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This Is Your Brain on Nature

When we get closer to nature—be it untouched wilderness or a backyard tree—we do our overstressed brains a favor.

When you head out to the desert, David Strayer is the kind of man you want behind the wheel. He never texts or talks on the phone while driving. He doesn’t even approve of eating in the car. A cognitive psychologist at the University of Utah who specializes in attention, Strayer knows our brains are prone to mistakes, especially when we’re multitasking and dodging distractions. Among other things, his research has shown that using a cell phone impairs most drivers as much as drinking alcohol does.

Strayer is in a unique position to understand what modern life does to us. An avid backpacker, he thinks he knows the antidote: Nature.

On the third day of a camping trip in the wild canyons near Bluff, Utah, Strayer is mixing up an enormous iron kettle of chicken enchilada pie while explaining what he calls the “three-day effect” to 22 psychology students. Our brains, he says, aren’t tireless three-pound machines; they’re easily fatigued. When we slow down, stop the busywork, and take in beautiful natural surroundings, not only do we feel restored, but our mental performance improves too. Strayer has demonstrated as much with a group of Outward Bound participants, who performed 50 percent better on creative problem-solving tasks after three days of wilderness backpacking. The three-day effect, he says, is a kind of cleaning of the mental windshield that occurs when we’ve been immersed in nature long enough. On this trip he’s hoping to catch it in action, by hooking his students—and me—to a portable EEG, a device that records brain waves.

“On the third day my senses recalibrate—I smell things and hear things I didn’t before,” Strayer says. The early evening sun has saturated the red canyon walls; the group is mellow and hungry in that satisfying, campout way. Strayer, in a rumpled T-shirt and with a slight sunburn, is definitely looking relaxed. “I’m more in tune with nature,” he goes on. “If you can have the experience of being in the moment for two or three days, it seems to produce a difference in qualitative thinking.”

Strayer’s hypothesis is that being in nature allows the prefrontal cortex, the brain’s command center, to dial down and rest, like an overused muscle. If he’s right, the EEG will show less energy coming from “midline frontal theta waves”—a measure of conceptual thinking and sustained attention. He’ll compare our brain waves with those of similar volunteers who are sitting in a lab or hanging out at a parking lot in downtown Salt Lake City.

While the enchiladas are cooking, Strayer’s graduate students tuck my head into a sort of bathing cap with 12 electrodes embedded in it. They suction-cup another 6 electrodes to my face. Wires sprouting from them will send my brain’s electrical signals to a recorder for later analysis. Feeling like a beached sea urchin, I walk carefully to a grassy bank along the San Juan River for ten minutes of restful contemplation. I’m supposed to think of nothing in particular, just watch the wide, sparkling river flow gently by. I haven’t looked at a computer or cell phone in days. It’s easy to forget for a few moments that I ever had them.

In 1865 the great landscape architect Frederick Law Olmsted looked out over the Yosemite Valley and saw a place worth saving. He urged the California legislature to protect it from rampant development. Olmsted had already designed Central Park in New York City; he was convinced that beautiful green spaces should exist for all people to enjoy. “It is a scientific fact,” he wrote, “that the occasional contemplation of natural scenes of an impressive character ... is favorable to the health and vigor of men and especially to the health and vigor of their intellect.”

Olmsted was exaggerating; his claim was based less on science than on intuition. But it was an intuition with a long history. It went back at least to Cyrus the Great, who some 2,500 years ago built gardens for relaxation in the busy capital of Persia. Paracelsus, the 16th-century German-Swiss physician, gave voice to that same intuition when he wrote, “The art of healing comes from nature, not from the physician.” In 1798, sitting on the banks of the River Wye, William Wordsworth marveled at how “an eye made quiet by the power / Of harmony” offered relief from “the fever of the world.” American writers such as Ralph Waldo Emerson and John Muir inherited that outlook. Along with Olmsted, they built the spiritual and emotional case for creating the world’s first national parks by claiming that nature had healing powers.

There wasn’t much hard evidence then—but there is now. Motivated by large-scale public health problems such as obesity, depression, and pervasive nearsightedness, all clearly associated with time spent indoors, Strayer and other scientists are looking with renewed interest at how nature affects our brains and bodies. Building on advances in neuroscience and psychology, they’ve begun to quantify what once seemed divine and mysterious. These measurements—of everything from stress hormones to heart rate to brain waves to protein markers—indicate that when we spend time in green space, “there is something profound going on,” as Strayer puts it.

In England researchers from the University of Exeter Medical School recently analyzed mental health data from 10,000 city dwellers and used high-resolution mapping to track where the subjects had lived over 18 years. They found that people living near more green space reported less mental distress, even after adjusting for income, education, and employment (all of which are also correlated with health). In 2009 a team of Dutch researchers found a lower incidence of 15 diseases—including depression, anxiety, heart disease, diabetes, asthma, and migraines—in people who lived within about a half mile of green space. And in 2015 an international team overlaid health questionnaire responses from more than 31,000 Toronto residents onto a map of the city, block by block. Those living on blocks with more trees showed a boost in heart and metabolic health equivalent to what one would experience from a $20,000 gain in income. Lower mortality and fewer stress hormones circulating in the blood have also been connected to living close to green space.

It’s difficult to tell from these kinds of studies why people feel better. Is it the fresh air? Do certain colors or fractal shapes trigger neurochemicals in our visual cortex? Or is it just that people in greener neighborhoods use the parks to exercise more? That’s what Richard Mitchell, an epidemiologist at the University of Glasgow in Scotland, thought at first. “I was skeptical,” he says. But then he did a large study that found less death and disease in people who lived near parks or other green space—even if they didn’t use them. “Our own studies plus others show these restorative effects whether you’ve gone for walks or not,” Mitchell says. Moreover, the lowest income people seemed to gain the most: In the city, Mitchell found, being close to nature is a social leveler.

What he and other researchers suspect is that nature works primarily by lowering stress . Compared with people who have lousy window views, those who can see trees and grass have been shown to recover faster in hospitals, perform better in school, and even display less violent behavior in neighborhoods where it’s common. Such results jibe with experimental studies of the central nervous system. Measurements of stress hormones, respiration, heart rate, and sweating suggest that short doses of nature—or even pictures of the natural world—can calm people down and sharpen their performance.

In Sweden physician Matilda van den Bosch found that after a stressful math task, subjects’ heart rate variability—which decreases with stress—returned to normal more quickly when they sat through 15 minutes of nature scenes and birdsong in a 3-D virtual reality room than when they sat in a plain room. A real-life experiment is under way at the Snake River Correctional Institution in eastern Oregon. Officers there report calmer behavior in solitary confinement prisoners who exercise for 40 minutes several days a week in a “blue room” where nature videos are playing, compared with those who exercise in a gym without videos. “I thought it was crazy at first,” says corrections officer Michael Lea. But he has experienced the difference. “There’s a lot of yelling really loud— it echoes horribly,” in the plain gym, he says. “In the blue room they tend not to yell. They say, ‘Hold on, I got to watch my video.’”

A 15-minute walk in the woods causes measurable changes in physiology. Japanese researchers led by Yoshifumi Miyazaki at Chiba University sent 84 subjects to stroll in seven different forests, while the same number of volunteers walked around city centers. The forest walkers hit a relaxation jackpot: Overall they showed a 16 percent decrease in the stress hormone cortisol, a 2 percent drop in blood pressure, and a 4 percent drop in heart rate. Miyazaki believes our bodies relax in pleasant, natural surroundings because they evolved there. Our senses are adapted to interpret in- formation about plants and streams, he says, not traffic and high-rises.

All this evidence for the benefits of nature is pouring in at a time when disconnection from it is pervasive, says Lisa Nisbet, a psychology professor at Canada’s Trent University. We love our state and national parks, but per capita visits have been declining since the dawn of email. So have visits to the backyard. One recent Nature Conservancy poll found that only about 10 percent of American teens spend time outside every day. According to research by the Harvard School of Public Health, American adults spend less time outdoors than they do inside vehicles—less than 5 percent of their day.

“People underestimate the happiness effect” of being outdoors, Nisbet says. “We don’t think of it as a way to increase happiness. We think other things will, like shopping or TV. We evolved in nature. It’s strange we’d be so disconnected.” But some people are starting to do something about it.

Visits to parks are down. So are visits to the backyard. One survey found only 10 percent of American teens spend time outside every day.

Nooshin Razani at UCSF Benioff Children’s Hospital in Oakland, California, is one of several doctors who have noticed the emerging data on nature and health. As part of a pilot project, she’s training pediatricians in the outpatient clinic to write prescriptions for young patients and their families to visit nearby parks. It’s not as simple as taking a pill. To guide the physicians and patients into a new mind-set, she says, “we have transformed the clinical space so nature is everywhere. There are maps on the wall, so it’s easy to talk about where to go, and pictures of local wilderness, which are healing to look at for both the doctor and patient.” The hospital is partnering with the East Bay Regional Parks District to provide transportation to parks and programs there for entire families.

In some countries governments are promoting nature experiences as a public health policy. In Finland, a country that struggles with high rates of depression, alcoholism, and suicide, government-funded researchers asked thousands of people to rate their moods and stress levels after visiting both natural and urban areas. Based on that study and others, Professor Liisa Tyrväinen and her team at the Natural Resources Institute Finland recommend a minimum nature dose of five hours a month—several short visits a week—to ward off the blues. “A 40- to 50-minute walk seems to be enough for physiological changes and mood changes and probably for attention,” says Kalevi Korpela, a professor of psychology at the University of Tampere. He has helped design a half dozen “power trails” that encourage walking, mindfulness, and reflection. Signs on them say things like, “Squat down and touch a plant.”

