_0\in L^2({\varOmega })\) . We note that with respect to ( 1 ), we should write \({\vartheta }= {\vartheta }(u-z)\) and \({\textbf{K}}= {\textbf{K}}(\theta (u-z))\) , however, we skip this notation for simplicity. We assume that the function \({\vartheta }(u)\) is non-negative, non-decreasing and Lipschitz continuous. Moreover, the tensor \({\textbf{K}}(u)\) is symmetric, positively semi-definite, and Lipschitz continuous.

In order to introduce the weak solution, we set \(H(\textrm{div},\varOmega )=\{v\in L^2({\varOmega })^d:\nabla \cdot v\in L^2({\varOmega })\}\) and define the spaces

where \({\vartheta }^\prime (u)=\partial _t{\vartheta }(u) = \frac{{\textrm{d}} {\vartheta }}{{\textrm{d}}u} \partial _tu\) denotes the time derivative (in the weak sense). Obviously, if \(v\in Y\) then \({\vartheta }(v)\in C([0,T],L^2({\varOmega }))\) . In order to shorten the notation, we set the physical flux

Definition 1

We say that \(u\in Y\) is the weak solution of ( 3 ) if \(u-u^* \in V\) and

where \(\big ({u},{v}\big )_{{\varOmega }}:=\int _{\varOmega }u v {\,{\mathrm d}x}\) and \(\big ({u},{v}\big )_{{\varGamma _\textrm{N}}}:=\int _{\varGamma _\textrm{N}}u v {\,{\mathrm d}S}\) .

The existence and uniqueness of the Richards equation is studied in [ 2 ], see also the later works [ 3 , 28 ].

3 Space-time discretization

We briefly describe the discretization of ( 6 ) by the space-time discontinuous Galerkin (STDG) method, for more details, see [ 13 , 14 ]. Let \(0=t_0<t_1<\ldots <t_r=T\) be a partition of the time interval (0,  T ) and set \(I_m=(t_{m-1},t_m)\) and \(\tau _m=t_m-t_{m-1}\) . For each \(m=0,\dots ,r\) , we consider a simplicial mesh \({{\mathcal {T}}_h^m}\) covering \({\overline{{\varOmega }}}\) . For simplicity, we assume that \({{\mathcal {T}}_h^m}\) , \(m=0,\dots ,r\) are conforming, i.e., neighbouring elements share an entire edge or face. However, this assumption can be relaxed by the technique from [ 12 ].

For each element \(K\in {{\mathcal {T}}_h^m}\) , we denote by \({\partial K}\) its boundary, \({n_K}\) its unit outer normal and \({h_K}=\text{ diam }(K)\) its diameter. In order to shorten the notation, we write \({{\partial K}_{\!N}}:={\partial K}\cap {\varGamma _\textrm{N}}\) . By the generic symbol \(\gamma \) , we denote an edge ( \(d=2\) ) or a face ( \(d=3\) ) of \(K\in {{\mathcal {T}}_h^m}\) and \(h_{\gamma }\) denotes its diameter. In the following, we speak only about edges, but we mean faces for \(d=3\) . We assume that

\({{\mathcal {T}}_h^m}\) , \(m=0,\dots , r\) are shape regular , i.e., \({h_K}/\rho _K\le C\) for all \(K\in {{\mathcal {T}}_h}\) , where \(\rho _K\) is the radius of the largest d -dimensional ball inscribed in K and constant C does not depend on \({{\mathcal {T}}_h^m}\) for \(h\in (0,h_0)\) , \(m=0,\dots ,r\) .

\({{\mathcal {T}}_h^m}\) , \(m=0,\dots , r\) are locally quasi-uniform , i.e., \({h_K}\le C h_{K^\prime }\) for any pair of two neighbouring elements K and \(K^\prime \) , where the constant C does not depend on \(h\in (0,h_0)\) , \(m=0,\dots ,r\) .

Let \(p_K\ge 1\) be an integer denoting the degree of polynomial approximation on \(K\in {{\mathcal {T}}_h^m}\) , \(m=0,\dots ,r\) and \(P_{p_K}(K)\) be the corresponding space of polynomial functions on K . Let

denote the spaces of discontinuous piecewise polynomial functions on \({{\mathcal {T}}_h^m}\) with possibly varying polynomial approximation degrees. Furthermore, we consider the space of space-time discontinuous piecewise polynomial functions

where \(q\ge 0\) denotes the time polynomial approximation degree and \(P_{q}(I_m,{S_{hp,m}})\) is the Bochner space, i.e., \(v\in P_{q}(I_m,{S_{hp,m}})\) can be written as \(v(x,t)=\sum _{j=0}^q t^j\,v_j(x)\) , \(v_j\in {S_{hp,m}}\) , \(j=0,\dots ,q\) .

For \(v\in {{S_{hp}^{\tau q}}}\) , we define the one-sided limits and time jumps by

where \(u_0\) is the initial condition. In the following, we use the notation

where M is either element \(K\in {{\mathcal {T}}_h^m}\) or its (part of) boundary \({\partial K}\) . The corresponding norms are denoted by \({\left\| \cdot \right\| }_{M} \) and \({\left\| \cdot \right\| }_{M,m} \) , respectively. By \(\sum _{K,m}=\sum _{m=1}^r\sum _{K\in {{\mathcal {T}}_h^m}}\) , we denote the sum over all space-time elements \(K\times I_m\) , where \(K\in {{\mathcal {T}}_h^m}\) and \(m=1,\ldots ,r\) .

Moreover, we define the jumps and mean values of \(v\in {S_{hp,m}}\) on edges \(\gamma \subset {\partial K},\ K\in {{\mathcal {T}}_h^m}\) by

where \(v^{\scriptscriptstyle (+)} \) and \(v^{\scriptscriptstyle (-)} \) denote the traces of v on \({\partial K}\) from interior and exterior of K , respectively, and \(u_D\) comes from the Dirichlet boundary condition. For vector-valued \(v\in [{S_{hp,m}}]^d\) , we set \([{v}] = (v^{\scriptscriptstyle (+)} -v^{\scriptscriptstyle (-)} )\cdot {n_K}\) for \(\gamma \in {\varOmega }\) and similarly for boundary edges.

For each space-time element \(K\times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\ldots ,r\) , we define the forms

where \(\alpha >0\) is a sufficiently large penalization parameter ( \(\alpha \sim p_K^2/h_K\) ) and \(\beta \in \{0,\tfrac{1}{2},1\}\) corresponds to the choice of the variants of the interior penalty discretization (SIPG with \(\beta =0\) , IIPG with \(\beta =1/2\) and NIPG with \(\beta =1\) ), see, e.g., [ 13 , Chapter 2].

We introduce the space-time discontinuous Galerkin discretization of ( 3 ).

Definition 2

The function \({u_h^{\tau }}\in {{S_{hp}^{\tau q}}}\) is called the approximate solution of ( 6 ) obtained by the space-time discontinuous Galerkin method (STDGM), if

with form \({A_{K,m}}\) given by ( 12 ) and \(\{\cdot \}\) defined by ( 9 ).

We note that \({u_h^{\tau }}\) is discontinuous with respect to time at \(t_m,\ m=1,\dots ,r-1\) . The solution between \(I_{m-1}\) and \(I_m\) is stuck together by the “time-penalty” term \(\big ({ \big \{{{\vartheta }(u)}\big \}_{m-1}},{v^{m-1}_+}\big )_{K}\) which also makes sense for u and v belonging to different finite element spaces.

Finally, we derive some identities that will be used later. Let \({{\mathcal {F}}_h^m}\) denote the set of all interior edges \(\gamma \not \subset {\varGamma }\) of mesh \({{\mathcal {T}}_h^m}\) and \({{\mathcal {F}}_D^m}\) the set of boundary edges of \({{\mathcal {T}}_h^m}\) lying on \({\varGamma _\textrm{D}}\) . Then, the identity

holds for a piecewise smooth vector-valued function w and a piecewise smooth scalar function z .

Using identity ( 15 ) and the following obvious formulas valid for interior edges \(\left\langle {\left\langle {{\textbf{K}}(u) \nabla u}\right\rangle }\right\rangle =\left\langle {{\textbf{K}}(u) \nabla u}\right\rangle \) , \(\left\langle {\alpha [{u}]}\right\rangle =\alpha [{u}]\) , \([{\left\langle {{\textbf{K}}(u) \nabla u}\right\rangle }]=0\) , \([{\alpha [{u}]}]=0\) , we gain

Consequently, from ( 12 ) and ( 16 ), we obtain the identity

4 A Posteriori Error Analysis

4.1 error measures.

In order to proceed to the derivation of error estimators, we define the spaces of piecewise continuous functions with respect to time by

Obviously, \(Y^0\subset Y\subset {Y^\tau }\subset X\) and \({{S_{hp}^{\tau q}}} \subset {Y^\tau }\) . Moreover, we have the following result.

Let \(u\in Y^0\) be the weak solution of ( 6 ). Then it satisfies

with \({a_{K,m}}\) given by ( 12 ) and the time jump \(\{\cdot \}_{m-1}\) defined by ( 9 ). Moreover, there exists a unique solution \(u\in Y^\tau \) such that \(u-u^*\in V^\tau \) and satisfies ( 19 ).

The proof follows directly by comparing formulas ( 19 )–( 20 ) with ( 6 ) and the fact that \(\big ({\{{\vartheta }(u)\}_{m-1}},{v^{m-1}_+}\big )_{K}=0\) for \(u\in Y^0\) . For the proof of uniqueness, we employ the fact that \(C_0^\infty (\varOmega )\) is dense in \(L^2(\varOmega )\) , i.e., there exists a sequence \(\{v_\varepsilon \}\subset C_0^\infty (\varOmega )\) for any \(v\in L^2(\varOmega )\) such that \(\Vert v_\varepsilon -v\Vert \rightarrow 0\) as \(\varepsilon \rightarrow 0\) , cf. [ 34 , Theorem 3.14]. We apply \(v=v_{s,\varepsilon _1}(x)v_{t,\varepsilon _2}(t)\) in ( 19 ), where the spatial component \(v_{s,\varepsilon _1}\in \{v\in H^1(\varOmega ):v|_{{\varGamma _\textrm{D}}}=0\}\) tends to \(\{{\vartheta }(u)\}_{m-1}\) as \(\varepsilon _1\rightarrow 0\) and the time component \(v_{t,\varepsilon _2}\) is given as 0 outside the interval \((t_{m-1},t_{m-1}+\varepsilon _2)\) and \(v_{t,\varepsilon _2}=1-(t-t_{m-1})/\varepsilon _2\) on \((t_{m-1},t_{m-1}+\varepsilon _2)\) , i.e., \(v_{t,\varepsilon _2}(t)\) tends to 0 as \(\varepsilon _2\rightarrow 0\) . Therefore, all the terms containing time integrals in ( 19 ) tend to 0 when \(\varepsilon _2\) tends to 0. Since \(v^{m-1}_+=v_{s,\varepsilon _1}\) , the remaining jump term tends to \(\Vert \{{\vartheta }(u)\}_{m-1}\Vert ^2\) as \(\varepsilon _1\) tends to 0. From this it follows that \(\{{\vartheta }(u)\}_{m-1}=0\) . Then it is possible to see that any solution of ( 19 ) satisfies the original weak formulation ( 6 ). Since the weak problem ( 6 ) has a unique solution, cf. [ 2 ], the extended problem ( 19 ) has a unique solution as well. \(\square \)

In virtue of [ 11 , § 2.3.1], we define a parameter \({d_{K,m}}\) associated with the space-time element \(K\times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots ,r\) . The parameter \({d_{K,m}}\) represents a user-dependent weight, typically with physical units \((\textrm{T}\,\textrm{L})^{1/2}\) so that the error measure has the same physical unit as the energy norm. In this paper, we use two choices

where \({\left\| \cdot \right\| }_{m,\infty } := {\left\| \cdot \right\| }_{L^\infty ({\varOmega }\times I_m)} \) . We note that the following error analysis is independent of the choice of \({d_{K,m}}\) . Moreover, we define the norm in the space \({V^\tau }\) (cf. ( 18 )) by

In virtue of ( 19 ), we introduce the error measure as a dual norm of the residual

where \({b_{K,m}}\) is given by ( 20 ). The residual \({{\mathcal {R}}}(v)\) represents a natural error measure for \(u-v\in {V^\tau }\) , cf. [ 11 , Remark 2.3]. In Sect.  4 , we estimate \({{\mathcal {R}}}({u_h^{\tau }})\) for \({u_h^{\tau }}\) being the solution of ( 13 ).

Since the approximate solution \({u_h^{\tau }}\) belongs to the space of discontinuous function \({{S_{hp}^{\tau q}}}\not \subset {V^\tau }\) , we introduce the second building block measuring the nonconformity of the solution in spatial variables. Therefore, similarly to [ 18 ], we define the form

where \( C_{K,m,{\textbf{K}},\alpha }=\alpha ^2 +{\left\| {\textbf{K}}({u_h^{\tau }})\right\| }_{L^\infty (K\times I_m)}^{2} \) . The scaling factors are chosen such that \({{\mathcal {J}}}(v)^{1/2}\) has the same physical unit as \({{\mathcal {R}}}({u_h^{\tau }})\) .

We note that \({{\mathcal {J}}}(v)\) measures also the violation of the Dirichlet boundary condition since \({{\mathcal {J}}}(v)\) contains the term \({\left\| v-u_D\right\| }_{{\partial K}\cap {\varGamma _\textrm{D}},m} \) , cf. ( 11 ).

The final error measure is then defined by

where \({{\mathcal {R}}}({u_h^{\tau }})\) is given by ( 23 ) and \({{\mathcal {J}}}({u_h^{\tau }})\) by ( 24 ).

The error measure \({{\mathcal {E}}}({u_h^{\tau }})=0\) if and only if \({u_h^{\tau }}=u\) is the weak solution given by ( 6 ).

Obviously, if \({u_h^{\tau }}=u\) , then \({{\mathcal {J}}}({u_h^{\tau }})=0\) and \({{\mathcal {R}}}({u_h^{\tau }})=0\) due to ( 19 ). On the other hand, if \({{\mathcal {J}}}({u_h^{\tau }})=0\) , then \({u_h^{\tau }}\in {Y^\tau }\) and \({u_h^{\tau }}-u^*\in {V^\tau }\) . Moreover, \({{\mathcal {R}}}({u_h^{\tau }})=0\) and the uniqueness of ( 19 ) imply that \({u_h^{\tau }}\) is the weak solution ( 6 ). \(\square \)

4.2 Temporal and Spatial Flux Reconstructions

Similarly as in [ 18 ], we define a temporal reconstruction \({R_h^{\tau }} = {R_h^{\tau }}(x,t)\) as a continuous function with respect to time that mimics \({\vartheta }({u_h^{\tau }})\) , \({u_h^{\tau }}\in {{S_{hp}^{\tau q}}}\) . Let \(r_m\in P_{q+1}(I_m)\) be the right Radau polynomial on \(I_m\) , i.e., \(r_m(t_{m-1})=1\) and \(r_m(t_m)=0\) , and \(r_m\) is orthogonal to \(P_{q-1}(I_m)\) with respect to the \(L^2(I_m)\) inner product. Then we set

where \(\big \{{\cdot }\big \}\) is given by ( 9 ). The temporal flux reconstruction \({R_h^{\tau }}(x,t)\) is continuous in time, namely \({R_h^{\tau }}\in H^1(0,T, L^2({\varOmega }))\) and it satisfies the initial condition due to

Moreover, by the integration by parts and the properties \(r_m(t_{m-1})=1\) , \(r_m(t_m)=0\) , we obtain

which together with definition ( 26 ) implies

Finally, using the orthogonality of \(r_m\) to \(P_{q-1}(I_m)\) , we obtain from ( 28 ), the formula

Consequently, if \({u_h^{\tau }}\) is the approximate solution given by ( 13 ), then it satisfies

Obviously, the reconstruction \({R_h^{\tau }}\) is local and explicit, so its computation is fast and easy to implement.