Perhaps no one has embraced the medicalization of nature with more enthusiasm than the South Koreans. Many suffer from work stress, digital addiction, and intense academic pressures. More than 70 percent say their jobs, which require notoriously long hours, make them depressed, according to a survey by electronics giant Samsung. Yet this economically powerful nation has a long history of worshipping nature spirits. The ancient proverb “ Shin to bul ee —Body and soil are one” (not body and soul) is still popular.

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At the Saneum Healing Forest, east of Seoul, a “health ranger” offers me elm bark tea, then takes me on a hike along a small creek, through shimmering red maples, oaks, and pine-nut trees. It’s autumn, and the changing foliage and crisp air have lured scores of urban refugees to the woods. Soon we come upon a cluster of wooden platforms arranged in a clearing. Forty middle-aged firefighters who have been diagnosed with post-traumatic stress disorder are paired off on the platforms as part of a free three-day program sponsored by the local government. In North America groups of men in the woods likely would be hunting or fishing, but here, after a morning of hiking, they practice partner yoga, rub lavender massage oil into each other’s forearms, and make delicate dried flower collages. Among them is Kang Byoung-wook, a weathered 46-year-old from Seoul. Recently returned from a big fire in the Philippines, he looks exhausted. “It’s a stressed life,” he says. “I want to live here for a month.”

Saneum is one of three official healing forests in South Korea, but 34 more are planned by 2017, meaning most major towns will be near one. Chungbuk University offers a “forest healing” degree program, and job prospects for graduates are good; the Korea Forest Service expects to appoint 500 health rangers in the next couple of years. It’s a cradle-to-grave operation: Programs include everything from prenatal forest meditation to woodcrafts for cancer patients to forest burials. A government-run “happy train” takes kids who’ve been bullied into the woods for two days of camping. A hundred-million-dollar healing complex is under construction next to Sobaeksan National Park.

Korea Forest Service scientists used to study timber yields; now they also distill essential oils from trees such as the hinoki cypress and study them for their ability to reduce stress hormones and asthma symptoms. In the new industrial city of Deajun, I pay a visit to the forest minister, Shin Won Sop, a social scientist who has studied the effects of forest therapy on alcoholics. Human well-being, he tells me, is now a formal goal of the nation’s forest plan. Thanks to the new policies, visitors to Korea’s forests increased from 9.4 million in 2010 to 12.8 million in 2013.

“Of course we still use forests for timber,” Shin says. “But I think the health area is the fruit of the forest right now.” His agency has data suggesting that forest healing reduces medical costs and benefits local economies. What’s still needed, he says, is better data on specific diseases and on the natural qualities that make a difference. “What are the main factors in the forest that are most responsible for the physiological benefits, and what types of forests are more effective?” Shin asks.

My own city brain, which spends much of the year in Washington, D.C., seems to like the Utah wilderness very much. By day, on David Strayer’s camping trip, we hike among flowering prickly pear cacti; by night we sit around the campfire. Strayer’s students seem more relaxed and sociable than they do in the classroom, he says, and they give much better presentations. What’s going on inside their brains and mine?

A lot of different things, judging from the neuroscience research that’s starting to come in. Korean researchers used functional MRI to watch brain activity in people viewing different images. When the volunteers were looking at urban scenes, their brains showed more blood flow in the amygdala, which processes fear and anxiety. In contrast, the natural scenes lit up the anterior cingulate and the insula—areas associated with empathy and altruism. Maybe nature makes us nicer as well as calmer.

It may also make us nicer to ourselves. Stanford researcher Greg Bratman and his colleagues scanned the brains of 38 volunteers before and after they walked for 90 minutes, either in a large park or on a busy street in downtown Palo Alto. The nature walkers, but not the city walkers, showed decreased activity in the subgenual prefrontal cortex—a part of the brain tied to depressive rumination—and from their own reports, the nature walkers beat themselves up less. Bratman believes that being outside in a pleasant environment (not the kind where you’re getting eaten alive by gnats or pummeled by hail) takes us outside of ourselves in a good way. Nature, he says, may influence “how you allocate your attention and whether or not you focus on negative emotions.”

Strayer is most interested in how nature affects higher order problem solving. His research builds on the attention restoration theory proposed by environmental psychologists Stephen and Rachel Kaplan at the University of Michigan. They argue that it’s the visual elements in natural environments—sunsets, streams, butterflies—that reduce stress and mental fatigue. Fascinating but not too demanding, such stimuli promote a gentle, soft focus that allows our brains to wander, rest, and recover from what Olmsted called the “nervous irritation” of city life. “Soft fascination ... permits a more reflective mode,” wrote the Kaplans—and the benefit seems to carry over when we head back indoors.

A few years ago, for example, in an experiment similar to Bratman’s, Stephen Kaplan and his colleagues found that a 50-minute walk in an arboretum improved executive attention skills, such as short-term memory, while walking along a city street did not. “Imagine a therapy that had no known side effects, was readily available, and could improve your cognitive functioning at zero cost,” the researchers wrote in their paper. It exists, they continued, and it’s called “interacting with nature.”

A few months after our Utah trip, Strayer’s team sent me the results of my EEG test. The colorful graph charted the power of my brain waves at a range of frequencies and compared them with samples from the two groups that had stayed in the city. My theta signals were indeed lower than theirs; the soft fascination of the San Juan River had apparently quieted my prefrontal cortex, at least for a while.

So far, says Strayer, the results are consistent with his hypothesis. But even if the study bears it out, it won’t offer anything like a full explanation of the brain-on-nature experience. Something mysterious will always remain, Strayer says, and maybe that’s as it should be. “At the end of the day,” he says, “we come out in nature not because the science says it does something to us, but because of how it makes us feel.”

At the end of the day, we come out in nature not because the science says it does something to us, but because of how it makes us feel.

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Article • 7 min read

What Is Problem Solving?

By the Mind Tools Content Team

We all spend a lot of our time solving problems, both at work and in our personal lives.

Some problems are small, and we can quickly sort them out ourselves. But others are complex challenges that take collaboration, creativity, and a considerable amount of effort to solve.

At work, the types of problems we face depend largely on the organizations we're in and the jobs we do. A manager in a cleaning company, for example, might spend their day untangling staffing issues, resolving client complaints, and sorting out problems with equipment and supplies. An aircraft designer, on the other hand, might be grappling with a problem about aerodynamics, or trying to work out why a new safety feature isn't working. Meanwhile, a politician might be exploring solutions to racial injustice or climate change.

But whatever issues we face, there are some common ways to tackle them effectively. And we can all boost our confidence and ability to succeed by building a strong set of problem-solving skills.

Mind Tools offers a large collection of resources to help you do just that!

How Well Do You Solve Problems?

Start by taking an honest look at your existing skills. What's your current approach to solving problems, and how well is it working? Our quiz, How Good Is Your Problem Solving? lets you analyze your abilities, and signposts ways to address any areas of weakness.

Define Every Problem

The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem – not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the real issue might be a general lack of training, or an unreasonable workload across the team.

Tools like 5 Whys , Appreciation and Root Cause Analysis get you asking the right questions, and help you to work through the layers of a problem to uncover what's really going on.

However, defining a problem doesn't mean deciding how to solve it straightaway. It's important to look at the issue from a variety of perspectives. If you commit yourself too early, you can end up with a short-sighted solution. The CATWOE checklist provides a powerful reminder to look at many elements that may contribute to the problem, keeping you open to a variety of possible solutions.

Understanding Complexity

As you define your problem, you'll often discover just how complicated it is. There are likely several interrelated issues involved. That's why it's important to have ways to visualize, simplify and make sense of this tangled mess!

Affinity Diagrams are great for organizing many different pieces of information into common themes, and for understanding the relationships between them.

Another popular tool is the Cause-and-Effect Diagram . To generate viable solutions, you need a solid understanding of what's causing the problem.

When your problem occurs within a business process, creating a Flow Chart , Swim Lane Diagram or a Systems Diagram will help you to see how various activities and inputs fit together. This may well highlight a missing element or bottleneck that's causing your problem.

Quite often, what seems to be a single problem turns out to be a whole series of problems. The Drill Down technique prompts you to split your problem into smaller, more manageable parts.

General Problem-Solving Tools

When you understand the problem in front of you, you’re ready to start solving it. With your definition to guide you, you can generate several possible solutions, choose the best one, then put it into action. That's the four-step approach at the heart of good problem solving.

There are various problem-solving styles to use. For example:

• Constructive Controversy is a way of widening perspectives and energizing discussions.
• Inductive Reasoning makes the most of people’s experiences and know-how, and can speed up solution finding.
• Means-End Analysis can bring extra clarity to your thinking, and kick-start the process of implementing solutions.

Specific Problem-Solving Systems

Some particularly complicated or important problems call for a more comprehensive process. Again, Mind Tools has a range of approaches to try, including:

• Simplex , which involves an eight-stage process: problem finding, fact finding, defining the problem, idea finding, selecting and evaluating, planning, selling the idea, and acting. These steps build upon the basic, four-step process described above, and they create a cycle of problem finding and solving that will continually improve your organization.
• Appreciative Inquiry , which is a uniquely positive way of solving problems by examining what's working well in the areas surrounding them.
• Soft Systems Methodology , which takes you through four stages to uncover more details about what's creating your problem, and then define actions that will improve the situation.