The spatial flux reconstruction needs to define a function \({\sigma _h^{\tau }}\in L^2(0,T,H(\textrm{div},\varOmega ))\) which mimics the flux \({\sigma }({u_h^{\tau }}, \nabla {u_h^{\tau }}) = {\textbf{K}}({u_h^{\tau }})\nabla {u_h^{\tau }}\) , cf. ( 5 ). In particular, \({\sigma _h^{\tau }}|_{K\times I_m} \in P_q(I_m, {\textrm{RTN}}_{p}(K))\) where

is the Raviart-Thomas-Nedelec finite elements, cf. [ 7 ] for more details. We assume that the reconstructed flux \({\sigma _h^{\tau }}\) has to be equilibrated with the temporal flux \({R_h^{\tau }}\)

and with the Neumann boundary condition

In Sect.  5 we present two possible constructions of \({\sigma _h^{\tau }}\) including the choice of the spatial polynomial degree p in ( 32 ).

4.3 Auxiliary Results

In the forthcoming numerical analysis, we need several technical tools. We will employ the scaled space-time Poincarè inequality , cf. [ 11 , Lemma 2.2]: Let \(\varphi _{K,m}\in P_0(K\times I_m)\) be the \(L^2\) -orthogonal projection of \(\varphi \in H^1(K\times I_m)\) onto a constant in each space-time element \(K \times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=0,\dots ,r\) . Then,

where \(C_{\textrm{P}}\) is the Poincarè constant equal to \(1/\pi \) for simplicial elements and the last equality follows from ( 22 ).

Moreover, we introduce the space-time trace inequality

Let \(\varphi _{\gamma ,m}\in P_0(\gamma \times I_m)\) be the \(L^2\) -orthogonal projection of \(\varphi \in H^1(K\times I_m)\) onto a constant on each \(\gamma \times I_m\) , where \(\gamma \subset {\partial K}\) is an edge of \(K\in {{\mathcal {T}}_h^m}\) . Then there exists a constant \(C_{\textrm{T}}>0\) such that

where \(C_{\textrm{T}}= \max (c_T,C_{\textrm{P}})\) , \(C_{\textrm{P}}\) is from ( 35 ) and \(c_T>0\) is the constant from the (space) trace inequality.

The proof is straightforward, we present it for completeness. Let \(\varphi \in H^1(K \times I_m)\) and, for all \(t\in I_m\) , set \({{\tilde{\varphi }}}(t):= |\gamma |^{-1} \int _{\gamma } \varphi (x,t){\,{\mathrm d}S}\) . Observing that \((\varphi -{{\tilde{\varphi }}})\) and \(({{\tilde{\varphi }}}-\varphi _{\gamma ,m})\) are \(L^2(\gamma \times I_m)\) -orthogonal, we have

Using the standard trace inequality (e.g., [ 21 , Lemma 3.32]), we have

where \(c_T>0\) is a constant whose values can be set relatively precisely, see the discussion in [ 37 , Section 4.6]. Hence, integrating the square of ( 38 ) over \(I_m\) and using the fact that \(h_{\gamma }\le {h_K}\) , \(\gamma \subset {h_K}\) , we estimate the first term on the right-hand side of ( 37 ) as

Using the fact that \(\varphi _{\gamma ,m} = \tau _m^{-1} \int _{I_m} {{\tilde{\varphi }}}(t){\,{\mathrm d}t}\) , the one-dimensional Poincarè inequality on \(I_n\) and the Cauchy–Schwarz inequality yield

Collecting bounds ( 37 ), ( 39 ), ( 40 ) and the definition of the norm ( 22 ) yields ( 36 ). \(\square \)

4.4 Reliability

We presented the upper bound of \({{\mathcal {R}}}({u_h^{\tau }})\) , cf. ( 23 ).

Let \(u\in Y\) be the weak solution of ( 6 ) and \({u_h^{\tau }}\in {{S_{hp}^{\tau q}}}\) be the approximate solution given by ( 13 ). Let \({R_h^{\tau }}\in H^1(0,T, L^2({\varOmega }))\) be the temporal reconstruction given by ( 26 ) and \({{\sigma _h^{\tau }}}\in L^2(0,T,H(\textrm{div},\varOmega ))\) be the spatial reconstruction satisfying ( 33 ). Then

where \(C_{\textrm{P}}\) is the constant from Poincarè inequality ( 35 ), \(C_{\textrm{T}}\) is the constant from the trace inequality ( 36 ) and the estimators \(\eta _{R,K,m}\) , \(\eta _{S,K,m}\) , \(\eta _{T,K,m}\) , and \(\eta _{N,K,m}\) are given by

The proof of Theorem  1 can be found in [ 19 ] for the case of the homogeneous Dirichlet boundary condition. For completeness, we present its modification including mixed Dirichlet-Neumann boundary conditions.

Starting from ( 20 ), adding the terms \(\pm \big ({{R_h^{\tau }}},{v}\big )_{K,m}\) and \(\pm \big ({\nabla \cdot {\sigma _h^{\tau }}},{v}\big )_{K,m}\) , and using the integration by parts, we obtain

The terms \(\xi _i\) , \(i=1,\dots ,4\) are estimated separately.

Let \({v_{K,m}}\in P_0(K\times I_m)\) be the piecewise constant projection of \(v\in {V^\tau }\) given by the identity \( \big ({{v_{K,m}}},{1}\big )_{K,m} = \big ({v},{1}\big )_{K,m} \) . Using the Cauchy–Schwarz inequality, assumption ( 33 ), the Poincarè inequality ( 35 ), and ( 22 ), we have

Furthermore, by the Cauchy–Schwarz inequality and ( 22 ), we obtain

The use of ( 29 ), and a similar manipulations as in ( 45 ), give

Hence, estimates ( 45 )–( 46 ), the Cauchy inequality and ( 22 ) imply

Furthermore, let \({v_{\gamma ,m}}\in P_0(\gamma \times I_m)\) , \(\gamma \subset {{\partial K}_{\!N}}\) be the \(L^2\) -orthogonal projection from Lemma  3 . Then using assumption ( 34 ), the Cauchy inequality and the space-time trace inequality ( 36 ), we have

The particular estimates ( 44 ), ( 47 ), and ( 48 ), together with the discrete Cauchy–Schwarz inequality, imply ( 41 ). \(\square \)

Obviously, if \({\partial K}\cap {\varGamma _\textrm{N}}\ne \emptyset \) , then \(\eta _{N,K,m}=0\) .

4.5 Efficiency

The aim is to show that the local individual error estimators \(\eta _{R,K,m}\) , \(\eta _{S,K,m}\) and \(\eta _{T,K,m}\) from ( 41 )–(42) are locally efficient, i.e., they provide local lower bounds to the error measure up to a generic constant \(C>0\) which is independent of u , \({u_h^{\tau }}\) , h and \(\tau \) , but may depend on data problems and the degrees of polynomial approximation p and q . A dependence of the estimate up to this generic constant we will denote by \(\lesssim \) .

In order to derive the local variants of the error measure, we denote by \({\omega _K}\) the set of elements sharing at least a vertex with \(K\in {{\mathcal {T}}_h^m}\) , i.e.,

Moreover, we define the functional sub-spaces \(V_{D,m} = \{ v \in {V^\tau }: \text{ supp }\,(v) \subset \overline{ D \times I_m } \}\) for any set \(D\subset {\varOmega }\) (cf. ( 18 )) and the corresponding error measures (cf. ( 23 ))

Obviously, the definition of \(V_{D,m}\) and \({{\mathcal {R}}}_{D,m}({u_h^{\tau }})\) together with the shape regularity implies

Moreover, for each space-time element \(K\times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots ,r\) , we introduce the \(L^2(K\times I_m)\) -projection of the non-polynomial functions, namely

Finally, for each vertex a of the mesh \({{\mathcal {T}}_h^m}\) , we denote by \({\omega _a}\) a patch of elements \(K\in {{\mathcal {T}}_h^m}\) that share this vertex. By \(p_a= \max _{K\in {\omega _a}} p_K\) we denote the maximal polynomial degree on \({\omega _a}\) . Then, for each a of \(K\in {{\mathcal {T}}_h^m}\) , we define a vector-valued function \({\overline{\sigma }}_a = {\overline{\sigma }}_a({u_h^{\tau }}, \nabla {u_h^{\tau }}) \in P_q(I_m, {\textrm{RTN}}_{p_a}(K))\) (cf. ( 32 )) by

where \(\left\langle {\cdot }\right\rangle \) denotes the mean value on \(\gamma \subset {\partial K}\) and \(\psi _a\) is a continuous piecewise linear function such that \(\psi _a(a)=1\) and it vanishes at the other vertices of K . Finally, we set \({\overline{\sigma }}|_{K\times I_m}=\sum _{{a\in K}}{\overline{\sigma }}_a\) .

The proof of the local efficiency of the error estimates presented is based on the choice of a suitable test function in ( 23 ). We set

where \(\chi _K(x)\) is the standard bubble function on K , \(\varPhi _m(t)\) is the Legendre polynomial of degree \(q+1\) on \(I_m\) (and vanishing outside) and \(P_h\big (\big \{{{\vartheta }({u_h^{\tau }})}\big \}_{m-1}\big )\in P_{p_K}(K)\) is the \(L^2(K)\) -projection weighted by \(\chi _K(x)\) given by

We note that

in general, compare with ( 52 ).

Using the inverse inequality, the polynomial function w given by ( 54 ) can be estimated as

Similarly as in [ 11 ] or [ 18 ], we introduce the oscillation terms

The goal is to prove the lower bounds of the proposed error estimates, namely to estimate \(\eta _{T,K,m}\) , \(\eta _{R,K,m}\) and \(\eta _{S,K,m}\) by \({{\mathcal {R}}}_{K,m}({u_h^{\tau }})\) and the oscillation terms ( 58 ), \(K\in {{\mathcal {T}}_h}\) , \(m=1,\dots ,r\) .

Let \(\eta _{T,K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots , r\) be the error estimates given by (42), then

where \({{\mathcal {R}}}_{K,m}\) are the local error measures defined by ( 49 )–( 50 ) and the oscillation terms \(\eta _{G,K,m}\) , \(\eta _{{\vartheta },K,m}\) and \(\eta _{{\vartheta }^\prime ,K,m}\) are given by ( 58 ).

We start the proof by the putting function w from ( 54 ) as the test function in ( 50 ), i.e.

since \(\text{ supp } (w) = K\times I_m\) , cf. ( 54 ). Then, using ( 20 ) and the fact that w vanishes on \({\partial K}\) , we have

The functions \(\overline{\vartheta ^\prime ({u_h^{\tau }})}\) , \( {\overline{g}}\) and \({\sigma _h^{\tau }}\) are polynomials of degree q in time whereas w and \(\nabla w\) are the (Legendre) polynomial of degree \((q+1)\) in time, cf. ( 54 ). Due to the \(L^2(I_m)\) -orthogonality of the Legendre polynomials, we have \(\xi _1=0\) , since

Moreover, using inequality ( 57 ), relations ( 54 )-( 55 ) and the equivalence of norms on finite dimensional spaces,

Furthermore, let \(w_{K,m} = \frac{1}{K\times I_m} \int _{K\times I_m} w {\,{\mathrm d}x}{\,{\mathrm d}t}\) be the mean value of w on the space-time element \(K\times I_m\) . Due to ( 52 ), the Cauchy–Schwarz inequality and ( 35 ), we have

Similarly, the Cauchy–Schwarz inequality and ( 22 ) imply

Collecting ( 61 )–( 66 ), we have

Moreover, using ( 42c ), ( 26 ), integration by parts, the boundedness of the Radau polynomials, the triangle inequality and ( 58 ), we have

Hence, ( 67 ) and ( 68 )

which proves the theorem. \(\square \)

Let \(\eta _{S,K,m}\) and \(\eta _{R,K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots , r\) be the error estimates given by (42), then

where \({{\mathcal {R}}}_{{\omega _K},m}\) is the local error measures defined by ( 49 )–( 50 ) and the oscillation terms \(\eta _{G,K,m}\) , \(\eta _{{\vartheta },K,m}\) and \(\eta _{{\vartheta }^\prime ,K,m}\) are given by ( 58 ).

The proof is in principle identical with the proof [ 18 , Lemmas 7-9], we present the main step for completeness. Let \({\overline{g}}\) and \({\overline{\sigma }}\) be the projection given by ( 52 ) and ( 53 ). Using the triangle inequality, the inverse inequality and ( 58 ), we obtain

The first term on the right-hand side of ( 72 ) can be estimated as in [ 36 , Theorem 4.10] by

where the resulting oscillation terms are estimated with the aid ( 58 ). Moreover, the last term on the right-hand side of ( 72 ) together with ( 42b ) and assumption ( 58 ), reads

which proves ( 70 ).

The proof of ( 71 ) is based on the decomposition

While the second term on the right-hand side of ( 75 ) can be estimated by assumption ( 58 ), the estimate of the first term is somewhat more technical. It depends on the flux reconstruction used. For the flux reconstruction in Sect.  5.2 , the proof is identical to the proof of [ 18 , Lemma 9], which mimics the stationary variant [ 24 , Theorem 3.12]. On the other hand, using the flux reconstruction from Sect.  5.1 , it is possible to apply the technique from [ 11 , Lemma 7.5], where the final relation has to be integrated over \(I_m\) . \(\square \)

5 Spatial Flux Reconstructions and Stopping Criteria

We present two ways of reconstructing the spatial flux \({\sigma _h^{\tau }}\in L^2(0,T,H(\textrm{div},\varOmega ))\) that satisfies the assumptions ( 33 )–( 34 ). The first one, proposed in [ 19 ] for the case of homogeneous Dirichlet boundary condition, is defined by the volume and edge momenta of the Raviart-Thomas-Nedelec (RTN) elements, cf. [ 7 ], and is easy to compute. The second approach is based on the solution of local Neumann problems on patches associated with each vertex of the mesh. This idea comes from, e.g., [ 24 ], its space-time variant was proposed in [ 18 ] for nonlinear convection-diffusion equations. Finally, in Sect.  5.3 , we discuss the errors arising from the solution of algebraic systems and introduce a stopping criterion for the appropriate iterative solver.

5.1 Element-Wise Variant

We denote by \(p_{K,\max }\) the maximum polynomial degree over the element K and its neighbours that share the entire edge with K and \(p_{\gamma ,\max }\) the maximum polynomial degree on neighbouring elements having a common edge \(\gamma \) . Let \({\textrm{RTN}}_{p_{K,\max }}(K)\) be the space of RTN finite elements of order \(p_{K,\max }\) for element \(K\in {{\mathcal {T}}_h^m}\) , cf. ( 32 ), and \({u_h^{\tau }}\in {{S_{hp}^{\tau q}}}\) be the approximate solution. The spatial reconstruction \({\sigma _h^{\tau }}\) is defined element-wise: for each \(K\in {{\mathcal {T}}_h^m}\) , find \({{\sigma _h^{\tau }}}|_{K\times I_m}\in P_q(I_m,{\textrm{RTN}}_{p_{K,\max }}(K))\) with \({{\sigma _h^{\tau }}\cdot n}|_{\gamma \times I_m}\in P_q(I_m,P_{p_{\gamma ,\max }}(\gamma ))\) such that

The edge momenta in ( 76 ) are uniquely defined and since \(p_{\gamma ,\max }\le p_{K,\max }\) , \({\sigma _h^{\tau }}\) in ( 76 ) is well defined as well. Here, the numerical flux \(\left\langle {{\textbf{K}}({u_h^{\tau }}) \nabla {u_h^{\tau }}}\right\rangle \cdot {n} - \alpha [{{u_h^{\tau }}}]\cdot {n}\) is conservative on interior edges, which implies that \({\sigma _h^{\tau }}\cdot {n}\) are the same on each interior edge \(\gamma \) and therefore the resulting reconstruction \({{\sigma _h^{\tau }}}\in L^2(0,T,H(\textrm{div},\varOmega ))\) globally.

Obviously, the first relation in ( 76 ) with \(p_K\le p_{\gamma ,\max }\) directly implies assumption ( 34 ). Moreover, using the Green theorem, ( 76 ), ( 12 ), ( 31 ) and \(p_K\le p_{\gamma ,\max }\le p_{K,\max }\) , we obtain

which justifies the assumption ( 33 ).

5.2 Patch-Wise Variant

For each vertex a of the mesh \({{\mathcal {T}}_h^m}\) , we denote by \({\omega _a}\) a patch of elements \(K\in {{\mathcal {T}}_h^m}\) sharing this vertex. By \(p_a= \max _{K\in {\omega _a}} p_K\) we denote the maximal polynomial degree on \({\omega _a}\) . Let \(P_{p_a}^{*}({\omega _a})\) be the space of piecewise polynomial discontinuous functions of degree \(p_a\) on \({\omega _a}\) with mean value zero for \(a \notin \partial {\varOmega }\) . We define the space

We set the local problems on patches \({\omega _a}\) for all vertices a : find \({\sigma _h^{\tau }}\in P_q(I_m, W^N_{{\textrm{RTN}},p_a}({\omega _a}))\) and \(r^\tau _a \in P_q(I_m, P_{p_a}^{*}({\omega _a}))\) such that

and \(\ell _{m,\gamma }: {S_{hp,m}}\rightarrow [{S_{h0,m}}]^d\) is the lifting operator defined by

Then the final reconstructed flux is obtained by summing up \({\sigma }^\tau _a\) on each element that contains vertex a , i.e.,

The assumption ( 34 ) follows directly from ( 78 ) and \(p_K\le p_a\) . Inserting the hat function \(\psi _a v\) for \(a\not \in \partial \varOmega \) and \(v\in P_q(I_m)\) in ( 17 ), using ( 5 ), ( 82 ) and omitting the zero terms, we have

Applying ( 13 ) and ( 31 ), we gain for \(a\not \in \partial \varOmega \) and \(v\in P_q(I_m)\)

From this it follows that the second relation in ( 79 ) holds element-wise, i.e.

Then ( 33 ) follows from

and from \(p_K\le p_a\) .

5.3 Stopping Criteria for Iterative Solvers

The space-time discretization ( 13 ) leads to a system of nonlinear algebraic equations for each time level \(m=1,\dots ,r\) . These systems have to be solved iteratively by a suitable solver, e.g., the Picard method, the Newton method or their variants. Therefore, it is necessary to set a suitable stopping criterion for the iterative solvers so that, on the one hand, the algebraic errors do not affect the quality of the approximate solution and, on the other hand, an excessive number of algebraic iterations is avoided.

However, the error estimates presented in Sect.  4 do not take into account errors arising from the inaccurate solution of these systems. Indeed, the aforementioned reconstructions fulfill assumption ( 33 ) only if the systems given by ( 13 ) are solved exactly. The full a posteriori error analysis including algebraic errors has been treated, e.g., in [ 8 , 23 , 29 ]. These error estimators are based on additional flux reconstructions that need to be evaluated at each iteration, and therefore, the overall computational time is increased.

To speed up the computations and control the algebraic errors, we adopt the technique of [ 17 ]. This approach offers (i) the measurement of algebraic errors by a quantity similar to the error measure ( 23 ), (ii) the setting of the stopping criterion for iterative solvers with one parameter corresponding to the relative error, and (iii) a fast evaluation of the required quantities.

For each \(m=1,\dots ,r\) , we define the estimators (cf. ( 23 ))

where the norm \({\left\| \cdot \right\| }_{{V^\tau }} \) is given by ( 22 ),

The space \({S_{hp+1}^{\tau q+1}}\) is an enrichment space of \({S_{hp}^{\tau q}}\) by polynomials of the space degree \(p_K+1\) and the time degree \(q+1\) for each \(K\times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots ,r\) . Finally, we define the global in time quantities

Obviously, if \({u_h^{\tau }}\) fulfills ( 13 ) exactly, then \({\eta _{\textrm{alg}}^m}({u_h^{\tau }})=0\) for all \(m=0,\dots ,r\) . Moreover, if \({u_h^{\tau }}\) is the weak solution ( 6 ) then \({\eta _{\textrm{spa}}^m}({u_h^{\tau }})=0\) for all \(m=0,\dots ,r\) . Comparing ( 88 ) with ( 23 ), the quantity \({\eta _{\textrm{spa}}}({u_h^{\tau }})\) exhibits a variant of the error measure \({{\mathcal {R}}}({u_h^{\tau }})\) . Nevertheless, \({\eta _{\textrm{spa}}}({u_h^{\tau }})\) is neither lower nor upper bound of \({{\mathcal {R}}}({u_h^{\tau }})\) since \({S_{hp+1}^{\tau q+1}}\not \subset {V^\tau }\) and \({V^\tau }\not \subset {S_{hp+1}^{\tau q+1}}\) .

The quantities ( 88 ) can be evaluated very fast since the suprema (maxima) are the sum of the suprema (maxima) for all space-time elements \(K\times I_m\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots ,r\) , which are computable separately, cf. [ 17 ] for details. Hence, we prescribe the stopping criterion for the corresponding iterative solver as

where \(c_A\in (0,1)\) is the user-dependent constant. The justification of this approach and the influence of algebraic errors on the error estimates are studied numerically in Sect.  6.1.1 .

6 Numerical Experiments

We present numerical experiments that justify the a posteriori error estimates ( 41 )–(42). Since the error measure ( 23 ) is the dual norm of the residual, it is not possible to evaluate the error even if the exact solution is known. Therefore, similarly to [ 18 ], we approximate the error by solving the dual problem given for each time interval \(I_m,\ m=1,\dots ,r\) : Find \( \psi _m \in {Y^{\tau }_{m}}=L^2(I_m,H^1({\varOmega })) \) ,

where (cf. ( 21a )–( 22 ))

Then we have \({{\mathcal {R}}}({u_h^{\tau }})^2 = \sum ^r_{m=1}{\left\| \psi \right\| }_{{Y^{\tau }_{m}}}^{2} \) . We solve ( 92 ) for each \(m=1,\dots ,r\) by linear conforming finite element on a global refinement of the space-time mesh \({{\mathcal {T}}_h^m}\times I_m\) which is proportional to the space and time polynomial approximation degrees. We denote this quantity by \(\widetilde{{{\mathcal {R}}}}({u_h^{\tau }})\) . The second error contribution \({{\mathcal {J}}}\) given by ( 24 ) is computable, so the total error \({{\mathcal {E}}}\) (cf. ( 25 )) is approximated by \(\widetilde{{{\mathcal {E}}}}({u_h^{\tau }}):= \left( \widetilde{{{\mathcal {R}}}}({u_h^{\tau }})^2+{{\mathcal {J}}}({u_h^{\tau }})\right) ^{1/2}\) .

Sometimes, this approximate evaluation of the (exact) error is not sufficiently accurate for fine grids and high polynomial approximation degrees. In this case, very fine global refinement is required and then the resulting algebraic systems are too large to be solved in a reasonable time.

All numerical experiments were carried out using the patch-wise reconstruction from Sect.  5.2 using the in-house code ADGFEM [ 10 ]. The arising nonlinear algebraic systems are solved iteratively by a Newton-like method, we refer to [ 14 ] for details. Each Newton-line iteration leads to a linear algebraic system that is solved by GMRES method with block ILU(0) preconditioner.

6.1 Barenblatt Problems

We consider two nonlinear variants of ( 3 ) following from the Barenblatt problem [ 4 ] where the analytical solution exists. The first variant reads

where the analytical solution is

Using the substitution \(v:=u^{1/m}\) , we have the second variant

having the solution

For both problems (( 94 )–( 95 ) and ( 96 )–( 97 )), the computational domain is \(\varOmega = (-6, 6)^2\) and the Dirichlet boundary condition is prescribed on all boundaries by ( 95 ) or ( 97 ). The final time is \(T=1\) .

We carried out computation using a sequence of uniform triangular grids (having 288, 1152, 4608 and 18432 triangles) with several combinations of polynomial approximation degrees with respect to space ( p ) and time ( q ). The time step has been chosen constant \(\tau =0.01\) . Besides the error quantities ( \(\widetilde{{{\mathcal {R}}}}({u_h^{\tau }})\) and \({{\mathcal {J}}}({u_h^{\tau }})\) ) and its estimators \(\eta \) , \(\eta _{R}:=\sum _{K,m}\eta _{R,K,m}\) , \(\eta _{S}:=\sum _{K,m}\eta _{S,K,m}\) and \(\eta _{T}:=\sum _{K,m}\eta _{T,K,m}\) , we evaluate the effectivity indices

In addition, we present the experimental order of convergence (EoC) of the errors and the estimators for each pair of successive meshes.

Tables  1 – 4 show the results for both Barenblatt problems (( 94 )–( 95 ) with \(m=0.25\) and ( 96 )–( 97 ) with \(m=2\) ) with two variants of the scaling parameter \({d_{K,m}}\) , \(K\in {{\mathcal {T}}_h^m}\) , \(m=1,\dots ,r\) given by ( 21a ) and ( 21b ). The quantity \(\#\textrm{DoF}\) represents the number of degrees of freedom in the space, that is, \(\#\textrm{DoF}=\dim {S_{hp,m}}\) , \(m=1,\dots ,r\) . We observe a good correspondence between \(\widetilde{{{\mathcal {R}}}}({u_h^{\tau }})\) and \(\eta \) , the effectivity index \(i_{\textrm{eff}}\) varies between 1 and 2.5 for all tested values of p and q and both variants of \({d_{K,m}}\) (( 21a ) and ( 21b )).

Finally, we note that the experimental orders of convergence EoC in Tables  1 – 4 ) of the error \(\widetilde{{{\mathcal {R}}}}({u_h^{\tau }})\) and its estimate \(\eta \) are \(O(h^p)\) for the choice ( 21b ) of the scaling parameter \({d_{K,m}}\) but only \(O(h^{p-1})\) for the choice ( 21a ). This follows from the fact that \(\tau _m \ll h_K\) for the computations of the Barenblatt problem and then the dominant part of \({d_{K,m}}\) is \(\tau _m^{-2} T {\left\| \tfrac{{\mathrm d}{\vartheta }}{{\mathrm d}u}\right\| }_{K,m,\infty } \) , cf. ( 21a ), which implies that \({d_{K,m}}= O(h^0)\) (the time step is the same for all computations). The dominant part of the error estimator is \(\eta _{S,K,m}\) , hence if \({\left\| {\sigma _h^{\tau }}- {\sigma }({u_h^{\tau }}, \nabla {u_h^{\tau }}) \right\| }_{K,m} = O(h^{p})\) then \(\eta _{S,K,m}= O(h^{p-1})\) , cf. ( 42b ). Nevertheless, comparing the pairs of Tables  1 – 2 and Tables  3 – 4 , we found that the effectivity indexes are practically independent of the choice of \({d_{K,m}}\) .

6.1.1 Justification of the Algebraic Stopping Criterion ( 91 )

We present the numerical study of the stopping criterion ( 91 ) which is used in the iterative solution of algebraic systems given by ( 13 ). We consider again the Barenblatt problem ( 94 )–( 95 ) with \(m=0.25\) and ( 96 )–( 97 ) with \(m=2\) . The user-dependent constant \(c_A\) in ( 91 ) has been chosen as \(10^{-1}\) , \(10^{-2}\) , \(10^{-3}\) and \(10^{-4}\) . Tables  5 and 6 show the estimators \(\eta \) , \({{\mathcal {J}}}({u_h^{\tau }})\) , \({\eta _{\textrm{alg}}}\) and \({\eta _{\textrm{alg}}}\) , cf. ( 90 ), for selected meshes and polynomial approximation degrees and the scaling parameter \({d_{K,m}}\) chosen by ( 21a ).

Additionally, we present the total number of steps of the Newton-like solver \({N_{\textrm{non}}}\) , the total number of GMRES iterations \({N_{\textrm{lin}}}\) and the computational time in seconds. The computational time has only an informative character.

We observe that the error estimators \(\eta \) , \({{\mathcal {J}}}({u_h^{\tau }})\) and also \({\eta _{\textrm{spa}}}\) converge to the limit values for decreasing \(c_A\) in ( 91 ) which mimic the case when the algebraic errors are negligible. Moreover, the relative differences between the actual values \(\eta \) and \({{\mathcal {J}}}({u_h^{\tau }})\) and their limits correspond more or less to the value of \(c_A\) . Obviously, smaller values of \(c_A\) cause prolongation of the computational time, due to a higher number of iterations, with a negligible effect on accuracy. Thus, the choice \(c_A=10^{-2}\) seems to be optimal in order to balance accuracy and efficiency.

The presented numerical experiments indicate that the estimator \({\eta _{\textrm{spa}}}({u_h^{\tau }})\) gives an upper bound of \({{\mathcal {R}}}({u_h^{\tau }})\) , however, this observation is not supported by the theory. The quantity \({\eta _{\textrm{spa}}}({u_h^{\tau }})\) is used only in the stopping criterion ( 91 ).

6.2 Tracy Problem

Tracy problem represents a standard benchmark, where the analytical solutions of the Richards equation are available [ 35 ]. We consider the Gardners constitutive relations [ 26 ]

where \(\psi = u - z\) is the pressure head, z is the vertical coordinate and the material parameters \({\textbf{K}}_s=1.2{\mathbb {I}}\) , \(\theta _s=0.5\) , \(\theta _r=0.0\) , and \(\alpha =0.1\) are the isotropic conductivity, saturated water content, residual water content, and the soil index parameter related to the pore-size distribution, respectively.

The computational domain is \({\varOmega }=(0,1)^2\) , the initial condition is set \(u = u_r: = -10\) in \({\varOmega }\) where \(u_r\) corresponds to the hydraulic head when the porous medium is dry. On the top part of the boundary \({\varGamma }_1:=\{(x, z),\ x\in (0,1),\ z= 1\}\) , we prescribe the boundary condition

and on the rest of boundary \({\varGamma }\) we set \(u = u_r\) . We note that this benchmark poses an inconsistency between the initial and boundary conditions on \({\varGamma }_1\) . Hence, the most challenging part is the computation close to \(t=0\) . In order to avoid the singularity at \(t=0\) , we investigate the error only on the interval \(t\in [1.0 \times 10^{-5}, 1.1 \times 10^{-4}]\) with the fixed time step \(\tau \) is \(1.0 \times 10^{-6}\) .

We perform a computation using a sequence of uniform triangular grids with several combinations of polynomial approximation degrees and the choice ( 21b ), the results are shown in Table  7 . We observe reasonable values of the effectivity indices except for the finest grids and the higher degrees of polynomial approximation, where the effectivity indices \(i_{\textrm{eff}}\) are below 1. Based on the values of EoC, we suppose that \(i_{\textrm{eff}}\) below 1 is not caused by the failure of the error estimator but due to an inaccurate approximation \(\widetilde{{{\mathcal {R}}}}({u_h^{\tau }})\) of the exact error; see Remark  3 .