Further Problem-Solving Strategies

Good problem solving requires a number of other skills – all of which are covered by Mind Tools.

For example, we have a large section of resources to improve your Creativity , so that you come up with a range of possible solutions.

By strengthening your Decision Making , you'll be better at evaluating the options, selecting the best ones, then choosing how to implement them.

And our Project Management collection has valuable advice for strengthening the whole problem-solving process. The resources there will help you to make effective changes – and then keep them working long term.

Problems are an inescapable part of life, both in and out of work. So we can all benefit from having strong problem-solving skills.

It's important to understand your current approach to problem solving, and to know where and how to improve.

Define every problem you encounter – and understand its complexity, rather than trying to solve it too soon.

There's a range of general problem-solving approaches, helping you to generate possible answers, choose the best ones, and then implement your solution.

Some complicated or serious problems require more specific problem-solving systems, especially when they relate to business processes.

By boosting your creativity, decision-making and project-management skills, you’ll become even better at solving all the problems you face.

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Creative Problem Solving

Finding Innovative Solutions to Challenges

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Research Article

Creativity in the Wild: Improving Creative Reasoning through Immersion in Natural Settings

Affiliation Department of Psychology, University of Kansas, Lawrence, Kansas, United States of America

* E-mail: [email protected]

Affiliation Department of Psychology, University of Utah, Salt Lake City, Utah, United States of America

• Ruth Ann Atchley, 
• David L. Strayer, 
• Paul Atchley

• Published: December 12, 2012
• https://doi.org/10.1371/journal.pone.0051474
• Reader Comments

Adults and children are spending more time interacting with media and technology and less time participating in activities in nature. This life-style change clearly has ramifications for our physical well-being, but what impact does this change have on cognition? Higher order cognitive functions including selective attention, problem solving, inhibition, and multi-tasking are all heavily utilized in our modern technology-rich society. Attention Restoration Theory (ART) suggests that exposure to nature can restore prefrontal cortex-mediated executive processes such as these. Consistent with ART, research indicates that exposure to natural settings seems to replenish some, lower-level modules of the executive attentional system. However, the impact of nature on higher-level tasks such as creative problem solving has not been explored. Here we show that four days of immersion in nature, and the corresponding disconnection from multi-media and technology, increases performance on a creativity, problem-solving task by a full 50% in a group of naive hikers. Our results demonstrate that there is a cognitive advantage to be realized if we spend time immersed in a natural setting. We anticipate that this advantage comes from an increase in exposure to natural stimuli that are both emotionally positive and low-arousing and a corresponding decrease in exposure to attention demanding technology, which regularly requires that we attend to sudden events, switch amongst tasks, maintain task goals, and inhibit irrelevant actions or cognitions. A limitation of the current research is the inability to determine if the effects are due to an increased exposure to nature, a decreased exposure to technology, or to other factors associated with spending three days immersed in nature.

Citation: Atchley RA, Strayer DL, Atchley P (2012) Creativity in the Wild: Improving Creative Reasoning through Immersion in Natural Settings. PLoS ONE 7(12): e51474. https://doi.org/10.1371/journal.pone.0051474

Editor: Jan de Fockert, Goldsmiths, University of London, United Kingdom

Received: April 23, 2012; Accepted: November 5, 2012; Published: December 12, 2012

Copyright: © 2012 Atchley et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors have no support or funding to report.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Our environment plays a critical role in how we think and behave. The modern environment experienced by most individuals living in urban or suburban settings can be characterized by a dramatic decrease in our exposure to natural settings and a correlated increase in exposure to a technology intense environment. Data suggest that children today spend only 15–25 minutes a day in outdoor play and sports [1] and this number continues to decline. There has been a 20% decline in per capita visits to national parks since 1988, and a 18–25% decline in nature-based recreation since 1981 [2] . Concurrently, eighty percent of kindergarten aged children are computer users (USDE, 2005) and the average 8–18 year old now spends over seven and a half hours per day using one or more types of media (TV, cell phones, computers) [3] , while adults likely spend more time engaged with different forms of media technology (for example see OFCOM Communications Market Report) [4] .

Attention Restoration Theory (ART) [5] suggests that nature has specific restorative effects on the prefrontal cortex-mediated executive attentional system, which can become depleted with overuse. High levels of engagement with technology and multitasking place demands on executive attention to switch amongst tasks, maintain task goals, and inhibit irrelevant actions or cognitions. ART suggests that interactions with nature are particularly effective in replenishing depleted attentional resources. Our modern society is filled with sudden events (sirens, horns, ringing phones, alarms, television, etc.) that hijack attention. By contrast, natural environments are associated with a gentle, soft fascination, allowing the executive attentional system to replenish. In fact, early studies have found that interacting with nature (e.g., a wilderness hike) led to improvements in proof reading [6] , control of Necker Cube pattern reversals [7] , [8] , and performance on the backwards digit span task [9] . Laboratory-based studies have also reported that viewing slides of nature improved sustained attention [10] and the suppression of distracting information [9] . However, the impact of more sustained exposure to natural environments on higher-level cognitive function such as creative problem solving has not been explored.

To empirically test the intriguing hypotheses that complex cognition is facilitated by prolonged exposure to natural settings and the parallel release from technology immersion, the current research utilized a simple and ecologically valid paradigm of measuring higher order cognitive production in a pre-post design looking at the cognitive facilitative effects of immersion in nature. To the best of our knowledge, this is the first attempt to examine changes in higher-order cognitive production after sustained exposure to nature, while participants are still in the natural environment. The higher order cognitive task used was the Remote Associates Test (RAT) developed by Mednick [11] , [12] , which has been widely used as a measure of creative thinking and insight problem-solving. Utilizing insight, problem solving, and convergent creative reasoning to effectively connect the cues provided through a mediated relationship (for example: SAME/TENNIS/HEAD = MATCH) is thought to draw on the same pre-frontal cortical structures that are hypothesized to be overtaxed by the constant demands on our selective attention and threat detection systems from our modern, technology-intensive environment.

Fifty-six (26 Female, average age = 28 years) adults involved in wilderness expeditions run by Outward Bound ( http://www.outwardbound.org/ ) participated in the study. Informed voluntary consent was provided in writing by the Outward Bound organization and was obtained for all participants in the study. The study utilized a between subjects design with 8 hiking groups (half randomly assigned to the pre-hike group and half to the in-hike group). The pre-hike groups backpacked in Alaska (n = 8), Colorado (n = 10), or Maine (n = 6) and the in-hike groups backpacked in Alaska (n = 9), Colorado (n = 14) or Washington (n = 9) and there was no communication between hiking groups. All hikes involved backpacking in the wilderness for 4–6 days and all participants were prohibited from using any electronic technology during the outing. A between-subjects design was selected to avoid unwanted carry-over effects (including collaboration between participants).

The pre-hike participant sample was composed of twenty-four participants (11 Female, average age = 34) and the in-hike group was made up of 32 participants (15 Female, average age = 24). Because age has an effect on the task, age was run as a covariate in subsequent analyses. The pre-hike group completed the RAT measure on the morning before they began their backpacking trip. The in-hike group completed the RAT measure in the morning of the fourth day or their trip. All participants were given an unlimited amount of time to complete 10 Remote Associate Items [13] and the primary dependent variable was the number of correct items provided out of 10 possible. All RAT tasks were completed independently and both analysis of the responses provided and Outward Bound councilors indicated that no collaboration happened between participants.

A simple between-participant ANOVA was utilized. As anticipated, age of participant did significantly influence hit rate for the RAT measure ( F (1,53) = 7.20, p <.01, MS  = 32.88) and therefore was included as a covariate in the analysis of Group effects. In this analysis we found that the pre-hike group were able to answer fewer RAT items ( M  = 4.14, SD  = .46) than the in-hike group ( M  = 6.08, SD  = .39), F (1,53) = 9.71, p <.01, MS  = 44.33, Cohen’s D = 0.86. This represents a 50% increase in performance after four days of exposure to nature.

Testing higher-order cognitive skills in a natural environment is a challenge. The current study is unique in that participants were exposed to nature over a sustained period and they were still in that natural setting during testing. Despite the challenging testing environment, the current research indicates that there is a real, measurable cognitive advantage to be realized if we spend time truly immersed in a natural setting. Further, unlike previous research in which cognitive changes were measured with laboratory tests of attentional function and/or laboratory surrogates for exposure to nature, the current work demonstrates that higher-order cognitive skills improve with sustained exposure to a natural environment. The current study lays the groundwork for further work examining the mechanism of this effect by providing evidence and a method by which improved cognitive performance can be examined in the wild.

There are multiple candidates for potential mechanisms underlying the effects observed here and in other studies. It is likely that the cognitive benefits of nature are due to a range of these mechanisms and it will require a sustained program of research to fully understand this phenomenon. One suggestion is that natural environments, like the environment that we evolved in, are associated with exposure to stimuli that elicit a kind of gentle, soft fascination, and are both emotionally positive and low-arousing [9] . It is also worth noting that with exposure to nature in decline, there is a reciprocal increase in the adoption of, use, and dependency upon technology [14] . Thus, the effects observed here could represent either removal of the costs associated with over-connection or a benefit associated with a return to a more positive/low-arousing restorative environment.