7 Mesh Adaptive Algorithm

We introduce the mesh adaptive algorithm which is based on the a posteriori error estimates \(\eta \) , cf. ( 41 ). Let \(\delta >0\) be the given tolerance, the goal of the algorithm is to define the sequence of time steps \(\tau _m\) , meshes \({{\mathcal {T}}_h^m}\) and spaces \({S_{hp,m}}\) , \(m=1,\dots ,r\) such that the corresponding approximate solution \({u_h^{\tau }}\in {{S_{hp}^{\tau q}}}\) given by ( 13 ) satisfies the condition

Another possibility is to require \(\left( \eta ^2 + {{\mathcal {J}}}({u_h^{\tau }})\right) ^{1/2} \le \delta \) , then the following considerations have to be modified appropriately.

The mesh adaptation strategy is built on the equi-distribution principle, namely the sequences \(\{\tau _m,\, {{\mathcal {T}}_h^m},\, {S_{hp,m}}\}_{m=1}^r\) should be generated such that

where \(\eta _m:= (\sum _{K\in {{\mathcal {T}}_h^m}} \eta _{K,m}^2 )^{1/2}\) is the error estimate corresponding to the time interval \(I_m\) , \(m=1,\dots ,r\) and \(\#{{\mathcal {T}}_h^m}\) denotes the number of elements of \({{\mathcal {T}}_h^m}\) . Obviously, if all the conditions in ( 102 ) are valid, then the criterion ( 101 ) is achieved.

Space-time mesh adaptive algorithm.

Based on ( 101 )–( 102 ), we introduce the abstract Algorithm 1. The size of \(\tau _m\) , \(m=1,\dots ,r\) (step  8 of the algorithm) are chosen to equilibrate estimates of the spatial and temporal reconstruction, \(\eta _{S,m}:= (\sum _{K\in {{\mathcal {T}}_h^m}} (\eta _{S,K,m})^2)^{1/2}\) and \(\eta _{T,m}:= (\sum _{K\in {{\mathcal {T}}_h^m}} (\eta _{T,K,m})^2)^{1/2}\) , cf. (42). Particularly, we set the new time step according to the formula

where \(c_F\in (0,1)\) is the security factor and \(q\ge 0\) is the polynomial degree with respect to time. Therefore, \(q+1\) corresponds to the temporal order of convergence.

The construction of the new mesh (step  11 in Algorithm 1) is based on the modification of the anisotropic hp -mesh adaptation method from [ 15 , 20 ]. Having the actual mesh \({{\mathcal {T}}_h^m}\) , for each \(K\in {{\mathcal {T}}_h^m}\) we set the new volume of K according the formula

where \(\delta _{K,m}\) is the local tolerance from ( 102b ), | K | is the volume of | K | and \(\varLambda :{\mathbb {R}}^+\rightarrow {\mathbb {R}}+\) is a suitable increasing function such that \(\varLambda (1) = 1\) . For particular variants of \(\varLambda \) , we refer to [ 15 , 20 ].

When the new volume of mesh elements is established by ( 104 ), the new shape of K and a new polynomial approximation degree \(p_K\) are optimized by minimizing the interpolation error. This optimization is done locally for each mesh element. In one adaptation level, we admit the increase or decrease of \(p_K\) by one. Setting the new area, shape, and polynomial approximation degree for each element of the current mesh, we define the continuous mesh model [ 16 ] and carry out a remeshing using the code ANGENER [ 9 ].

The generated meshes are completely non-nested and non-matching, hence the evaluation of the time-penalty term (cf. Remark  1 ) is delicate. We refer to [ 20 ] where this aspect is described in detail and numerically verified. The presented numerical analysis takes into account the errors arising from the re-meshing in the temporal reconstruction \({R_h^{\tau }}\) , which contains term \( \big \{{{\vartheta }({u_h^{\tau }})}\big \}_{m-1}\) , cf. ( 26 ). The following numerical experiments show that the error estimator is under the control also after each remeshing.

7.1 Barenblatt Problem

We apply Algorithm 1 to the Barenblatt problem ( 96 ) with \(m=2\) . Table  8 shows the error estimators obtained by adaptive computation for three different tolerances \(\delta \) . Compared with the error estimators from Table  4 , we observe that the adaptive computations achieve significantly smaller error estimates using a significantly smaller number of degrees of freedom. We note that we are not able to present the quantity \(\widetilde{{{\mathcal {R}}}}\) (cf. ( 92 )–( 93 )) approximating the error since the finite element code used for the evaluation of \(\widetilde{{{\mathcal {R}}}}\) supports only uniform grids.

Figure  1 shows the performance of Algorithm 1, where each dot corresponds to one time step \(m=1,\dots ,r\) . We plot the values of the accumulated estimators \({\overline{\eta }}_m =\sum _{i=1}^m \eta _i\) for all \(m=1,\dots ,r\) . The red nodes correspond to all computed time steps, including the rejected ones (steps  11 – 12 of Algorithm 1) whereas the blue nodes mark only the accepted time steps. The rejected time steps indicate the re-meshing. Moreover, we plot the “accumulated” tolerance \(\delta (t_m/T)^{1/2}\) , cf. ( 101 ) and ( 102a ). We observe that the resulting estimator \(\eta \) at \(t=T\) is below the tolerance \(\delta \) by a factor of approximately 2.5 since conditions ( 102 ) are stronger than ( 101 ).

Barenblatt problem, ( 96 )–( 97 ), \(m=2\) , performance of Algorithm 1, accumulated error estimator \({\overline{\eta }}_m\) and the “accumulated” tolerance \(\delta (t_m/T)^{1/2}\) for \(m=1,\dots ,r\)

Figure  2 , left, shows the hp -mesh obtained by Algorithm 1 at the final time \(T=1\) , each triangle is highlighted by a color corresponding to the polynomial degree used \(p_K\) , \(K\in {{\mathcal {T}}_h^m}\) . We observe a strong anisotropic refinement about the circular singularity of the solution when \(u\rightarrow 0^+\) , see the analytical formula ( 97 ). Outside of this circle, large triangles with the smallest polynomial degree ( \(p=1\) ) are generated. On the other hand, due to the regularity of the solution in the interior of the circle, the polynomial degrees \(p=2\) or \(p=3\) are generated.

Barenblatt problem, hp -mesh obtained by Algorithm 1 (left) and the error estimators \(\eta _{K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) at \(T=1\)

Moreover, Fig.  2 , right, shows the error estimator \(\eta _{K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) at \(T=1\) . The elements in the exterior of the circle have small values of \(\eta _{K,m}\approx 10^{-17}\) – \(10^{-14}\) due to a constant solution and negligible errors. On the other hand, the values of \(\eta _{K,m}\) for the rest of elements \(K\in {{\mathcal {T}}_h^m}\) are in the range \(10^{-13}\) – \(10^{-11}\) due to the equidistant principle used.

7.2 Single Ring Infiltration

We deal with the numerical solution of the single ring infiltration experiment, which is frequently used for the identification of saturated hydraulic conductivity, cf. [ 32 , 39 ] for example. We consider the Richards equation ( 3 ) where the active pore volume \(\vartheta \) is given by ( 2 ), the water content function \(\theta \) is given by the van Genuchten’s law [ 27 ] and the conductivity \({\textbf{K}}(u) ={\textbf{K}}_s {{{\mathcal {K}}}}_r(u)\) is given by the Mualem function [ 31 ], namely

where \(\psi = u - z\) is the pressure head, z is the vertical coordinate and the material parameters \({\textbf{K}}_s=0.048\,{\mathbb {I}}\,\,\mathrm {m\cdot hours^{-1}}\) , \(\theta _s=0.55\) , \(\theta _r=0.0\) , \(\alpha =0.8\,\mathrm {m^{-1}}\) , \(n=1.2\) , \(m=1/6\) and \(S_s=10^{-3}\,\mathrm {m^{-1}}\) (cf. ( 2 )).

The computational domain together with the boundary parts is sketched in Fig.  3 a. On the boundary part \(\varGamma _D\) we set the Dirichlet boundary condition \(u= 1.05\,\textrm{m}\) , and on \({\varGamma _\textrm{N}}={\varGamma }{\setminus }{\varGamma _\textrm{D}}\) we consider the homogeneous Neumann boundary condition. The smaller “magenta” vertical lines starting at \({\varGamma _\textrm{D}}\) belong to \({\varGamma _\textrm{N}}\) . At \(t=0\) , a dry medium with \(u= \psi + z =-2\,\textrm{m}\) is prescribed. We carried out the computation until the physical time \(T=2\,\textrm{hours}\) . The inconsistency of the initial and boundary condition on \({\varGamma _\textrm{D}}\) makes the computation quite difficult for \(t\approx 0\) .

Single ring infiltration problem

Figure  3 b verifies the conservativity of the adaptive method. We plot the quantities

where F ( t ) is the total flux of the water through the boundary \({\varGamma }\) till time t and \(\varDelta V(t)\) is the changes of the water content in the domain between times 0 and t . From equation ( 3 ) and the Stokes theorem, we have the conservation law \(F(t) = \varDelta V(t)\) for all \(t\in [0,T]\) . Therefore, we also show the relative difference between these quantities \(|F(t) - \varDelta V(t)|/ \varDelta V(t)\) for \(t>0\) in Fig.  3 b the vertical label on the right. We observe that, except for the time close to zero, where the inconsistency between initial and boundary conditions is problematic, the relative difference is at the level of several percent.

Furthermore, Fig.  4 shows the accumulated estimators \({\overline{\eta }}_m =\sum _{i=1}^m \eta _i\) for time levels \(t_m\) , \(m=1,\dots ,m\) . The red nodes correspond to all computed time steps, including the rejected steps whereas the blue line connects only the accepted time steps. The rejected time steps are followed by the remeshing which is carried out namely for small t . We observe that the elimination of the rejected time steps causes that the errors arising from the remeshing do not essentially affect the total error estimate \(\eta \) .

Single ring infiltration, performance of Algorithm 1, accumulated error estimator \({\overline{\eta }}_m\) with respect to \(t_m\) , \(m=1,\dots ,r\)

Moreover, Fig.  5 shows the hp -meshes, the hydraulic head and the error estimator \(\eta _{K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) at selected time levels obtained from Algorithm 1 with \(\delta =5.0 \times 10^{-3}\) . We observe the mesh adaptation namely at the (not sharp) interface between the saturated and non-saturated medium and also in the vicinity of the domain singularities. The error estimators \(\eta _{K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) indicate an equi-distribution of the error.

Single ring infiltration, hydraulic head (top ), the corresponding hp -meshes obtained by Algorithm 1 (center) and the error estimators \(\eta _{K,m}\) , \(K\in {{\mathcal {T}}_h^m}\) (bottom) at \(t=0.4\) , \(t=0.8\) and \(t=2\) \(\,\textrm{hours}\) (from left to right)

8 Conclusion

We derived reliable and efficient a posteriori error estimates in the residual-based norm for the Richards equation discretized by the space-time discontinuous Galerkin method. The numerical verification indicates the effectivity indexes between 1 and 2.5 for the tested examples. Moreover, we introduced the hp -mesh adaptive method handling varying non-nested and non-matching meshes and demonstrated its efficiency for simple test benchmark and its applicability for the numerical solution of the single ring infiltration experiment.

It will be possible to generalize the presented approach to genuinely space-time hp -adaptive method, where the (local) polynomial order q in time is varied as well. However, the question is of potential benefit. Based on our experience, the setting \(q=1\) gives sufficiently accurate approximation for the majority of tested problems.

On the other hand, the choice \(q=0\) would be sufficient only in subdomains of \({\varOmega }\) where the solution is almost constant in time. Therefore, we suppose that the benefit of local varying of polynomial order in time will be low.

Although the presented numerical examples are two-dimensional, it would be possible to apply the presented error estimates and mesh adaptation to three-dimensional problems as well. We refer, e.g., to [ 1 ] and the references therein, where the anisotropic mesh adaptation techniques are developed for time-dependent 3D problems.

Data Availability

No datasets were generated or analysed during the current study.

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Open access publishing supported by the National Technical Library in Prague. This work has been supported by the Czech Science Foundation Grant No. 20-01074 S (V.D.), the Charles University grant SVV-2023-260711, and the Grant Agency of Charles University Project No. 28122 (H.S.), European Development Fund-Project “Center for Advanced Aplied Science” No. CZ.02.1.01/0.0/0.0./16 019/0000778 (M.V.). V.D. acknowledges the membership in the Nečas Center for Mathematical Modeling ncmm.karlin.mff.cuni.cz.

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Dolejší, V., Shin, HG. & Vlasák, M. Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation. J Sci Comput 101 , 11 (2024). https://doi.org/10.1007/s10915-024-02650-x

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• Published: 19 August 2024

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• Xiangdang Huang 2 , 4 &
• Qiuling Yang 2 , 4  

Scientific Reports volume  14 , Article number:  19140 ( 2024 ) Cite this article

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In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

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Introduction.

In reality, many optimization problems involve multiple conflicting objectives, such as the design of urban public transport routes 1 , production scheduling 2 , securities portfolio management 3 and so on. These types of optimization problems are called multi-objective optimization problems (MOPs). This means that there is no one solution to make all the objectives reach the optimum simultaneously, that is, the optimization of one objective may lead to the deterioration of other objectives 4 , 5 . Consequently, the solutions of MOPs are usually a set of compromise solutions that weigh all objectives. The definition of MOPs is as follows:

Among them, \(f\left(x\right)\) is the m-dimensional objective vector, which contains m conflicting objective functions; \({f}_{i}\left(x\right)\) represents the i-th objective function; x represents the n-dimensional decision variable; \(\Omega \) represents decision space; R m represents the objective space.

In the field of multi-objective optimization, problems with 2 or 3 optimization objectives are called general multi-objective optimization problems (GMOPs). Problems with more than 3 optimization objectives are called many-objective optimization problems (MaOPs) 6 , 7 , 8 . GMOPs aren’t the focus of our attention, as there have been many reports about GMOPs 9 , 10 . On the contrary, MaOPs are the focus of our attention, as there are still some challenges to be solved. The fundamental difference between GMOPs and MaOPs is the number of optimization objectives. Assuming that the number of optimization objectives is m, the probability that one individual dominates another is \(1/{2}^{m-1}\) in theory 11 , 12 . This means that with the increase of the number of optimization objectives, traditional Pareto dominance will fail, Pareto resistance will occur, and most multi-objective optimization algorithms will lose selection pressure in terms of convergence.

In recent years, with the research and exploration of MaOPs, many-objective optimization technology has been developed to a certain extent, and basically 4 mainstream many-objective optimization algorithms have been formed 13 . The first is many-objective optimization algorithm based on dominance. The algorithm modifies the definition of traditional Pareto domination by domination relaxation technique to enhance the selection pressure of the algorithm in terms of convergence. \(\alpha\) -Dominance, \(\upepsilon \) -Dominance and Cone \(\upepsilon \) -Dominance are all common domination relaxation techniques. Compared with traditional Pareto dominance, the effectiveness of dominance relaxation technology has been reported in many works. Therefore, dominance relaxation technology has been widely used to solve MaOPs. However, the current domination relaxation technique also faces two problems: (1) With the increase of the number of optimization objectives, the effect of the domination relaxation technique is getting worse and worse; (2) The domination relaxation technique tends to make the population converge to a certain sub-region of the real Pareto front (PF).

The second is many-objective optimization algorithm based on index. The algorithm guides the selection and evolution of the population by integrating convergence and diversity into one index (such as IGD, HV). Its representative work includes: HypE, MaOEA/IGD, SMS-EMOA. However, the algorithm faces some problems when it is used to solve MaOPs, such as complex index calculation, difficult selection of reference point or reference PF.

The third is many-objective optimization algorithm based on decomposition. The algorithm transforms MaOPs into several single-objective optimization sub-problems through an aggregation function, and then drives the individuals in the neighborhood to update by neighborhood strategy, finally realizes the evolution of the whole population. Its representative work includes: MOEA/D, MOEA/D-D, MOEA/D-DU. However, many-objective optimization algorithm based on decomposition is only suitable for MaOPs with regular PF (such as the DTLZ1 problem). When dealing with MaOPs with irregular PF, many-objective optimization algorithm based on decomposition often performs poorly.

The fourth is many-objective optimization algorithm based on hybrid strategy. The algorithm adopts different search strategies in different environments (different stages or different sub-populations), and uses the advantages of their respective search strategies to deal with complex MaOPs. Its representative work includes: AHM, eMOFEOA, CPSO. In many reports, many-objective optimization algorithm based on hybrid strategy is more suitable for solving MaOPs.

According to the above analysis, this paper considers using the many-objective optimization algorithm based on hybrid strategy, and further proposes the many-objective evolutionary algorithm based on three states (MOEA/TS). The innovations and contributions of this paper are as follows: (1) A feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. (2) Based on the Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. (3) A repulsion field method is proposed. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF. (4) Design a new concurrent algorithm framework. In the framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can freely switch among these three states according to its own evolution.

The remainder of this paper is organized as follows: Sect. " Preparatory work " introduces the basic definition, related work and research motivation. Sect. " Basic definition " introduces each part of the MOEA/TS algorithm in detail. Sect. " Related work " introduces the test results of MOEA/TS algorithm and 7 advanced many-objective optimization algorithms on various test problems, and then analyzes and summarizes them according to the test results. Sect. " Many-objective optimization algorithm based on dominance "summarizes this article and looks forward to future work.

Preparatory work

Basic definition.

In this section, we will introduce some basic definitions related to many-objective optimization technology.

Definition of dominance: if solution x isn’t worse than solution y in all objectives and solution x is better than solution y in at least one objective, it is said that x dominates y . That is, if \(\forall i\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{i}\left(x\right)\le {f}_{i}\left(y\right)\) and \(\exists j\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{j}\left(x\right)<{f}_{j}\left(y\right)\) , it is said that x dominates y .

Definition of non-dominated solution: if there are no solutions that can dominate x in the decision space, then x is called a Pareto optimal solution or a non-dominated solution. That is, if \(\nexists {x}^{*}\in \Omega \) makes x* dominate x , then x is called a Pareto optimal solution or a non-dominated solution.

Definition of Pareto optimal solution set: the set composed of Pareto optimal solutions is called the Pareto optimal solution set (PS). The mathematical description of PS is as follows:

Definition of Pareto front: the mapping of PS in the objective space is called Pareto front (PF). The mathematical description of PF is as follows:

The goal of the many-objective optimization technology is to find a set of non-dominated solutions that are close to the real PF (convergence) and make them well distributed on the real PF (diversity).

Related work

In recent years, many scholars have conducted in-depth research and exploration in the many-objective optimization technology.

Many-objective optimization algorithm based on dominance

Considering the limitations of Pareto dominance relationship in high-dimensional objective space, Zhou et al 14 proposed a many-objective optimization algorithm based on dominance relation selection. Firstly, they introduced an angle domination relationship with higher selection pressure based on the traditional Pareto domination relationship, and designed a new dominance selection strategy. Additionally, they proposed an angle-based individual distribution method to ensure even population distribution in the objective space. The algorithm shows strong competitiveness in solving MaOPs. Wang et al 15 believed that as the number of objectives increased, the traditional dominance relationship would become invalid. Therefore, they proposed a modified dominance relation. That is, they used penalty-based adaptive matrix regions to assist the traditional dominance relationship. Further, for MaOPs with irregular Pareto fronts, they introduced a population-based adaptive adjustment method to replace the predefined weight vector. On this basis, for MaOPs, they developed a many-objective optimization algorithm based on modified dominance relation and adaptive adjustment method. Zhang et al 16 believed that the current many-objective optimization algorithms focused too much on convergence, which would cause the population to converge to a certain sub-region of the real Pareto front. In order to solve this problem, they proposed a many-objective optimization algorithm based on double distance domination. In this algorithm, double distance can not only measure the convergence of the algorithm to adapt to different Pareto fronts, but also combine angle-based niche technology to emphasize the diversity of the algorithm. In addition, they also designed a special mutation operator. This operator can generate high-quality individuals in sparse areas to improve the diversity of the algorithm.

Many-objective optimization algorithm based on index

Aiming at the high complexity problem of hypervolume computation, Shang et al 17 proposed a new multi-objective evolutionary algorithm (MOEA) based on R2 index, namely the R2HCA-EMOA algorithm. The core idea of this algorithm is to use R2 index variables to approximate the contribution of hypervolume. The basic framework of the proposed algorithm is similar to that of SMS-EMOA. In order to improve the calculation efficiency of the algorithm, the utility tensor structure is introduced to calculate R2 index variables. In addition, the normalization mechanism is incorporated into the R2HCA-EMOA algorithm to improve its performance. Zhang et al 18 believed that the loss of selection pressure was the core reason for the poor performance of the algorithm. In order to solve this problem, they proposed a many-objective optimization algorithm based on fitness evaluation and hierarchical grouping. The fitness evaluation method combined the convergence measure based on the cos function and the diversity measure based on angle to create the selection pressure of convergence and diversity. In order to further strengthen the selection pressure, they proposed a hierarchical grouping strategy. Firstly, individuals are divided into different layers by front index, and then individuals in the same layer are divided into different groups by R2 index. Although some indexes can approximate the contribution of HV, However, Nan et al 19 believed that the key of performance evaluation was to find the worst solution rather than accurately approaching the HV value of each solution. In order to improve the ability to identify the worst solution, they proposed a two-stage R2 index evaluation method. In the first stage, the R2 indexes of all individuals are roughly evaluated to select some candidate solutions. In the second stage, these candidate solutions are accurately evaluated. Finally, they proposed a many-objective optimization algorithm based on the two-stage R2 index.

Many-objective optimization algorithm based on decomposition

In order to balance the convergence and diversity of the decomposition-based algorithm and reduce its dependence on the real PF direction, Wu et al 20 developed a many-objective optimization algorithm based on antagonistic decomposition method. This method utilizes the complementary characteristics of different sub-problems in a single example. Specifically, two populations are co-evolved by two sub-problems with different contours and opposite search directions. In order to avoid allocating redundant computing resources to the same area of PF, two populations are matched into one-to-one pairing according to their working areas on PF. In mating selection, each solution pair can only contribute one parent at most. In order to improve the performance of decomposition-based algorithms, Fan et al 21 proposed a differential multi-objective optimization algorithm based on decomposition. Firstly, they designed a neighborhood intimacy factor to improve the diversity of the algorithm based on the characteristics of neighborhood search. Then, they introduced a Gaussian mutation operator with dynamic step size to enhance the algorithm’s ability to escape from local optimal regions and improve convergence. Finally, they combined a difference strategy with the decomposition-based multi-objective optimization algorithm to further strengthen its evolutionary ability. Peng et al 22 believed that data dimensionality reduction could be applied to the objective space. Based on this consideration, they proposed a many-objective optimization algorithm based on projection. Firstly, they used the idea of data dimensionality reduction and spatial decomposition to divide the objective space into projection plane and free dimension. Then, a double elite strategy was used to maintain the balance between convergence and diversity of the algorithm. Finally, the algorithm based on decomposition was used as the algorithm of free dimension to solve MaOPs.

Many-objective optimization algorithm based on hybrid strategy

Aiming at convergence problem and diversity problem of the algorithm, Sun et al 23 proposed a many-objective optimization algorithm based on two independent stages. The algorithm deals with convergence and diversity problems in two independent and successive stages. Firstly, they introduced a non-dominated dynamic weight aggregation method, which is capable of identifying the Pareto optimal solutions of MaOPs. Then, they used these solutions to learn the Pareto optimal subspace in order to solve the convergence problem. Finally, the diversity problem was solved by using reference lines in the Pareto optimal subspace. Considering the advantages of the multi-objective and multi-population (MPMO) framework in solving MaOPs, Yang et al 24 proposed an algorithm based on the MPMO framework. The algorithm adopts the deviation sorting (BS) method to solve MaOPs, so as to obtain good convergence and diversity. In terms of convergence, the BS method is applied to each population in the MPMO framework, and the effect of non-dominant sorting is enhanced by the optimization objectives of the corresponding population. In terms of diversity, the maintenance method based on reference vector is used to save the diversity solutions. Aiming at the five-objective job shop scheduling problem (JSSP), Liu et al 25 proposed a new genetic algorithm based on the MPMO framework. Firstly, five populations are used to optimize five objectives, respectively. Secondly, in order to prevent each population from focusing only on its corresponding single objective, an archive sharing technology (AST) is proposed to store the elite solutions collected from five populations, so that the population can obtain the optimization information of other objectives from the archive. Thirdly, the archive updating strategy (AUS) is proposed to further improve the quality of the solutions in the archive.

Research motivation

Based on the related work, we believe that there are still the following problems in the current many-objective optimization technology:

(1) The diversity and convergence of the algorithm are difficult to balance. Most algorithms can’t coordinate the balance between them well, and they either emphasize convergence or diversity too much, which leads to poor quality of the non-dominated solution set.

(2) It is difficult to maintain the convergence of the algorithm. When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail.

(3) It is difficult to maintain the diversity of the algorithm. Especially when the real PF is complex or the latitude of the objective space is high, individuals may have the clustering effect, and the population may not be evenly distributed on the real PF.

(4) The evolution efficiency of the algorithm is low. The traditional evolution operators have strong randomness and low evolution efficiency, and aren’t suitable for dealing with MaOPs.

Therefore, solving these problems and providing a good many-objective optimization algorithm constitute the research motivation of this paper.

For problem 1, some work attempts to separate the convergence optimization and diversity optimization of the algorithm, thus designing a concurrent algorithm architecture. Concurrent algorithm architecture means that only one of convergence or diversity is considered in one iteration instead of considering both convergence and diversity simultaneously. In order to solve GMOPs, Professor Ye Tian 26 tried to design a concurrent algorithm architecture and proposed the MSEA algorithm, and the experimental results were satisfactory. Therefore, it seems to be a feasible path to solve MaOPs by using concurrent algorithm architecture. However, recent research 23 shows that in MaOPs, the concurrent algorithm architecture seems to be unstable, and the experimental results fluctuate greatly (such as MaOEA/IT algorithm). Because when the algorithm only considers the convergence of the population, it often affects the diversity of the population; Similarly, when the algorithm only considers the diversity of the population, it often affects the convergence of the population. If a coordination intermediary can be added to the concurrent algorithm architecture to alleviate the contradiction between diversity and convergence, the concurrent algorithm architecture will become stable and its superiority will be truly reflected. Based on this motivation, this paper proposes a new concurrent algorithm framework. In the new algorithm framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the population convergence; Diversity maintenance state is responsible for improving population diversity; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution.

For problem 2, some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. If we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Obviously, compared with domination relaxation technique, individual importance degree has greater advantages.

For problem 3, the traditional diversity maintenance technology isn’t suitable for high-dimensional objective space. For instance: the niche method, the density evaluation method, and the weight vector method. In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination. Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

For problem 4, traditional evolution operators aren’t suitable for dealing with MaOPs. Because traditional evolution operators have strong randomness and low evolution efficiency. For instance: the binary crossover operator, the polynomial mutation operator, and the differential evolution operator. In principal component analysis, the decomposition of the covariance matrix and correlation matrix is a very important step. By decomposing the covariance matrix or the correlation matrix, we can obtain a set of orthogonal bases. These orthogonal bases are the most important features of the original data 27 . Therefore, this paper designs a feature extraction operator based on Cholesky decomposition 28 . The feature extraction operator can be understood as a feature extractor. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Obviously, compared with traditional evolution operators, the feature extraction operator has higher evolution efficiency.

MOEA/TS algorithm

Feature extraction operator.

The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. The workflow of the feature extraction operator is shown in Fig.  1 .

The workflow of feature extraction operator.

In the first step, W high-quality solutions \({x}^{1},{x}^{2},{x}^{3},...,{x}^{W}\) are selected from the population. These W solutions will form the high-quality solution set S.

In the second step, calculate the mean \(\overline{x}\) and covariance matrix A of the high-quality solution set S:

Among them, \({x}^{i}={\left({x}_{1}^{i},{x}_{2}^{i},{x}_{3}^{i},...,{x}_{n}^{i}\right)}^{T}, i\in (1,...,W); {x}_{j}={\left({x}_{j}^{1},{x}_{j}^{2},{x}_{j}^{3},...,{x}_{j}^{W}\right)}^{T}, j\in (1,...,n)\)

In the third step, Cholesky decomposition is performed on the covariance matrix A. That is, the covariance matrix A is decomposed into the product of the lower triangular matrix and the transposition of the lower triangular matrix. Assuming that the lower triangular matrix is L, there is

Through formula \(A=L*{L}^{T}\) , we can calculate \({a}_{11}={l}_{11}^{2}\) , that is, \({l}_{11}=\sqrt{{a}_{11}}\) . Then, according to \({a}_{i1}={l}_{i1}*{l}_{11}\) , we can get \({l}_{i1}={a}_{i1}/{l}_{11}\) , so we can get the first column element of matrix L .

Assuming that we have calculated the first k-1 column elements of the matrix L. Through

In this way, we can solve the k-th column element of matrix L through the first k-1 column elements of matrix L. Then, we can solve matrix L by recursion.

In the fourth step, the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by Gaussian distribution \(N\left(\text{0,0.7}\right)\) . Then, a feature solution is generated.

Among them, \({x}^{feature}={\left({x}_{1}^{feature},...,{x}_{n}^{feature}\right)}^{T}\)

It should be noted that the standard deviation std is an important parameter of the Gaussian distribution. In this paper, the standard deviation std is set to 0.7. The parameter analysis verifies that 0.7 is a reasonable standard deviation. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

In the fifth step, assuming that the selected individual is \({x}^{i}({x}_{1}^{i},...,{x}_{n}^{i})\) . Based on binary crossover operator 29 and feature solution, the formula of generating offspring individual is as follows:

Among them, \({c({c}_{1},...,{c}_{n})}^{T}\) is the offspring individual. \({\beta }_{k}\) is dynamically determined by the feature factor \(\mu \) :

Among them, \(rand\) is used to generate a random number between 0 and 1; \(r and i(\text{0,1})\) is used to generate 0 or 1 randomly.

For the design principle of formula ( 15 ), please browse the Supplementary Information Document.

In the sixth step, the individual \(c\) is detected and repaired. When some components in individual \(c\) exceed the upper bound or lower bound, these components need to be repaired. The repair formula is as follows:

Among them, \({c}_{i}^{u}\) , \({c}_{i}^{l}\) represent the upper bound and lower bound of the i-th component of individual \(c\) , respectively. \({{c}{\prime}({c}_{1}{\prime},...,{c}_{n}{\prime})}^{T}\) represents the repaired individual.

Individual importance degree based on the Pareto front layer

When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail. Some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. Figure  2 shows 4 non-dominant individuals. Among them, individual B is the closest to Origin O, individual C is second, individual A is third, and individual D is the farthest from Origin O. This means that in the population, individual B is the most important, individual C is the second most important, individual A is the third most important, and individual D is the least important. In addition, we can also find from Fig.  2 that there is a significant difference between the objective values of individual B and the objective values of other individuals, that is, \(\sum_{X\in \left\{A,C,D\right\}}\sum_{i=1}^{2}{f}_{i}(B)-{f}_{i}(X)\) is the smallest. This shows that there is a special relationship between the importance of individuals and the difference of the objective values. Based on this discovery, if we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, we propose the concept of “individual importance degree”.