Exposure to nature may also engage what has been termed the “default mode” networks of the brain, which an emerging literature suggests may be important for peak psychosocial health [15] . The default mode network is a set of brain areas that are active during restful introspection and that have been implicated in efficient performance on tasks requiring frontal lobe function such as the divergent thinking task used here [16] . On a hike or during exposure to natural stimuli which produce soft-fascination, the mind may be more able to enter a state of introspection and mind wandering which can engage the default mode. Interestingly, engaging the default mode has been shown to be disrupted by multimedia use, which requires an external attentional focus, again pointing to the possibility that natural environments such as those experienced by the current participants may have both removed a cost (technology) and added a benefit (activation of brain systems that aid divergent thinking).

This study is the first to document systematic changes in higher-level cognitive function associated with immersion in nature. There is clearly much more research to be done in this area, but the current work shows that effects are measurable, even in completely disconnected natural environments, laying the groundwork for further studies. Much about our cognitive and social experience has changed in our current technology-rich society and it is challenging to fully assess the health costs associated with these changes. Nevertheless, the current research establishes that there are cognitive costs associated with constant exposure to a technology-rich, suburban or urban environment, as contrasted with exposure to the natural environment that we experience when we are immersed in nature. When our research participants spent four days in a natural setting, absent all the tools of technology, the surrounding natural setting allowed them to bring a wide range of cognitive resources to bear when asked to engage in a task that requires creativity and complex convergent problem solving.

A limitation to the current research is the inability to determine if the effects are due to an increased exposure to nature, to a decreased exposure to technology, or to other factors associated with spending three days immersed in nature. In the majority of real-world multi-day hiking experiences, the exposure to nature and technology are inversely related and we cannot determine if one factor has more influence than another. From a scientific perspective, it may prove theoretically important to understand the unique influences of nature and technology on creative problem solving; however, from a pragmatic perspective these two factors are often so strongly interrelated that they may be considered to be different sides of the same coin. We suggest that attempts to meaningfully dissociate the highly correlated real-world effects of nature and technology may be like asking Gestalt psychologists whether figure or ground is more important in perceptual grouping.

In principle, a 2×2 factorial study with high or low levels of nature (N+ or N−, respectively) and high or low levels of technology (T+ or T−, respectively) could shed light on the issue of dissociating the effects of nature and technology on complex problem solving. In the majority of real-world urban environments, T+N− is the norm whereas T−N+ is more common in the outdoor settings. Our research demonstrates that interacting for three days in T−N+ environments (i.e., the in-hike group) results in significant improvements in creative problem solving compared to T+N− environments (i.e., the pre-hike group). The T+N+ condition reflects an interesting situation where the interloper brings technology with them on the hike (assuming there is service and power) and, based on ART, we predict that interacting in this sort of environment would not benefit creative problem solving. The T−N− condition reflects a different scenario in which people interact in urban settings without the use of technology – a condition that is becoming increasingly rare in the modern world. Based upon ART, which places an emphasis on natural environments for maximal restoration, we predict that T−N+ condition would result in superior creative problem solving compared to T−N− condition (assuming that we could convince people to part with their digital technology for three full days). Future research will be required to evaluate these latter predictions.

Acknowledgments

We wish to thank Mr. Jon Frankel and the Outward Bound Organization for their valuable contributions to this work and for their willingness to collaborate with us on this project.

Author Contributions

Conceived and designed the experiments: RAA DLS PA. Performed the experiments: RAA DLS PA. Analyzed the data: RAA PA DLS. Wrote the paper: RAA DLS PA.

• 1. Juster FT, Stafford F, Ono H. Changing Times of American Youth: 1981–2003. Ann Arbor, MI: Institute for Social Research, University of Michigan. Available at: http://www.ns.umich.edu/Releases/2004/Nov04/teen_time_report.pdf . Accessed 11 November 2012.
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• 4. OFCOM Communications Market Report. (2011). Available at http://stakeholders.ofcom.org.uk/market-data-research/market-data/communications-market-reports .Accessed 11 November 2012.
• 15. Immordino-Yang MH, Christodoulou JA, Singh V (2012) Rest Is Not Idleness. Perspectives on Psychological Science, 7(4), 352–364.
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Greater Good Science Center • Magazine • In Action • In Education

Parenting & Family Articles & More

Six ways nature helps children learn, spending time in nature helps kids do better in school, in a number of surprising ways. .

Some years ago, Richard Louv made the case in his book,  Last Child in the Woods , that kids were spending so little time in nature that they had “nature deficit disorder.” The consequences they suffered were dire: more stress and anxiety, higher rates of obesity and ADHD, and more.

Many parents probably recognize that being outside in nature is good for their children’s health. But they may also see a tradeoff: Encouraging their kids to get outside means less time hitting the books. And less time studying must mean less academic success, right?

Wrong. Remarkably, it turns out that the opposite may be true. As research has grown in this area—including my own—we’ve discovered that nature is not just good for kids’ health; it improves their ability to learn, too. Even small doses of nature can have profound benefits.

The evidence for this comes from hundreds of studies, including experimental research. In one study , fifth-grade students attended school regularly at a local prairie wetlands, where science, math, and writing were taught in an integrated, experiential way as students participated in onsite research. When compared to peers attending regular schools, those who’d attended school outside had significantly stronger reading and writing skills (as measured by standardized tests) and reported feeling more excited about school because of the experience. Students at the outdoor school who’d previously had low attendance rates ended up with higher attendance, too.

Other studies echo these findings. One  study  found that students at schools with more tree cover performed better academically—especially if they came from lower socioeconomic backgrounds. Still another  compared students randomly assigned to take science lessons either in a classroom or in a school garden and found outdoor lessons more effective for learning—and the more time they spent in the garden, the greater their gains.

How do green space and nature help kids learn? In a surprising variety of ways, we’re discovering . Nature improves children’s psychological and physical well-being, for sure—and that can impact learning. But it also seems to affect how they attend to and engage in the classroom, how much they can concentrate, and how well they get along with teachers and peers. Here is what we know so far.

Nature restores children’s attention

Attention is clearly important for learning, but many kids have trouble paying attention in the classroom, whether it be because of distractions, mental fatigue, or ADHD. Luckily, spending time in nature— talking a walk in a park  and even having a view of nature  out the window—helps restore kids’ attention, allowing them to concentrate and perform better on cognitive tests.

Nature relieves children’s stress

Just like adults, children are less stressed when they have green spaces to retreat to occasionally, helping them to be more resilient. Studies have  found  that holding a class outdoors one day a week can significantly improve the daily cortisol patterns of students—reflecting less stress and better adaptation to stress—when compared to kids with indoor-only instruction. Also, in a study  looking at children in rural environments, those with more nature nearby recovered better from stressful life events in terms of their self-worth and distress.

Nature helps children develop more self-discipline

Many children—particularly those with ADHD—have trouble with impulse control, which can get in the way of school learning. My colleagues and I have found that green space near kids’ homes helps them to have more self-discipline and concentrate better— especially girls . Also, parents of kids with ADHD report that when their kids participate in activities outdoors versus indoors, it  reduces  their ADHD symptoms. Since self-discipline and impulse control are tied to academic success, it’s perhaps no surprise that…

Outdoor instruction makes students more engaged and interested

Kids seem to like classes outdoors. Unfortunately, many teachers fear bringing kids outside to learn, worrying that they’ll be “bouncing off the walls” afterward and less engaged in the next (indoor) lesson. Luckily, research seems to suggest that kids are more engaged in learning not only during outdoor classes but also  upon returning  to their classroom afterward—even if the subject they return to is not nature-related.

Time outdoors may increase physical fitness

While physical fitness is important for children for many reasons, one that may not immediately come to mind is the role it plays in learning. In particular, cardiorespiratory fitness seems to support efficient cognitive processing, and kids with higher fitness levels do better academically . Though it’s  not clear  that nature affects physical fitness directly, it is true that the more time kids spend in nature, the better their cardiorespiratory fitness . Having access to nature may encourage children to be more physically active and keep in shape longer as they age.

Nature settings may promote social connection and creativity

The  social  and physical environment in which children learn can make a difference in their academic success. Letting kids spend time in settings with natural elements or giving them structured nature experiences can make for a calmer, socially safe, and fun learning environment. And being outdoors can also enhance  peer-to-peer  relationships and  student/teacher  relationships needed for learning,  even  for students who otherwise feel marginalized socially.

Some argue that nature provides a rich tapestry of “loose parts”—sticks, stones, mud—that encourage pretend play and exploration, creativity and problem solving. Indeed, teachers’ and principals’ observations suggest that children’s play becomes strikingly more creative, physically active, and social in the presence of loose parts.    It’s clear to me that we need to do more to bring this important resource into our schools. Architects and city planners should keep trees and green areas in and near schoolyards. And teachers and principals should incorporate lessons outdoors and use recess not as a reward for good behavior, but as a way to rejuvenate students’ minds for the next lesson.    By doing so, we won’t only be benefitting our kids’ psychological well-being—though that’s reason enough! We will likely help them perform better in school, too. And, as a connection to nature breeds more care for nature, we may also be inspiring the future stewards of our natural world. 

Humans evolved to grow and thrive in natural environments, and research is showing the costs of indoor childhoods. It’s time to cure “nature deficit disorder” in our kids by giving “nature time”—not just studying and extracurricular time—the importance it deserves.