Schematic diagram of individual importance.

Assuming that there are n solutions in a certain Pareto front layer, the objective function values of these solutions are normalized to [0,1] based on the maximum and minimum values of each objective function. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Define the Pareto dominance function

The trend of the Pareto dominance function is shown in Fig.  3 .

The trend of the Pareto dominance function.

Pareto dominance function can be used to reflect the dominance degree among different individuals. For example, \(PDF({f}_{k}{\prime}({x}^{i})-{f}_{k}{\prime}({x}^{j}))\) represents the dominance degree of individual \({x}^{i}\) to individual \({x}^{j}\) on the k-th objective function; \(PDF({f}_{k}{\prime}({x}^{j})-{f}_{k}{\prime}({x}^{i}))\) represents the dominance degree of individual \({x}^{j}\) to individual \({x}^{i}\) on the k-th objective function; Obviously, the greater the dominance degree, the better one individual is than another on one objective function. Therefore, on one objective function, the dominance degree of one individual to another can be expressed as:

On this basis, the dominance degree of one individual to another can be expressed as:

Further, the importance degree of one individual to another can be expressed as:

Importance degree can indicate the importance of one individual to another. The greater the importance degree, the more important one individual is than another.

Since a certain Pareto front layer has n solutions, each solution needs to be compared with other n-1 solutions, so as to construct n-1 competing pairs. Assuming that an individual is \({x}^{i}\) , then the n-1 competing pairs are \(\left({x}^{i},{x}^{1}\right),\left({x}^{i},{x}^{2}\right),...,\left({x}^{i},{x}^{j}\right),...,\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ). Thus, the importance degree of individual \({x}^{i}\) to the other n-1 individuals is \(Imp\left({x}^{i},{x}^{1}\right),Imp\left({x}^{i},{x}^{2}\right),...,Imp\left({x}^{i},{x}^{j}\right),...,Imp\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ).

Finally, the importance degree of the individual \({x}^{i}\) can be expressed as:

The importance degree of one individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. The greater the importance degree of one individual, the more important it is in the same Pareto front layer.

Figure 4 shows the use of individual importance degree. Firstly, based on a certain Pareto front layer, the competition pools and competition pairs are constructed. Then, the individual importance degree of different individuals is calculated by Formula ( 24 ). Finally, the importance of different individuals in the same Pareto front layer is obtained.

The use of individual importance degree.

Taking the two-objective optimization problem as an example, it is assumed that there are 4 non-dominant individuals. They are \(A\left(\text{17,5}\right)\) , \(B\left(\text{9,7}\right)\) , \(C\left(\text{7,15}\right)\) and \(D\left(\text{5,25}\right)\) , respectively. It means that these 4 individuals belong to the first non-dominant layer, and their advantages and disadvantages can’t be compared by the non-dominant rank. The distribution of 4 individuals in the objective space is shown in Fig.  5 (a).

The distribution of 4 individuals.

In order to better compare the advantages and disadvantages of these 4 individuals, we use the individual importance degree to deal with these 4 individuals. Firstly, the objective function values of these 4 individuals are normalized to [0,1]. After normalization, the coordinates of these 4 individuals are \(A\left(\text{1,0}\right)\) , \(B\left(\text{0.333,0.1}\right)\) , \(C\left(\text{0.167,0.5}\right)\) and \(D\left(\text{0,1}\right)\) , respectively. The distribution of 4 individuals in the normalized objective space is shown in Fig.  5 (b). Next, according to Fig.  4 , the competition pools and competition pairs are constructed. Then, according to formula ( 22 ) and formula ( 23 ), the \(P\left({x}^{i},{x}^{j}\right)\) and \(Imp\left({x}^{i},{x}^{j}\right)\) of each competition pair are calculated. The calculation results are shown in Table 1 . Finally, according to the formula ( 24 ), the importance degree of these 4 individuals is 0.1918, 0.9488, 0.6673 and 0.1921, respectively. The results show that individual B is the most important, individual C is the second most important, individual D is the third most important, and individual A is the least important. This result is consistent with the intuitive perception that we get from Fig.  5 (b). Based on the above example, we believe that the concept of individual importance degree and related process are effective and can achieve the desired goals.

Repulsion field method

In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination (As shown in Fig.  6 ). Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

The uniform distribution of microscopic particles.

Firstly, according to the maximum and minimum values of each objective function, the objective function values of all solutions in the population are normalized to [0,1]. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Then, a repulsion potential field with repulsion radius r is constructed around each individual. Assuming that a repulsion potential field has been constructed for individual \({x}^{i}\) , then all individuals within the repulsion potential field will be subject to the repulsion potential from individual \({x}^{i}\) . The magnitude of the repulsion potential depends on the distance between other individuals and individual \({x}^{i}\) . When other individuals are outside the repulsion potential field of individual \({x}^{i}\) , the repulsion potential is 0. When other individuals are within the repulsion potential field of individual \({x}^{i}\) , the closer the other individuals are to individual \({x}^{i}\) , the greater the repulsion potential that they obtain. Assuming that there is individual \({x}^{j}\) , then the repulsion potential that individual \({x}^{j}\) obtains is

Among them, \(\rho \) is the gain coefficient of the repulsion potential field, usually set to 1; \(r\) is the radius of the repulsion potential field; \(dis\left({x}^{j},{x}^{i}\right)\) represents the euclidean distance between individual \({x}^{j}\) and individual \({x}^{i}\) in the objective space. The formula is as follows:

Further, the repulsion \(Rep\left({x}^{j},{x}^{i}\right)\) that individual \({x}^{j}\) obtains is the negative gradient of the repulsion potential \(Repfield\left({x}^{j},{x}^{i}\right)\) . The formula is as follows:

It means that when \(dis\left({x}^{j},{x}^{i}\right)\le r\) , the smaller \(dis\left({x}^{j},{x}^{i}\right)\) is, the larger \(Rep\left({x}^{j},{x}^{i}\right)\) is. when \(dis\left({x}^{j},{x}^{i}\right)>r\) , \(Rep\left({x}^{j},{x}^{i}\right)=0\) .

Based on the repulsion potential field, the total repulsion potential that individual \({x}^{j}\) obtains is

Finally, the total repulsion that individual \({x}^{j}\) obtains is

It should be noted that the repulsion potential and repulsion proposed in this paper are both vectors. It means that repulsion potential and repulsion have both magnitude and direction. The addition of different repulsion is the vector synthesis of repulsion, rather than the pure numerical addition. This is also an obvious feature that the repulsion field method is different from other scalar function methods (such as niche method). Figure 7 shows the vector synthesis process of repulsion in a two-dimensional space environment. Among them, F SUM is the total repulsion that individual A obtains; F BA is the repulsion generated by individual B to individual A; F CA is the repulsion generated by individual C to individual A.

The vector synthesis process of repulsion.

In the repulsion field method, the individual with large repulsion usually means that the individual is located in the multiple repulsion potential field that other individuals construct. It indicates that the individual is located in a dense area in the objective space and is close to other individuals. Therefore, individuals with large repulsion aren’t conducive to maintaining population diversity. Naturally, we hope that individuals with large repulsion can move away from dense areas in the objective space along the direction of repulsion. Based on this idea, firstly, we need to find some individuals closest to the direction of the repulsion to construct a high-quality solution set. Then, the feature extraction operator is used to extract the location features of the high-quality solution set. Finally, based on these features, individuals with large repulsion can evolve along the direction of repulsion. As shown in Fig.  8 , individual D and individual E are the individuals closest to the direction of repulsion. The feature extraction operator is used to extract the position features of these two individuals. Based on these features, individual A evolves into individual A*, which is far away from the previous dense area.

Individual A is far away from the dense area.

It should be noted that the feature extraction operator has the randomness caused by Gaussian sampling. Therefore, the evolution of individuals also has a certain degree of randomness.

Framework of MOEA/TS algorithm

The framework of the MOEA/TS algorithm is shown in Fig.  9 . Firstly, the relevant parameters of the algorithm are initialized; secondly, judge which state the algorithm is in. If the algorithm is in the convergence maintenance state, the following steps are adopted to improve the convergence of the algorithm: (1) Randomly select the parent individual. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the diversity maintenance state, the following steps are adopted to improve the diversity of the algorithm: (1) Select the individual with the worst diversity in the population. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the coordination state, the following steps are adopted to coordinate the convergence and diversity of the algorithm: (1) Randomly select the parent individual. (2) Use the Gaussian mutation operator to generate offspring individuals. (3) If the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual. Then, it is judged whether the algorithm has completed the i-th iteration. If the algorithm doesn’t complete the i-th iteration, the corresponding maintenance step or coordination step is re-executed. If the algorithm completes the i-th iteration, the current state of the algorithm is updated. Finally, it is judged whether the algorithm ends. If the algorithm doesn’t end, the corresponding maintenance step or coordination step is performed according to the current state of the algorithm. If the algorithm is finished, the population is output.

The framework of MOEA/TS algorithm.

Description of MOEA/TS algorithm

Main framework.

This section describes the main framework of the MOEA/TS algorithm. The pseudo-code of the main framework of the MOEA/TS algorithm is shown in Algorithm 1. The main steps include: in line (1), initializing population P, repulsion field radius r, and state value (state=1 means that the algorithm is in convergence maintenance state, state=2 means that the algorithm is in diversity maintenance state, and state=3 means that the algorithm is in coordination state.); In line (2), the Front value, Imp value and Rep value of each solution in population P are calculated (The Front value is calculated by the fast non-dominated sorting method.); In line (3), it is judged whether the algorithm meets the termination condition (The termination condition is usually the maximum iterations.); In line (4), the count value is initialized. The count value is used to count the number of updated solutions in the i-th iteration; in lines (5)-(11), according to the current state of the algorithm, the update way of the population is selected. When state=1, the convergence of the population is updated. When state=2, the diversity of the population is updated. When state=3, the convergence and diversity of the population are coordinated; in line (12), the state value of the algorithm is updated according to the current state of the algorithm and the count value.

Main framework.

Convergence maintenance

This section mainly describes the convergence maintenance of the population. The pseudo-code of convergence maintenance is shown in Algorithm 2. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In lines (3)–(4), based on Front, Imp, the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the parent individual; In line (6), the individual with the worst convergence in the population is found; In lines (7)–(13), if the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst convergence in the population, the flag is marked as 0. Among them, the flag value is used to indicate whether the population P has changed (flag=0 means that the population P hasn’t changed, flag=1 means that the population P has changed.); In lines (14)–(16), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Convergence maintenance.

Diversity maintenance

This section mainly describes the diversity maintenance of the population. The pseudo-code of diversity maintenance is shown in Algorithm 3. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), the individual with the worst diversity in the population is found; In line (3), according to the direction of total repulsion that the worst individual obtains, the distance \({dis}_{j}\) can be calculated; In line (4), based on \({dis}_{j}\) , the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the worst individual; In lines (6)–(12), if the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst diversity in the population, the flag is marked as 0; In lines (13)–(15), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Diversity maintenance.

Coordination of convergence and diversity

This section mainly describes the coordination of convergence and diversity of the population. The pseudo-code of coordination of convergence and diversity is shown in Algorithm 4. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In line (3), based on the parent individual, the Gaussian mutation operator is used to generate the offspring solution; In lines (4)–(10), if the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the parent individual in convergence and diversity, the flag is marked as 0; In lines (11)–(13), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Coordination.

This section mainly describes the feature extraction operator. The pseudo-code of the feature extraction operator is shown in Algorithm 5. The main steps include: in line (1), the features \(\overline{x},L\) of the high-quality solution set S are extracted by formula ( 4 ) and formula ( 13 ); In line (2), the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by the Gaussian distribution \(N\left(\text{0,0.7}\right)\) ; In line (3), based on \(\text{s},\overline{x},L\) , the feature solution \({x}^{feature}\) is generated by formula ( 14 ); In line (4), based on parent, \({x}^{feature}\) , the offspring solution O is generated by formula ( 15 ).

Feature extraction.

Update of algorithm state

In this paper, the algorithm state is further updated according to the current state of the algorithm and the stability of the population. The pseudo-code of the update of the algorithm state is shown in Algorithm 6. When the algorithm is in the convergence maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of convergence, then the algorithm turns to the diversity maintenance state; When the algorithm is in the diversity maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of diversity, then the algorithm turns to the coordination state; When the algorithm is in the coordination state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of coordination, then the algorithm turns to the convergence maintenance state. It should be noted that the threshold value T is a key parameter in measuring whether the population tends to be stable or not. In this paper, the threshold value T is set to 5%. The parameter analysis verifies that 5% is a reasonable threshold. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

Determination state.

Computational complexity of one iteration of MOEA/TS algorithm

Assuming that the size of the population is N, the number of the objective function is m, the dimension of the decision variable is n, and the size of the high-quality solution set is W, then the computational complexity of Rep is O(mN2), the computational complexity of Front is O(mN2), and the computational complexity of Imp is O(mN2). The core steps of the feature extraction operator (Algorithm 5) include the construction of the covariance matrix and Cholesky decomposition. The computational complexity of covariance matrix construction is O(Wn2) and the computational complexity of Cholesky decomposition is O(n3/6). Therefore, the computational complexity of the feature extraction operator (Algorithm 5) is O(Wn2+n3/6). The core steps of convergence maintenance (Algorithm 2) include population ranking, feature extraction operator, selection of the worst individual, and updating of Front, Imp and Rep. Their computational complexity is O(N2), O(Wn2+n3/6), O(N), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of convergence maintenance (Algorithm 2) is O(N(N2+Wn2+n3/6+N+3mN2)). The core steps of diversity maintenance (Algorithm 3) include selection of the worst individual, distance calculation, population ranking, feature extraction operator, and updating of Front, Imp and Rep. Their computational complexity is O(N), O(nN), O(N2), O(Wn2+n3/6), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of diversity maintenance (Algorithm 3) is O(N(N+nN+N2+Wn2+n3/6+3mN2)). The core steps of coordination of convergence and diversity (Algorithm 4) include the Gaussian mutation operator, and updating of Front, Imp and Rep. Their computational complexity is O(n), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of coordination of convergence and diversity (Algorithm 4) is O(N(n+3mN2)). The computational complexity of Determination State (Algorithm 6) is O(1). Based on the above computational complexity analysis, the computational complexity of one iteration of the MOEA/TS algorithm is max{O(N(N2+Wn2+n3/6+N+3mN2)), O(N(N+nN+N2+Wn2+n3/6+3mN2)), O(N(n+3mN2))}+O(1)≈max{O(NWn2+Nn3+mN3), O(NWn2+Nn3+mN3), O(mN3)}= O(NWn2+Nn3+mN3). In this paper, N>>max{W, n, m}. Therefore, the computational complexity of the MOEA/TS algorithm is O(mN3). As a reference algorithm, the computational complexity of the NSGA-III algorithm is O(mN2). The computational complexity of the MOEA/TS algorithm is an order of magnitude higher than that of the NSGA-III algorithm. This shows that the MOEA/TS algorithm is an expensive many-objective optimization algorithm.

It should be noted that although MOEA/TS algorithm has a higher computational complexity. But compared with the NSGA-III algorithm, the MOEA/TS algorithm also has greater advantages. In terms of convergence optimization, the NSGA-III algorithm adopts the traditional definition of Pareto domination. Obviously, the traditional definition can’t solve the problem of Pareto resistance. MOEA/TS algorithm uses the concept of “individual importance degree”. Individual importance degree can solve the problem of Pareto resistance. In terms of diversity optimization, the NSGA-III algorithm uses predefined reference points. The predefined reference points can’t solve the problem that the population can't be evenly distributed on the real PF in the high-dimensional objective space. MOEA/TS algorithm uses the repulsion field method. The repulsion field method can solve the problem that the population can’t be evenly distributed on the real PF in the high-dimensional objective space. In terms of algorithm architecture, the NSGA-III algorithm adopts the serial algorithm architecture. The serial algorithm architecture is difficult to coordinate the convergence optimization and diversity optimization of the algorithm. MOEA/TS algorithm adopts the concurrent algorithm architecture. The concurrent algorithm architecture can coordinate the convergence optimization and diversity optimization of the algorithm. In terms of operators, the NSGA-III algorithm uses the traditional binary crossover operator and polynomial mutation operator. The evolutionary ability of these two operators is weak. MOEA/TS algorithm uses feature extraction operator. Feature extraction operator has strong evolutionary ability. Therefore, the MOEA/TS algorithm has better performance than the NSGA-III algorithm. The comparison results support our conclusion. For the comparison results of these two algorithms, please browse Supplementary Information Document.

Experimental results and analysis

Experimental settings, configuration of experimental software and hardware.

The hardware and software configurations of the experiment are shown in Table 2 . Among them, PlatEMO 30 is a professional many-objective optimization experiment platform. The platform includes multiple test function sets and many-objective optimization algorithms.

Test function

The test functions used in the experiment include: DTLZ test function set (DTLZ1-7), MAF test function set (MAF1-6) and WFG test function set (WFG1-9). Literature 31 describes the characteristics of related test functions. The parameter settings of the related test functions are shown in Table 3 .

Comparison algorithm

In order to verify the performance of MOEA/TS algorithm in the many-objective optimization field, this paper compares MOEA/TS algorithm with 7 advanced many-objective optimization algorithms. These 7 many-objective optimization algorithms include: VMEF 32 , BiGE-BEW 33 , MOEA/DG 34 , MOEA/D 35 , LSMaODE 36 , MaOEA/IT 23 and MaOEA/IGD 37 .

For all test cases, Wilcoxon rank sum test at 5% significance level 38 is used to compare the significance of the difference between the MOEA/TS algorithm and the comparison algorithms. The symbol “+” indicates that the comparison algorithms are significantly better than the MOEA/TS algorithm; the symbol “-“indicates that the comparison algorithms are significantly inferior to the MOEA/TS algorithm. The symbol “=” indicates that there is no significant difference between the MOEA/TS algorithm and the comparison algorithms.

Performance evaluation

In the aspect of performance evaluation, this paper uses inverted generational distance plus (IGD+) and hypervolume (HV) 39 to measure the performance of many-objective optimization algorithm. The smaller the IGD+ value that the algorithm obtains, the better the performance of the algorithm. The larger the HV value that the algorithm obtains, the better the performance of the algorithm.

In order to facilitate observation, we provide the normalized HV value of each algorithm relative to the best HV result. This normalization makes all the results lie in the range [0,1], and 1 represents the best value.

Considering the length of the paper, we only show the IGD+ values of different algorithms in the experiment chapter. For the HV values of different algorithms, please browse the Supplementary Information Document.

Parameter setting

In terms of algorithm parameters, according to some existing parameter research results 13 , 40 , the feature factor \(\mu \) is set to 20 in this paper. According to the parameter sensitivity analysis, the number of high-quality solutions W is set to 9 in this paper. The parameter sensitivity analysis of W is detailed in the subsequent chapters.

The algorithm parameters of the 7 comparison algorithms are determined according to the best parameters provided by the corresponding literature.

Performance comparison under benchmark test functions

Performance comparison under dtlz test function set.

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 4 . As can be seen from Table 4 , MOEA/TS algorithm wins the first place in 15 test cases; BiGE-BEW algorithm wins the first place in 5 test cases; MOEA/D algorithm wins the first place in 15 test cases. In the 35 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 21, 27, 25, 16, 32, 35 and 31, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 5, 5, 15, 1, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 8, 3, 5, 4, 2, 0 and 4, respectively. Therefore, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm have the best performance. The performance of VMEF algorithm, MOEA/DG algorithm, BiGE-BEW algorithm and LSMaODE algorithm decreases in turn. The performance of MaOEA/IGD algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 4 , we further analyze the performance of these algorithms. In the DTLZ test function set, MOEA/TS algorithm performs poorly on DTLZ1, DTLZ5 and DTLZ6 test functions. The possible reasons are that the DTLZ1 test function has multiple local optima, and the DTLZ5 and DTLZ6 test functions have a narrow convergence curve. In the DTLZ1 test function, although the repulsion field method of the MOEA/TS algorithm makes the population widely distributed. However, its population distribution isn’t uniform and regular. The population distribution of some algorithms using predefined weight vectors is uniform and regular. In the DTLZ5 and DTLZ6 test functions, the coordination mechanism of MOEA/TS algorithm fails. The narrow convergence curve makes the population more concentrated, but the repulsion field method will disperse the population. The coordination mechanism is difficult to play a role.

The real Pareto front of DTLZ test function set is regular and the function complexity isn’t high. Therefore, algorithms with better diversity may be more popular. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. Therefore, it has good performance. VMEF algorithm uses different convergence ranking methods to deal with different test problems. Therefore, VMEF algorithm is good in convergence and poor in diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. In theory, the algorithm should perform well. However, there are defects in its convergence and diversity measurement formula. Finally, the experimental results of the algorithm aren’t as good as the expected results. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence and good in diversity. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because the real Pareto front of DTLZ test function set isn’t complex, the advantage of this multi-population algorithm architecture isn’t obvious. Therefore, compared with other algorithms, its performance is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm's performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under MAF test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 5 . As can be seen from Table 5 , MOEA/TS algorithm wins the first place in 10 test cases; BiGE-BEW algorithm wins the first place in 8 test cases; MOEA/DG algorithm wins the first place in 2 test cases; MOEA/D algorithm wins the first place in 5 test cases; LSMaODE algorithm wins the first place in 5 test cases. In the 30 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 18, 25, 21, 20, 27 and 30, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 11, 2, 5, 9, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 2, 1, 3, 4, 1, 2 and 0, respectively. Therefore, in the MAF test function set, MOEA/TS algorithm has the best performance. The performance of BiGE-BEW algorithm, LSMaODE algorithm, VMEF algorithm, MOEA/D algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. The performance of MaOEA/IGD algorithm is the worst.

Based on Table 5 , we further analyze the performance of these algorithms. In the MAF test function set, MOEA/TS algorithm performs poorly on MAF2 and MAF3 test functions. The possible reasons are that the MAF2 test function greatly increases the difficulty of convergence on the basis of the DTLZ2 test function, and the MAF3 test function has a convex Pareto front and many local fronts. In the MAF2 test function, although the MOEA/TS algorithm can recognize the advantage and disadvantage of different individuals in the same front layer, the evolutionary efficiency of the MOEA/TS algorithm isn’t ideal. In other words, after the algorithm is finished, the population still has the large evolution potential in convergence. In the MAF3 test function, MOEA/TS algorithm can effectively deal with the convex Pareto front. However, MOEA/TS algorithm is difficult to deal with multiple local fronts because feature extraction operator of MOEA/TS algorithm is difficult to extract features of multiple local fronts.

MAF test function set is the variety of DTLZ test function set. It adds a lot of characteristics to the DTLZ test function set. For example, degenerate, convex, concave, partial, multimodal, deceptive, et al. Therefore, the MAF test function set is more difficult in terms of convergence and diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. VMEF algorithm uses different convergence ranking methods to deal with different test problems. However, the complex Pareto fronts and diversified characteristics still pose a great challenge to VMEF algorithm. Therefore, the performance of VMEF algorithm is mediocre. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. MOEA/D algorithm can easily deal with the DTLZ test function set. However, its performance isn’t ideal when dealing with more complex MAF test function set. Surprisingly, LSMaODE algorithm shows good performance. We speculate that the possible reason is that the real Pareto front of the MAF test function set is complex, and then the advantages of multi-population algorithm architecture can be reflected. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under WFG test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 6 . As can be seen from Table 6 , MOEA/TS algorithm wins the first place in 27 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 6 test cases; MOEA/DG algorithm wins the first place in 1 test case; LSMaODE algorithm wins the first place in 3 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 26, 29, 42, 45, 39, 45 and 43, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 10, 9, 3, 0, 3, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 9, 7, 0, 0, 3, 0 and 2, respectively. Therefore, in the WFG test function set, MOEA/TS algorithm has the best performance. The performance of VMEF algorithm, BiGE-BEW algorithm, LSMaODE algorithm and MOEA/DG algorithm decreases in turn. The performance of MaOEA/IGD algorithm, MOEA/D algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 6 , we further analyze the performance of these algorithms. MOEA/TS algorithm performs well in all WFG test functions. The possible reason is that the problem characteristics of the WFG test function set are bias, fraud and degradation. The WFG test function set is more difficult than the DTLZ test function set. However, the problem characteristics of the WFG test function set don’t include multiple local fronts (From the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with multiple local fronts.). MOEA/TS algorithm can deal with these problem characteristics. Therefore, MOEA/TS algorithm performs well in all WFG test functions. It should be noted that the WFG3 test function has a narrow convergence curve, but the performance of MOEA/TS algorithm is still the best. This is an interesting phenomenon. Because from the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with test functions with narrow convergence curves (such as DTLZ5 and DTLZ6 test functions). Based on the convergence difficulty of the WFG test function set, we speculate that the performance of the other 7 algorithms is worse, thus highlighting the performance of MOEA/TS algorithm.

Compared with the DTLZ test function set, the MAF test function set is more difficult in terms of convergence and diversity. VMEF algorithm uses different convergence ranking methods to deal with different test problems. This approach helps VMEF algorithm to deal with different problem characteristics. Therefore, the performance of VMEF algorithm is good. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. This approach isn’t suitable for dealing with test functions with bias characteristic. Therefore, the performance of MOEA/D algorithm is the worst. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because most WFG test functions have bias characteristic, LSMaODE algorithm doesn’t consider the bias problem. Therefore, the performance of LSMaODE algorithm is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Comparison and analysis

By synthesizing Tables 4 , 5 , 6 , we can obtain the data shown in Table 7 . As can be seen from Tables 4 , 5 , 6 , MOEA/TS algorithm wins the first place in 52 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 19 test cases; MOEA/DG algorithm wins the first place in 3 test cases; MOEA/D algorithm wins the first place in 20 test cases; LSMaODE algorithm wins first place in 8 test cases. As can be seen from Table 7 , in the 110 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 69, 74, 92, 82, 91, 107 and 104, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 25, 10, 20, 13, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 19, 11, 8, 8, 6, 2 and 6, respectively. Based on the above data, we can get the following conclusions: MOEA/TS algorithm has the best performance; the performance of BiGE-BEW algorithm, VMEF algorithm, MOEA/D algorithm, LSMaODE algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. MaOEA/IGD algorithm has the worst performance.

In addition to the above conclusions, we can also observe 3 interesting phenomena:

(1) In the MAF test function set and WFG test function set, MOEA/TS algorithm has no competitors. However, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm are competitors, and they have similar performance. This is because most DTLZ test functions have regular PF, while most MAF test functions and WFG test functions have more complex PF. It can be seen from Sect. " Introduction " that MOEA/D algorithm is suitable for MaOPs with regular PF. Therefore, in the DTLZ test function set, MOEA/D algorithm can compete with MOEA/TS algorithm. In the MAF test functions and WFG test functions, only MOEA/TS algorithm shows excellent performance.

(2) The performance of MOEA/TS algorithm is better on the test cases with 10 objectives, 15 objectives and 20 objectives. The performance of MOEA/TS algorithm is relatively ordinary on the test cases with 5 objectives and 8 objectives. This is because when the number of optimization objectives is small, most many-objective optimization algorithms perform well. Compared with other many-objective optimization algorithms, the advantages of MOEA/TS algorithm aren’t obvious. However, with the increase of the number of optimization objectives, the performance of other many-objective optimization algorithms becomes worse and worse. In contrast, the performance of MOEA/TS algorithm isn’t significantly affected. Therefore, compared with other many-objective optimization algorithms, MOEA/TS algorithm has obvious advantages. This shows that MOEA/TS algorithm is more suitable for solving MaOPs with more than 10 objectives.

(3) Without considering MOEA/TS algorithm, MOEA/D algorithm has the best performance in the DTLZ test function set. BiGE-BEW algorithm has the best performance in the MAF test function set. VMEF algorithm has the best performance in the WFG test function set. This shows that different many-objective optimization algorithms are suitable for different test function sets. However, MOEA/TS algorithm can show excellent performance on three test function sets. This indicates that MOEA/TS algorithm has strong universality and applicability.

Distribution diagram of solutions in the objective space

In order to describe the distribution of solutions in the high-dimensional objective space more intuitively, this paper draws the distribution diagram of solutions in the objective space. Considering the length of the paper, it is unrealistic to show the distribution diagrams of all test functions. Therefore, this section only shows the distribution diagrams of 3 representative test cases. These 3 test cases are DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives, respectively.

Figure 10 shows the distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives. It can be seen from Fig.  10 that distribution diagrams of MOEA/TS algorithm, BiGE-BEW algorithm, MOEA/DG algorithm and MOEA/D algorithm are similar, which indicates that these 4 algorithms are excellent in convergence and diversity; VMEF algorithm and LSMaODE algorithm are good in diversity, but poor in convergence; MaOEA/IT algorithm and MaOEA/IGD algorithm are very poor in convergence and diversity.

Distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives.

Figure 11 shows the distribution diagrams of each algorithm on MAF1 test case with 15 objectives. It can be seen from Fig.  11 that MOEA/TS algorithm and VMEF algorithm are good in convergence, but poor in diversity; BiGE-BEW algorithm and LSMaODE algorithm are good in diversity, but poor in convergence. MOEA/DG algorithm, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on MAF1 test case with 15 objectives.

Figure 12 shows the distribution diagrams of each algorithm on WFG3 test case with 10 objectives. It can be seen from Fig.  12 that MOEA/TS algorithm has the best convergence and diversity; LSMaODE algorithm is also excellent, only slightly worse than MOEA/TS algorithm in terms of diversity; BiGE-BEW algorithm and MOEA/DG algorithm are good in diversity, but poor in convergence. VMEF algorithm is good in convergence, but poor in diversity. MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on WFG3 test case with 10 objectives.

Evolution curve analysis of the algorithm

This section takes DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives as examples to display the evolution curves of 8 algorithms (as shown in Figs.  13 , 14 , 15 ).

Evolution curve of each algorithm on DTLZ2 test case with 20 objectives.

In Figure 13 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of MOEA/DG algorithm, BiGE-BEW algorithm, MOEA/D algorithm, LSMaODE algorithm, VMEF algorithm and MaOEA/IGD algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on DTLZ2 test case with 20 objectives, and is suitable for solving DTLZ2 test problem with 20 objectives.

In Figure 14 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of BiGE-BEW algorithm, VMEF algorithm, LSMaODE algorithm, MaOEA/IGD algorithm, MOEA/D algorithm and MOEA/DG algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on MAF1 test case with 15 objectives, and is suitable for solving MAF1 test problem with 15 objectives.

Evolution curve of each algorithm on MAF1 test case with 15 objectives.