About the Author

Ming Kuo, Ph.D. , leads the Landscape and Human Health Laboratory at the University of Illinois at Urbana-Champaign. Her research convincingly links healthy urban ecosystems to stronger, safer neighborhoods, lower crime, reduced AD/HD symptoms, reduced aggression, and an array of other mental and physical health indicators. Dr. Kuo’s work has spurred increased urban forestry efforts in Wales, Germany, the Netherlands, the Caribbean, and the United States, and, in 2018, she was awarded the Heinz Award for the Environment.

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• Published: 24 April 2024

Energy-efficient superparamagnetic Ising machine and its application to traveling salesman problems

• Jia Si   ORCID: orcid.org/0000-0003-0737-4905 1 , 2 ,
• Shuhan Yang 1 ,
• Yunuo Cen 1 ,
• Jiaer Chen 1 ,
• Yingna Huang 1 ,
• Zhaoyang Yao 1 ,
• Dong-Jun Kim 1 ,
• Kaiming Cai 1 ,
• Jerald Yoo   ORCID: orcid.org/0000-0002-3150-1727 1 ,
• Xuanyao Fong   ORCID: orcid.org/0000-0001-5939-7389 1 &
• Hyunsoo Yang   ORCID: orcid.org/0000-0003-0907-2898 1  

Nature Communications volume  15 , Article number:  3457 ( 2024 ) Cite this article

Metrics details

• Electrical and electronic engineering
• Magnetic devices

The growth of artificial intelligence leads to a computational burden in solving non-deterministic polynomial-time (NP)-hard problems. The Ising computer, which aims to solve NP-hard problems faces challenges such as high power consumption and limited scalability. Here, we experimentally present an Ising annealing computer based on 80 superparamagnetic tunnel junctions (SMTJs) with all-to-all connections, which solves a 70-city traveling salesman problem (TSP, 4761-node Ising problem). By taking advantage of the intrinsic randomness of SMTJs, implementing global annealing scheme, and using efficient algorithm, our SMTJ-based Ising annealer outperforms other Ising schemes in terms of power consumption and energy efficiency. Additionally, our approach provides a promising way to solve complex problems with limited hardware resources. Moreover, we propose a cross-bar array architecture for scalable integration using conventional magnetic random-access memories. Our results demonstrate that the SMTJ-based Ising computer with high energy efficiency, speed, and scalability is a strong candidate for future unconventional computing schemes.

Introduction

The demands for future data-intensive and energy-efficient computing tasks overwhelm the computational power of conventional von Neumann architectures 1 . For example, NP-hard problems are often encountered in combinatorial optimizations 2 , resource allocation 3 , cryptography 4 , finance 5 , image processing 6 , tour planning 7 , and job sequencing 8 , and their computational time and hardware resources increase exponentially with the problem size, which makes them very difficult or impossible to be solved by conventional computers in a finite time. These problems can be mapped to the Ising model, a mathematical model to characterize interactions between magnetic spins 9 . The dynamics of the model is algorithm- based, i.e. by constructing a proper coupling matrix and allowing the system to evolve utilizing an intrinsic convergence property of the Ising model, the ground state could be obtained as a solution to the corresponding problems. However, as the system might be trapped in many local minima, the annealing process has usually been adopted in Ising computers to address such limitations. It is commonly agreed that adding fluctuations prevents the Ising computer from being stuck at the local minima.

Efficient algorithms and hardware systems for finding an optimal or near-optimal solution of an Ising model at a fast speed and low power have been sought. Adiabatic quantum computing (AQC) 10 , 11 and quantum computing 12 , 13 , 14 , 15 based on superconducting qubits are capable of converging the Ising model by tunneling out of local minima to the global minima. A 100-node Maxcut problem was solved using a quantum computer of 2048 spins with huge power consumption 16 . Besides the high cost and complexity of cryogenic temperature, this proof-of-concept system was limited by the sparse connections only between the nearest neighbors, which leads to sub-optimal outcomes 17 . Simulated annealing based on CMOS implementations was exploited for parallel Ising computing, including central processing units (CPU) 18 , 19 , graphics processing units (GPU) 20 , and field-programmable gate array (FPGA) 21 , 22 . These hardware have reported as large as 16,384 spins, however, it requires huge hardware resources for generating random numbers to introduce stochasticity to escape from the local minima 4 , 18 , 23 , 24 . Coherent Ising machine (CIM) is an optical scheme with competitive energy efficiency. However, it requires a long fiber ring cavity and relies on external FPGA for implementing coupling 25 , 26 . The temporal multiplexing process is also time-consuming and hard to expand to large systems. Recently, experiments and simulation works have investigated various devices to emulate the behavior of Ising spins by taking advantage of their intrinsic physics. An 8-spin asynchronous probabilistic computer based on superparamagnetic tunnel junctions for solving integer factorization tasks of values up to 945 was demonstrated 4 . SPICE simulations of 16-city TSP using simulated annealing method were presented 27 . Other works such as 8-spin phase-transition nano-oscillators 28 , multiferroic oxide devices with a high thermal stability 29 , and magnetoresistive random access memory (MRAM) 30 , 31 have also conceptually proved that spin-based devices are suitable for representing Ising units. However, these works have encountered challenges in either partially-connected Ising spins or small scalability which limit the Ising computer from solving practical problems.

TSP discussed in this paper is a well-known problem which is much beyond the limitation of locally connected Ising models. Other combinatorial optimization problems, such as knapsack problems, coloring problems, and number partitioning, need all-to-all connection to satisfy specific constraints 9 . In practice, an additional graph embedding process is often required when mapping to 2-dimensional CMOS circuitry which only considered the coupling between adjacent spins 32 , 33 , 34 . Since the embedding increases the required number of auxiliary spins and causes spin connections to change, the annealing accuracy is degraded significantly, especially when the problem size is large. This means that supporting a fully connected Ising model is highly recommended for dealing with a wide range of problems. Another problem is the rapidly increasing connectivity when considering large-scale systems, which usually results in huge energy consumption and latency. Since the number of spins that a particular annealing processor can handle limit the scale of the problem that can be solved, how to solve complex problems with limited hardware in an energy-efficient way has also drawn significant attention.

In this work, we experimentally report a scalable Ising computer based on 80 SMTJs with all-to-all connections and successfully solve the 4761-node TSP problem. The intrinsic stochasticity in SMTJ enables ultra-fast and low-power Ising annealing without using extra resources for random number generation and Metropolis determining process 7 . By combining global annealing with intrinsic annealing in SMTJ, the convergence of the Ising problem is guaranteed especially in large-scale Ising problems. The method to determine parameters of global annealing is discussed. With an all-to-all connection among Ising spins, the combinatorial optimization of 9-city TSP is solved with the optimal solution. We further develop the algorithm for constrained TSP (CTSP) with no extra auxiliary Ising bits both in algorithm and hardware, indicating the superiority and flexibility of this Ising computer. Furthermore, we propose an optimization strategy based on graph partitioning (GP) and CTSP and experimentally solved a 70-city TSP, which typically needs 4761 nodes, on our 80-node Ising computer with a near-optimal solution. The system can obtain the lowest power consumption of 0.64 mW as well as high energy efficiency of 39 solutions per second per watt among state-of-art Ising annealers. We have experimentally demonstrated that large-scale Ising problems can be solved by small-scale hardware in an energy-efficient way.

SMTJ-based artificial Ising spin

Various NP-hard problems can be solved by constructing corresponding Ising models and observing the ground states during evolution processes. Figure  1a shows an all-to-all connected Ising model, whose Ising Hamiltonian can be written as

where \(H\) is the total energy of the system, \(N\) is the total number of spins, \({s}_{i}\) is the \(\,i\) -th spin with one of two states; “+1” (Ising spin up) or “−1” (Ising spin down), \({J}_{i,j}\) is the coefficient of coupling between the \(i\) -th and the \(j\) -th spins, and \({h}_{i}\) is the external field of the \(\,i\) -th spin. For a fixed configuration of other spins than \({s}_{k}\) , the probability of \({s}_{k}\) staying in the down-state is given by

where \(\Lambda=\frac{\partial H}{\partial {s}_{k}}\) (see Supplementary Note  1 ).

a All-to-all connected 12-spin Ising model with s represents the spin and J 1,6 represents the coupling between s1 and s6. b Sigmoidal fit of probability of AP state ( \({p}_{{AP}}\) ) of an SMTJ under different input currents ( I ). \({p}_{{{{{{\rm{AP}}}}}}}=\frac{1}{1+{e}^{-4.672\times (I-3.905{{{{{\rm{\mu }}}}}}{{{{{\rm{A}}}}}})}}\) . Inset: diagram of an SMTJ. A tunneling barrier layer is sandwiched by a reference layer and a free layer. c Time-dependent resistance of an SMTJ under different input currents ( I ). d Photograph and schematic diagram of SMTJ-based Ising computer. The system contains 8 processing elements (PEs), 4 digital-to-analog converters (DACs), a comparator array, a multiplexer and a microcontroller unit (MCU). Each PE has 10 SMTJ computing units. Each computing unit includes a transistor and a resistor to adjust the property into stochastic. Blue lines and orange arrows represent the control and data flow, respectively.