In Fig.  15 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of LSMaODE algorithm, MOEA/DG algorithm, VMEF algorithm, BiGE-BEW algorithm, MaOEA/IGD algorithm and MOEA/D algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of the MaOEA/IT algorithm, VMEF algorithm, MaOEA/IGD algorithm, BiGE-BEW algorithm and VMEF algorithm are all greater than the initial IGD+ values. This shows that the performance of these 5 algorithms deteriorates during evolution, and they aren’t suitable for dealing with WFG3 test problem with 10 objectives. The initial IGD+ value of MOEA/DG algorithm is close to the final IGD+ value, and the IGD+ value of MOEA/DG algorithm fluctuates little during the evolution. This shows that MOEA/DG algorithm is insensitive to evolution. Only the final IGD+ values of LSMaODE algorithm and MOEA/TS algorithm are less than the initial IGD+ values. This shows that LSMaODE algorithm and MOEA/TS algorithm have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm have greater fluctuation. This shows that these 3 algorithms aren’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on WFG3 test case with 10 objectives, and is suitable for solving WFG3 test problem with 10 objectives.

Evolution curve of each algorithm on WFG3 test case with 10 objectives.

In addition, we can also observe an interesting phenomenon from Fig.  13 to Fig.  15 : the IGD+ values of some algorithms sometimes increase significantly with the increase of iterations. That is, the performance of some algorithms sometimes deteriorates seriously with the increase of iterations. The reasons for this phenomenon may include three aspects: (1) The algorithm doesn’t adopt the elite preservation strategy. Some high-quality solutions may gradually disappear; (2) Due to the complexity of the optimization problems, the evolutionary direction of the population may be misled by some pseudo-elite individuals; (3) The convergence optimization and diversity optimization of the algorithm aren’t coordinated. The optimization of convergence may affect the optimization of diversity or the optimization of diversity may affect the optimization of convergence. It can be seen from the pseudo-code of the algorithm in Section 3.5 that the MOEA/TS algorithm proposed in this paper considers the above three aspects. Therefore, MOEA/TS algorithm can effectively alleviate this phenomenon.

Effectiveness verification of innovation part

In order to verify the effectiveness of the innovative parts, 4 variants are designed in this section. As follows:

MOEA/TS-1 algorithm: The feature extraction operator in MOEA/TS algorithm is changed to the binary crossover operator and polynomial mutation operator;

MOEA/TS-2 algorithm: The repulsion field method in MOEA/TS algorithm is removed;

MOEA/TS-3 algorithm: The concurrent architecture in MOEA/TS algorithm is changed to serial architecture;

MOEA/TS-4 algorithm: The individual importance degree in MOEA/TS algorithm is removed.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 5 algorithms. In this paper, 5 algorithms are executed 30 times to get the average data as shown in Table 8 . As can be seen from Table 8 , MOEA/TS algorithm wins the first place in 24 test cases; MOEA/TS-1 algorithm wins the first place in 13 test cases; MOEA/TS-2 algorithm wins the first place in 7 test cases; MOEA/TS-3 algorithm wins the first place in 1 test case. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 21, 30, 40 and 45, respectively. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 11, 6, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 13, 9, 5 and 0, respectively. The average ranking of MOEA/TS algorithm is about 1.64; the average ranking of MOEA/TS-1 algorithm is about 2.02; the average ranking of MOEA/TS-2 algorithm is about 2.62; the average ranking of MOEA/TS-3 algorithm is about 3.71; the average ranking of MOEA/TS-4 algorithm is 5.

Therefore, we think that the 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm. This shows that our innovations are effective. In addition, based on the above data, we can also find that “individual importance degree” has the greatest influence on the algorithm; the algorithm architecture ranks second; the repulsion field method ranks third; the feature extraction operator ranks fourth.

Ablation experiment of selection approach

In the feature extraction operator, we select W high-quality solutions. To prove the effectiveness of this selection approach over random selection, the ablation experiment will be performed in this sect. " Introduction " variant is designed in this section. As follows:

MOEA/TS-5 algorithm: W solutions are randomly selected in the feature extraction operator.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 2 algorithms. In this paper, 2 algorithms are executed 30 times to get the average data as shown in Table 9 . As can be seen from Table 9 , MOEA/TS algorithm wins the first place in 45 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-5 algorithm is 42. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-5 algorithm is 0. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-5 algorithm is 3. Therefore, we believe that the performance of MOEA/TS algorithm is better than MOEA/TS-5 algorithm in the WFG test function set. It proves that the selection approach that we use is better than random selection in the feature extraction operator.

In addition, the performance of MOEA/TS-5 algorithm isn’t as good as that of MOEA/TS-1 algorithm. It means that the performance of the feature extraction operator based on random selection is even worse than that of some classical operators. The possible reason is that the randomly selected solution set will cause the feature extraction operator to extract many bad features. These bad features hinder individual evolution, which makes the convergence maintenance state and diversity maintenance state of MOEA/TS algorithm fail for a long time, and only the coordination state can play some role. The architecture of the MOEA/TS algorithm is undermined by some bad features.

Parameter sensitivity analysis.

The algorithm parameters analyzed in this paper are mainly the number of high-quality solutions W, threshold value T, standard deviation std. Due to the high complexity of the WFG3 test case with 10 objectives, it is difficult for the population of each algorithm to cover the real Pareto front, so this paper considers the WFG3 test case with 10 objectives as the main function of parameter analysis.

The initial value and value range of each parameter are shown in Table 10 .

As shown in Fig.  16 , when \(W<9\) , the IGD + value of the algorithm decreases significantly with the increase of W . It means that when \(W<9\) , the performance of the feature extraction operator is greatly improved with the increase of W . This is because the features extracted by the feature extraction operator are closer to the ideal situation. When \(W=9\) , the IGD + value of the algorithm is minimum. This shows that when \(W=9\) , the feature extraction operator performs best. When \(W>9\) , the IGD + value of the algorithm increases slowly. It means that when \(W>9\) , the performance of the feature extraction operator deteriorates gradually with the increase of W . This is because some features are over-extracted by feature extraction operators. Therefore, for WFG3 test case with 10 objectives, \(W=9\) is the best parameter selection.

The corresponding relationship between IGD + value and W.

As shown in Fig.  17 , when \(T<5\%\) , the IGD + value of the algorithm decreases significantly with the increase of T . This is because if the threshold value T is too small, the algorithm will remain in the same state for a long time, and it is difficult to be adjusted to other states. Convergence and diversity of algorithm will also be difficult to balance. This situation will be improved with the increase of T . When \(T=5\%\) , the IGD + value of the algorithm is minimum. This shows that when \(T=5\%\) , the algorithm has the best performance. When \(T>5\%\) , the IGD + value of the algorithm increases gradually with the increase of T . This is because if the threshold value T is too large, the algorithm’s state will be adjusted frequently. Even if the population isn’t stable in one state (convergence, diversity, coordination), the algorithm will also be adjusted to other states. This isn’t conducive to improving the convergence and the diversity of the algorithm. The efficiency of the algorithm will also be affected. Therefore, for WFG3 test case with 10 objectives, \(T=5\%\) is the best parameter selection.

The corresponding relationship between IGD + value and T.

As shown in Fig.  18 , when \(std<0.7\) , the IGD + value of the algorithm decreases significantly with the increase of std . This is because if std is too small, the results of Gaussian sampling are too concentrated in the middle region, and the randomness of the sampling vector is weak, which isn’t conducive to the use of features and generation of diversified feature solutions. When \(std=0.7\) , the IGD + value of the algorithm is minimum. This shows that when \(std=0.7\) , the feature extraction operator performs best. When \(std>0.7\) , the IGD + value of the algorithm increases significantly with the increase of std . This is because if the std is too large, the result of Gaussian sampling is too scattered, the randomness of the sampling vector is strong, some components are easy to exceed the upper bound or lower bound, and some features are easy to be eliminated by the repair operator. Therefore, for WFG3 test case with 10 objectives, \(std=0.7\) is the best parameter selection.

The corresponding relationship between IGD + value and std.

Based on the above analysis of algorithm parameters, we think \(W=9, T=5\%, std=0.7\) are the best parameter combinations in WFG3 test case with 10 objectives. Further, we test the performance of the above parameter combinations in more test cases. The experimental results show that the above parameter combinations perform well in most test cases. Therefore, this paper sets the number of high-quality solutions \(W\) , the threshold value \(T\) and the standard deviation \(std\) to 9, 5% and 0.7, respectively.

Practical problem testing

This section mainly explores the performance of MOEA/TS algorithm in practical problems. The practical problem selected in this section is the industrial internet optimization problem based on the blockchain provided in reference 40 .

The industrial internet can support effective control of the physical world through a large amount of industrial data, but data security has always been a challenge due to various interconnections and accesses. Blockchain technology supports the security and privacy protection of industrial internet data with its trusted and reliable security mechanism. Fragmentation technology can help improve the overall throughput and scalability of the blockchain network. However, due to the uneven distribution of malicious nodes, the effectiveness of fragmentation is still challenging. In addition, the conflict between multiple industrial network indicators is also a problem we have to consider. Therefore, the industrial internet optimization problem based on blockchain is an important research problem.

In this section, the industrial internet optimization problem based on blockchain has the following 4 optimization objectives:

(1) Minimizing the shard invalidation probability (SIP);

(2) Minimizing the transmission delay (TD);

(3) Maximizing the throughput (TP);

(4) Minimizing the load of Malicious Nodes (LMN).

The research background of the industrial internet based on blockchain and the calculation formulas of these 4 objectives are detailed in reference 40 .

In this section, we set the population size to 220, the number of iterations to 300, and the number of function evaluations to 66000. We still use inverted generational distance plus (IGD+) to measure the performance of many-objective optimization algorithms. However, the real PF of the practical problem is unknown. Therefore, we run these algorithms many times to obtain the different non-dominated solution sets. The non-dominated union set of the different non-dominated solution sets is considered as the real PF. The relevant parameters of these algorithms are shown in Section 4.1.

In this section, each algorithm is executed 30 times to get the data as shown in Table 11 . As can be seen from Table 11 , MOEA/TS algorithm has absolute advantages. The performance of BiGE-BEW algorithm and MOEA/DG algorithm is good and similar. The performance of VMEF algorithm and MOEA/D algorithm in practical problems is obviously not as good as that in benchmark test functions. This is because the real PF of the practical problem is more complex. The performance of LSMaODE algorithm is close to that of MOEA/D algorithm. The performance of MaOEA/IT algorithm and MaOEA/IGD algorithm is the worst. Based on the above observations and analysis, we believe that MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Considering that the solutions obtained by the many-objective optimization algorithms are the population, it is unrealistic to compare different network indicators of different algorithms intuitively. However, in practical applications, we only need to make choices according to the specific needs or preferences of users or enterprises. In this section, we first select the individuals with the largest throughput in each algorithm, and then compare the MOEA/TS algorithm with other algorithms on the basis of ensuring the maximum throughput. The network indicators obtained by these 8 algorithms are shown in Table 12 . As can be seen from Table 12 , in terms of SIP and TP, MOEA/TS algorithm has the best performance; In terms of TD, MOEA/TS algorithm ranks second; In terms of LMN, MOEA/TS algorithm ranks third. Therefore, we believe that the MOEA/TS algorithm has the best comprehensive performance in the industrial internet optimization problem based on blockchain, and various network indicators are at the forefront.

Based on the experimental analysis from Section 4.2 to Section 4.8, we can obtain the following conclusions:

(1) In the benchmark test cases, MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms.

(2) MOEA/TS algorithm is more suitable for dealing with the MaOPs with more than 10 objectives.

(3) MOEA/TS algorithm can show excellent performance in different test function sets, and has strong universality and applicability.

(4) MOEA/TS algorithm has the best convergence and diversity, the strongest evolution ability and the fastest convergence speed.

(5) The 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm.

(6) MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Summary and future work

Aiming at some difficulties in the many-objective optimization field, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, in terms of convergence maintenance, this paper doesn’t consider using domination relaxation technique, because the current domination relaxation technique still faces some problems. Based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Then, in terms of diversity maintenance, this paper considers maintaining the diversity of the population in the objective space by repulsion field, so that the population can be evenly distributed on the real PF. Finally, a new concurrent algorithm framework is designed. In the framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the convergence of population; Diversity maintenance state is responsible for improving the diversity of population; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution. The experimental results show that MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms. In addition, the effectiveness of the innovation parts is further verified.

However, MOEA/TS algorithm also has obvious defects: MOEA/TS algorithm isn’t good at dealing with test problems with narrow convergence curves or multiple local fronts. Therefore, in the future, we will further improve MOEA/TS algorithm, so that MOEA/TS algorithm can deal with test problems with narrow convergence curve or multiple local fronts. In addition, constrained MOPs and high-dimensional MOPs are also the focus of our future research.

Data availability

All data generated or analysed during this study are included in this published article.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 62362026 and 62162021; in part by the specific research fund of The Innovation Platform for Academicians of Hainan Province under grant YSPTZX202314; in part by the Key Project of Hainan Province under Grant ZDYF2023GXJS158.

National Natural Science Foundation of China, 62362026, 62362026, Specific research fund of The Innovation Platform for Academicians of Hainan Province, YSPTZX202314, YSPTZX202314, Key Project of Hainan Province, ZDYF2023GXJS158, ZDYF2023GXJS158.

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Huanhuan Yu, Hansheng Fei, Xiangdang Huang & Qiuling Yang

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Huijie Zhang

Innovation Platform for Academicians of Hainan Province, Hainan University, Haikou, 570228, China

Jiale Zhao, Huanhuan Yu, Hansheng Fei, Xiangdang Huang & Qiuling Yang

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Conceptualization, J.L.Z., H.J.Z. and X.D.H.; Methodology, J.L.Z., H.J.Z. and X.D.H.; Software, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Validation, H.H.Y. and H.S.F.; Formal analysis, J.L.Z. and H.J.Z.; Investigation, H.H.Y. and H.S.F.; Resources, Q.L.Y.; Data curation, H.H.Y. and H.S.F.; Writing-original draft, J.L.Z.; Writing-review&editing, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Visualization, J.L.Z. and H.J.Z.; Supervision, H.H.Y. and H.S.F.; Project administration, J.L.Z. and Q.L.Y.; Funding acquisition, Q.L.Y.; All authors have read and agreed to the published version of the manuscript.

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Zhao, J., Zhang, H., Yu, H. et al. A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem. Sci Rep 14 , 19140 (2024). https://doi.org/10.1038/s41598-024-70145-8

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DOI : https://doi.org/10.1038/s41598-024-70145-8

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12 Aug 2024  ·  Liuqing Chen , Yaxuan Song , Shixian Ding , Lingyun Sun , Peter Childs , Haoyu Zuo · Edit social preview

TRIZ, the Theory of Inventive Problem Solving, is derived from a comprehensive analysis of patents across various domains, offering a framework and practical tools for problem-solving. Despite its potential to foster innovative solutions, the complexity and abstractness of TRIZ methodology often make its acquisition and application challenging. This often requires users to have a deep understanding of the theory, as well as substantial practical experience and knowledge across various disciplines. The advent of Large Language Models (LLMs) presents an opportunity to address these challenges by leveraging their extensive knowledge bases and reasoning capabilities for innovative solution generation within TRIZ-based problem-solving process. This study explores and evaluates the application of LLMs within the TRIZ-based problem-solving process. The construction of TRIZ case collections establishes a solid empirical foundation for our experiments and offers valuable resources to the TRIZ community. A specifically designed workflow, utilizing step-by-step reasoning and evaluation-validated prompt strategies, effectively transforms concrete problems into TRIZ problems and finally generates inventive solutions. Finally, we present a case study in mechanical engineering field that highlights the practical application of this LLM-augmented method. It showcases GPT-4's ability to generate solutions that closely resonate with original solutions and suggests more implementation mechanisms.

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