One natural implementation of this Ising spin is based on a stochastic nanomagnet. The inset of Fig.  1b shows the sketch of an SMTJ, consisting of a tunneling barrier sandwiched by a reference layer and a free layer (see Methods section). Because of the small device diameter (~50 nm), the energy barrier of the free layer between the anti-parallel (AP) and parallel (P) states is low that the retention time of either state is in the range of μs to ms, similar to previous studies 4 , 35 . The SMTJ resistance, measured as a function of time in Fig.  1c , shows preferred AP states at high currents and P states at low currents. When the current ( I ) is ~4 μA, SMTJ shows an equal chance of AP and P states. The probability of the AP state under different input currents over 0.1 s is fitted in Fig.  1b by a sigmoid function:

where \({{{{{\rm{a}}}}}}=4.67 \, {{{{{\rm{and\; b}}}}}}=3.9 \, {{{{{\rm{\mu }}}}}}{{{{{\rm{A}}}}}}.\) In order to emulate Ising spin \({s}_{k}\) with our SMTJ device, we only need to make the probability of the down-state of \({s}_{k}\) to be equal to that for the AP state of SMTJ, namely \({p}{\_}{{{{{{\rm{\_}}}}}}{{{{{\rm{AP}}}}}}}={p}{\_}{{{{{{\rm{\_}}}}}}\downarrow }\) , with two calibration coefficients. Thus, we can derive the form of the current \({{I\_}}_{k}\) injected to SMTJ as (see Supplementary Note  1 ):

where \(c=1/{kT}\) is the effective inverse temperature which can be conducted for global annealing.

Intrinsic annealing in SMTJs-based Ising computer

By integrating 80 SMTJs with a peripheral circuit and a microcontroller unit (MCU), we build an 80-node Ising computer (see Supplementary Note  2 ). Each Ising spin in Eq. ( 1 ) is emulated by an SMTJ with intrinsic randomness, where P (AP) state represents spin-up (down). Figure  1d shows the photograph of the printed circuit board (PCB) and the diagram of the system (see Methods section). The system contains 8 processing elements (PEs); each PE has 10 SMTJ computing units. Each SMTJ computing unit includes a transistor and a resistor to adjust the state of SMTJ into stochastic. During the computing process, an MCU examines the states of all SMTJs by reading the output of comparator arrays through multiplexers and generates new input voltages for digital-to-analog converters (DACs) according to the updating rule in Eq. ( 4 ) (see Supplementary Note  3 for calibration of 80 SMTJ computing units).

During the evolution process, an Ising solver could be easily trapped in a local minimum state. To avoid this non-optimal solution, annealing algorithms such as simulated annealing (SA) or quantum annealing (QA) were developed. The general idea of SA is to make the system evolve from a high temperature to a low temperature gradually 7 . The convergence and relaxation of SA can be mathematically provable 36 . During each iteration, a random number is generated for stochasticity and introduced to determine whether the result in this iteration should be accepted or not. In QA, quantum fluctuations cause quantum tunneling between states 17 . In both SA and QA, stochasticity needs to be introduced into the annealing process. In contrast, our Ising system utilizes the intrinsic stochastic behaviors of SMTJ to perform the Metropolis process of standard SA in hardware, which greatly saves the solution time and hardware resources for generating randomness (see Supplementary Note  4 ). Besides, our Ising computer has an all-to-all connection which has wider application scenarios, as well as a better capability of escaping from local minima.

Ising mapping of N-city TSP and CTSP

We have applied our Ising computer to the TSP problem, one of the combinatorial optimization problems, which applies to various sectors, such as vehicle routing, logistics, planning, and scheduling. The goal is to find the shortest route that visits all listed cities once and only once given distances between the cities in the list. In order to solve this problem, we first map N -city-TSP to an \({N}^{2}\) -spin Ising model, or \({(N-1)}^{2}\) -spin model assuming a fixed starting city. Figure  2a shows the coordinates of 9 cities and Fig.  2b shows the 81-spin Ising model, whose rows indicate the cities and columns indicate the visiting order. We define the binary spin, s , as \({s}_{i,j}\)  = 1 if city i is visited as j -th city or \({s}_{i,j}\)  = −1 otherwise. The total Hamiltonian of TSP is expressed by 9

where the first term is a constraint that represents only one city is visited at the j -th visit, and the second term represents one city is visited only one time. \(w\) is a constant small enough ( \(0 \, < \, w \, < \, 1\) ) not to violate the two constraints of the TSP cycle. \({d}_{i,{i{{\hbox{'}}}}}\) is the distance between city \(i\) and city \({i{{\hbox{'}}}}\) . According to Eqs. ( 1 ) and ( 5 ), coupling matrix \(J\) of 81 spins could be obtained, as shown in Fig.  2c (see Supplementary Note  5 ). It shows that spins in the same row or column have strong coupling, as indicated by the first two terms in Eq. ( 5 ).

a Coordinates of all 9 cities used in this problem which are the first 9 cities in the dataset Burma14 from TSPLIB. b Ising spin representation for 9-city TSP (81 spins). Rows indicate names of cities and columns indicate the visiting order. Each spin can be 1 (visited) or −1 (not visited) in each iteration. c Color map of the coupling matrix J TSP of 9-city TSP, and the color bar represents an effective energy with the unit of kT . Here, k is the Boltzmann constant and T is the temperature. d Constrained TSP (CTSP) with a fixed vising sequence from city 2 to city 7 or from city 7 to city 2. The arrows represent the visiting sequence. e The Ising spin representation for CTSP with the fixed visiting sequence in d . Arrows represent possible vising sequences. f Color map of the difference of coupling matrix between TSP (J TSP ) in a and CTSP (J CTSP ) in d . Arrows represent the fixed vising sequences from city 2 to city 7 or from city 7 to 2.

We define CTSP as the visiting orders of some cities are enforced during the traveling. This is quite useful in real-life scenarios. For example, a delivery man collects food and drinks at shop A and must deliver hot drinks to B first even though the total cost is higher than optimal. We propose an algorithm for solving CTSP by adding negative “distance” to the Hamiltonian. For example, suppose that city A and city B are required to be connected in the CTSP as city 2 and city 7 shown in Fig.  2d , and then we add the term

such that the energy of a path, where city A and city B are connected, is always lowered by \(\theta .\) When \(\theta\) is sufficiently large, the optimal path must have city 2 and city 7 connected. Thus, the total Hamiltonian of the CTSP is expressed by

Constructing an Ising model for CTSP is exactly the same as TSP except for extra allowed visiting sequences, as shown in Fig.  2e . This would lead to a modification of the coupling matrix of \(J\) according to Eq. ( 7 ) (see the deduction of \({J}_{{CTSP}}\) in Supplementary Note  6 ). From Fig.  2f we can clearly see the differences between \({J}_{{CTSP}}\) and \({J}_{{TSP}}\) . This algorithm of CTSP fits for arbitrary constraints of visiting sequences as well as their combinations.

Experimental demonstration of 9-city TSP

We first run a 9-city TSP in the 80 SMTJ-based Ising computer at a relatively low but non-zero effective temperature to examine the intrinsic annealing in SMTJ. The iteration time is set comparable to the longest retention time of SMTJs to avoid reading previous spin states. In our experiments, we set the iteration time as 0.1 ms. As shown in Fig.  3a , as the effective inverse temperature ( c ) is increased quickly to 0.5, the system converges rapidly to a low energy state within 50 iterations and reaches the ground state after 4000 iterations. It should be noted that the intrinsic stochasticity in SMTJs helps the system escape from local minima without an extra annealing process, as shown in the right inset of Fig.  3a . Figure  3b illustrates the evolution of 9 spins out of 81 spins. The evolution of all 81 spins can be found in Supplementary Note  7 .

a Total energy transition of 9-city TSP with 5000 iterations (the optimal solution with the energy of 18.23 corresponds to the dashed horizontal line). Insets: effective inverse temperature ( c ) and total energy within 3500–4500 iterations. b Evolution of 9 representative SMTJ states in 5000 iterations. An offset is used in the y -axis to show each SMTJ clearly. c Visiting routes of state A, B, C, and D in a . d Corresponding Ising spins of state A, B, C, and D in a . The yellow squares represent ‘visited ( \({s}_{i,j}=1\) )’ and the purple squares represent ‘not visited ( \({s}_{i,j}=-1\) )’. e Total energy transition with increasing c from 0.2 to 1.8. Left inset: zoom-in view of total energy transition with increasing c from 0.392 to 0.52. Right inset: transition of c with iterations. The red dashed line represents the optimal path (success). f Success probability of solving TSP with varying the node size. The data points and shadows represent the median value and the interquartile range (IQR), respectively.

We choose four states in Fig.  3a to inspect the traveling path in Fig.  3c and their Ising spins, namely \({s}_{i,j}\) , as shown in Fig.  3d . The yellow square in Fig.  3d represents \({s}_{i,j}=1\) (visited) and the blue square represents \({s}_{i,j}=-1\) (not visited). In an initial state A, the spin states are randomly set and then converge to a relatively low energy at state B. State C is an intermediate solution during the annealing process. State D is the optimal solution satisfying two constraints of the TSP. Because we anneal the system to a relatively low but non-zero temperature so that the convergence to a sub-optimal state could be guaranteed, and at the same time, the intrinsic randomness in SMTJ helps the system to escape from local minima and find a ground state quickly. We test 10 different random initial states each with 5000 iterations and find that in all cases the system can obtain a relatively small energy, as shown in Supplementary Note  8 . However, there is a probability that the system jumps out of the ground state because of the non-zero temperature. If we continue to observe the evolution in a large timescale, the system would move back to the global minimum state. In some cases, where the speed and near-optimal solution matter but the accurate optimal solution is not, the number of iterations can be chosen to be small.

Further global annealing of the system to a lower effective temperature may guarantee the convergence of the computation. Here we use linear annealing as an example to examine the convergence of this algorithm in a very large-iteration limit. The initial temperature should be chosen sufficiently high to ensure that the thermal energy exceeds any energy barrier ( \(\Delta H{=H}_{\max }-{H}_{\min }\) ) within the system, while still adhering to the fundamental constraints of the specific Ising model. For a given N-city TSP, \({H}_{\max }\) in Eq. ( 5 ) can be estimated as \(w\times N\times \bar{d}\) , assuming that the distance between any two cities is the same as the average distance \(\bar{d}\) . Similarly, \({H}_{\min }\) can be estimated as \(w\times N\times {d}_{\min }\) . Therefore, the initial \(c\) of 9-city TSP in our experiment can be estimated as \({c}_{{{{{{\rm{initial}}}}}}}\, \sim 1/\Delta H=0.07\) , where \(w=0.5\) , \(N=9\) for a total of 9 cities, \(\bar{d}=4\) and \({d}_{\min }=0.8\) for the average and shortest distance of each two cities, respectively in Fig.  3c . We then choose \({c}_{{{{{{\rm{initial}}}}}}}\)  = 0.2 which is sufficiently safe for annealing. As the temperature linearly decreases, the dynamical system gradually stabilizes. The final temperature should be low enough i.e., \({c}_{{{{{{\rm{final}}}}}}} \, \gg \, 1/\Delta H\) , to freeze all possible fluctuations. Here we set \({c}_{{{{{{\rm{final}}}}}}}=1.8\) which is at least one order larger than \(1/\Delta H\) . This can also be verified by observing randomly generated states under \({c}_{{{{{{\rm{final}}}}}}}\) for long iterations. Regarding the annealing speed, if several changes in the spin configuration are observed under each value of c , then this annealing speed is valid. Plenty trials are required to find the proper annealing speed (details in Supplementary Note  8 ).

In Fig.  3e we can find the first global minimum energy appears after 16,500 iterations, and converge to the ground state after 40,000 iterations. Temperature schedules can be optimized to reduce iteration numbers, e.g. increase the effective temperature in the first few time steps, and then decrease gradually, or learned by the reinforcement learning method 37 . In practice, we use one memory to store the minimum energy state during the computation, and another memory to record the final energy state. We take the minimum value of these two results as the solution. Figure  3f shows the success probability (defined as finding the optimal path) of TSP with various node sizes. The success probability of 9-city TSP reaches 95% after 10 4 iterations. The success probability with the parameter \(w\) in Eq. ( 5 ) which determines the relative strength of the constrain term and distance term is also discussed. If the \(w\) is too large, then the probabilities of violations, namely the invalid path, would increase, as shown in Supplementary Note  8 . If \(w\) is too small, then the effect of the distance term is small, which results in a slower convergence to the ground state.

The advantages of this annealer are threefold: (1) Selective working modes by using different temperature schemes. One is the probabilistic sampling mode working at a constant temperature, which is similar to an asynchronous probabilistic computer 4 ; the other is the annealing mode conducted by reducing the effective temperature. (2) Fast speed and low power consumption to find the ground state because of the intrinsic annealing properties in SMTJ. (3) Global annealing outperforms probabilistic sampling in achieving efficient convergence, especially for large-scale problems.

We have implemented a synchronous design with a lower requirement on the speed of peripheral circuits. This design also effectively mitigates issues such as leakage, sneak currents, and parasitic resistances which might encountered in asynchronous hardware with a memristive (or resistive) crossbar array.

Compressing 70-city TSP to 80-node Ising computer

Generally, the number of spins required for an N -city TSP is ( N -1) 2 , which limits the scalability of TSP on state-of-the-art computing systems. Here, we propose a graph Ising compressing algorithm based on CTSP that can significantly reduce the number of spins and interactions for solving a TSP. Figure  4a is an example of how we apply this algorithm to our 80-node SMTJ Ising computer for solving a 70-city TSP (4761 nodes, st70 data set from TSPLIB 38 ). The major steps of this algorithm can be described as follows: (a) divide the cities into several smaller groups until the number of cities in each group is less than 10 by GP method; (b) solve TSP within each group separately; (c) integrate neighboring groups to obtain an initial path of the whole group; and (d) optimize the path in (c) by a CTSP window sliding over the whole map.

a Optimization algorithm for 70-city TSP. b Number of required SMTJs for various problems using different methods. Burma14, berlin52, eil76, and eil101 are TSP of 14, 52, 76, and 101 cities, respectively. c Comparison of total Ising energy (path) and total clock cycles for final solution with different SA-based algorithms, including symbiotic organisms search 40 , ant colony optimazation 41 , multi-offspring genetic algorithm 42 , and gene-expression programming 7 . Our method is tested on our Ising system and others are tested on Intel Core-i7 PC. In this comparison, our system runs at a main frequency of 10 kHz.

It is worth mentioning that GP is also an Ising problem. When converting a global TSP into local TSPs, using GP would be more hardware-friendly for our Ising computer compared to other clustering algorithms. It is based on the idea that the original graph can be separated into multiple sub-graphs depending on the Euclidean distance. The number of spins required for solving GP is ~ N and thus, GP is quite efficient for local TSPs since the problem size can be reduced to ~ \({\left(N-1\right)}^{2}/a\) , where \(a\) is the number of groups, and each TSP can be optimized independently (see GP mapping in Supplementary Note  9 ).

The final step (d) is based on CTSP, where a rectangular window slides over the path and cuts it into several disconnected lines, among which the two longest lines are chosen and the edge cities are connected as a circular path (Supplementary Note  10 ). The CTSP is solved within each window for sub-area optimization without changing the visiting order of edge cities. After this, the two lines at the edge cities are opened and CTSP is carried out again after sliding to the next window. GP-CTSP-based optimization algorithm provides an efficient way of finding near-optimal solutions for large-scale TSP on limited hardware resources.

Figure  4b shows the comparison of numbers of spins for different TSPs by a conventional Ising method 9 , cluster Ising method 39 , and our method. The required number of spins in our method is relatively unchanged for various TSPs, while that of other methods increases substantially with the scale of the problem. Figure  4c shows the total path of 70-city TSP as a function of iteration number using different SA-based algorithms, including symbiotic organisms search 40 , ant colony optimization 41 , multi-offspring genetic algorithm 42 , and gene-expression programming 7 . Finally, we obtain the near-optimal path with a total energy of 700.71, which is slightly higher than the optimal solution of 675. However, the iteration number for an optimized solution is 4.9 \(\times\) 10 6 by our method, which is two to three orders lower than that of SA-based algorithms running on Intel Core-i7 CPU 7 with the main frequency of 3 GHz, as shown in Fig.  4c .

Ising computer scaling and cross-bar architecture

The above experimental demonstration shows our Ising computer with 80 SMTJs is capable of finding a near-optimal solution to a medium-scale NP-hard problem. We then explore the performance with increasing from 70 to 200 cities. The simulation of complete TSP task is carried out using MATLAB, incorporating a stochastic model of the SMTJ employed in our experiment (details in Supplementary Note  11 ). The solution quality is defined as

Figure  5a illustrates the solution quality of the best results obtained for each TSP task (Supplementary Note  12 for the best solutions). Notably, as the number of SMTJ (M) increases, higher quality solutions can be attained. It is worth emphasizing that the shortest path obtained for the 101-city TSP is 640.9755 in our study, surpassing the optimal path of 642.3095 provided by TSPLIB (Eil101.opt.tour). This outcome serves as evidence of the superiority of our method. The utilization of more SMTJs solving TSP per sliding window leads to improved optimization of CTSP annealing, resulting in an enhanced solution quality, as depicted in Fig.  5b . Consequently, the time to convergence s would also increase with the use of more SMTJS. When dealing with a fixed hardware capacity, an appropriate number of SMTJs for CTSP optimization can be assigned, taking into account both the solution quality and convergence speed. Figure  5c showcases the success rate (defined as achieving 95% solution quality) as the problem size increases. The success probability of 200-city TSP, whose complexity is ~40,000 nodes, can reach as high as 90%, demonstrating the scalability of our method compared to typical TSP (without GP and CTSP) 9 .

a Solution quality of various problems using different number of SMTJs (M) in the array. The datasets used are St70, Eil101 and KroA200, for 70, 101 and 200 cities, respectively. b Total length of KroA200 TSP at different convergence speeds using different number of SMTJs. The dashed line represents the best demonstrated solution. c Success probability of different TSP algorithm (without/with GP and CTSP) as the number of cities increases after running for 50 times. A total of 512 SMTJs are used. Here we define the success as achieving the solution quality of 95%. d SMTJ cross-bar array which contains row decoder, SMTJ, select transistor and read sense amplifier (RSA). BL represents bit line, WL represents word line, Vin, Vout and Vdd represent the input voltage, output voltage and supply voltage of RSA. e Circuit of one RSA which contains a current mirror, voltage equalization circuit (VEC, with a control signal of EQ which initializes the voltages in Q and QB points, under a reference voltage of Vdd/2), voltage sense amplifier (VSA, with a control signal of SEN), reference resistance ( \({{{{{\rm{Rref}}}}}}=\frac{1}{2}({{{{{\rm{Rap}}}}}}+{{{{{\rm{Rp}}}}}})\) , Rap and Rp represent SMTJ’s resistance in AP and P state respectively), and control transistors. f Signals of writing/reading two adjacent SMTJ cells in one BL, selected by WL0 and WL1 in sequence. All signals are defined in e and f .

We also propose a cross-bar architecture for large-scale Ising computer implementation, which can be integrated by using modern MRAM and CMOS technologies. The core part of this architecture consists of SMTJ bit cells organized as a cross-bar array, integrated with row decoders and read sense amplifiers (RSA), as shown in Fig.  5d . Each SMTJ bit cell contains one select transistor and one SMTJ (1T1SMTJ), whereas the gate of the select transistor is driven by word lines (WL), and the source of all bit cells are connected to the ground. Each bit line is assigned with an RSA. The current flows through SMTJ can be continuously adjusted by Vin of RSA, and the state of SMTJ can be read by RSA at the same time. Figure  5e illustrated the circuit of RSA, in which two clamp transistors control the current flow through the bit cell path and reference path by the gate voltage (Vin), and a current mirror is used to guarantee the same current of the above two paths. Then different voltages would show in the Q and QB point when the resistance of SMTJ is higher or lower than the reference resistor (Rref). By utilizing an enabled voltage sense amplifier (VSA), the voltages at the Q and QB points are sensed, allowing the SMTJ state to be determined as either Vdd (P state) or 0 V (AP state). Particularly, a voltage equalization circuit (VEC) is designed for initializing VSA to avoid incorrect readout. Electrical coupling through a resistance change 43 is evaluated to have neglectable effects (details in Supplementary Note  11 ). Figure  5f shows the signals to control and read bit cells. In phase 0 (PH0), one row of SMTJs is selected by WL, and Vin prepared by peripheral circuit is applied to the corresponding RSA. EQ is set high to initialize Q, QB and Vout as Vdd/2. In phase 1 (PH1), the SMTJ fluctuates from the falling edge to next rising edge of EQ. Finally, in phase 2 (PH2), RSAs read the data of one row in parallel at the falling edge of SEN. After the first row has been retrieved, the partial sum starts to be computed. Meanwhile, the same process for the second row can be started, so and so forth. To avoid reading the previous state, the duration of PH1 is preferred to be comparable with the retention time of SMTJ, which limits the main frequency of the system (see details in Supplementary Note  11 ).

We compare our system with other state-of-art Ising solvers, including CMOS annealer (Intel Core i7 processor) 7 , quantum annealer (D-Wave 2000Q) 16 , 17 , CIM with FPGA 26 , memristor Hopfield neural networks (mem-HNN) 44 , and phase-transition nano-oscillators (PTNO) 28 in solving 4761-node TSP70, as shown in Table  1 . We use the experimental data for benchmarking from literature, and two kinds of SMTJs for comparison. One is our perpendicular anisotropy SMTJ device and the other is assuming recently reported in-plane anisotropy SMTJ with a retention time of 8 ns 45 , 46 . The major attributes are the main frequency (defined as 1/iteration time), power, time-to-solution as well as energy efficiency (defined as solutions per second per watt). As quantum computers, CIM, mem-HNN, and PTNO only demonstrated ~100-node max-cut problems, we estimate the time-to-solution for solving TSP70 by assuming that the algorithm and the total number of spins to find a near-optimal solution is the same as our work (details in Supplementary Note  13 ). Here, we set 80-spin Ising computer as a standard and fix the number of iterations of 400,000 for a good solution to TSP70. Only Ising computing parts are calculated for power consumption.

In Table  1 , although the main frequency of CPU is the highest among all candidates, the energy efficiency is lower than our SMTJ-based approach. This is due to the redundant logic and data transfer delay between the memory and PEs in a conventional von-Neumann architecture. The SMTJ-based approach currently outperforms the quantum annealer both in the power consumption as well as time to solution. The power of quantum annealer is huge which needs to be optimized further for real applications. CIM is another promising architecture with a fast speed and acceptable power consumption. Current CIM systems are proof-of-concept systems which are not at present optimized for energy efficiency. Mem-HNN has a relatively fast speed assuming the 180-nm CMOS technology. However, the required number of devices is large, which limits the integrated density. The PTNO approach uses capacitors or resistors to mimic spin coupling, whose main frequency would be limited by the system scale and parasitic effects. It is reported that the ideal main frequency would decrease from 500 to 87 MHz when the system scale increases from 8-node to 100-node 28 . Our SMTJ-based Ising computer outperforms other approaches with low power consumption with 0.64 mW (details in Supplementary Note  13 ).

We experimentally demonstrate perpendicular MTJs with a retention time of ~0.1 ms and solve TSP70 Ising problems at an energy efficiency of 39 solutions per second per watt. Furthermore, we simulate an Ising computer with 4 Kb SMTJs using 40 nm commercial CMOS technology. The simulated energy efficiency for solving TSP70 by using the same SMTJ can reach 68 solutions per second per watt. By using reported in-plane SMTJ 45 and advanced CMOS, the system could obtain the highest energy efficiency of \(5.4\times {10}^{3}\) , which shows several orders of magnitude improvement over other approaches. This result suggests that an SMTJ-based Ising computer can be a good candidate for solving dense Ising problems in a highly energy-efficient and fast way.

In summary, we have experimentally demonstrated an intrinsic all-to-all Ising computer based on 80 SMTJs, and solved 9-city TSP with the optimal solution. Furthermore, a compressing strategy based on CTSP and GP is proposed to experimentally solve 4761-node 70-city TSP on an 80-node system with a near-optimum solution as well as ultra-low energy consumption. A cross-bar architecture is then proposed for large-scale Ising computers and the 200 city TSP task is simulated. Our system provides a feasible solution to fast, energy-efficient, and scalable Ising computing schemes to solve NP-hard problems.

Sample growth and device fabrication

Thin film samples of substrate/[W (3)/Ru (10)] 2 /W (3)/Pt (3)/Co (0.25)/Pt (0.2)/[Co (0.25)/Pt (0.5)] 5 /Co (0.6)/Ru (0.85)/Co (0.6)/Pt (0.2)/Co (0.3)/Pt (0.2)/Co (0.5)/W (0.3)/CoFeB (0.9)/MgO (1.1)/CoFeB (1.5)/Ta (3)/Ru (7)/Ta (5) were deposited via DC (metallic layers) and RF magnetron (MgO layer) sputtering on the Si substrates with thermal oxide of 300 nm with a base pressure of less than \(2\times {10}^{-8}\) Torr at room temperature. The numbers in parentheses are thicknesses in nanometers. To fabricate the superparamagnetic tunnel junctions, bottom electrode structures with a width of 10 µm were firstly patterned via photolithography and Ar ion milling. MTJ pillar structures with a diameter of ~50 nm for the superparamagnetic behavior were patterned by using e-beam lithography. The encapsulation layer of Si 3 N 4 was in-situ deposited after ion milling without breaking vacuum by using RF magnetron sputtering, and top electrode structures with a width of 10 µm were patterned via photolithography and top electrodes of Ta (5 nm)/Cu (40 nm) were deposited by using DC magnetron sputtering.

MTJ characterization by probe station

The setup includes a source meter (Keithley 2400) for supplying DC bias currents and a data acquisition card (NI-DAQmx USB-6363) for the read operation. A single SMTJ operation cycle comprises two steps (i.e. bias and read). A small DC input current with an amplitude of 1–20 μA is applied to SMTJ. Simultaneously, the DAQ card reads the voltage signal across the SMTJ at a maximum sampling rate of 2 MHz. The MTJ switching probability varies in accordance with the amplitude of applied currents. The retention time of MTJ is determined from random telegraph noise measurements over 250 ms. The expectation values of event time τ is determined by fitting an exponential function to the experimental results.

80 SMTJ arrays and peripheral circuits are integrated on a 12 cm × 15 cm PCB, controlled by an MCU (Arduino Mega 2560 Rev3). Four 12-bit rail-to-rail DACs (AD5381) with 160 output channels in total are used to generate analog DC inputs for PE and comparator arrays. Half of the DAC output channels are used to provide stimulation to the gate terminal of NMOSs (2N7002DW-G), and others are used to provide reference voltages to comparators (AD8694). The drain voltages of NMOS are compared with reference voltages and generate outputs in parallel. Outputs of comparator arrays are read by MCU through four multiplexers (FST16233) and then are calculated to obtain new inputs for DACs. The supply voltage of the PCB board and SMTJs is 5 V and 0.8 V, respectively. The value of resistors in each computing unit can be designed to adjust the center of sigmoidal curves.

Data availability

The data generated during this study are available within the article and the  Supplementary Information file.  Source data are provided with this paper.

Code availability

The codes that support this study can be available from the corresponding author upon request.

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Acknowledgements

This work was supported by National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its Competitive Research Programme (NRF-000214-00 to H.Y.), Advanced Research and Technology Innovation Center (ARTIC to H.Y.), the National University of Singapore under Grant (project number: A-0005947-19-00 to H.Y.), and Ministry of Education, Singapore, under Tier 2 (T2EP50123-0025 to H.Y.). We thank Yuqi Su, and Chne-Wuen Tsai from National University of Singapore and Zhi-Da Song from Peking University for useful discussions.

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Jia Si, Shuhan Yang, Yunuo Cen, Jiaer Chen, Yingna Huang, Zhaoyang Yao, Dong-Jun Kim, Kaiming Cai, Jerald Yoo, Xuanyao Fong & Hyunsoo Yang

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J.S. and H.Y. conceived and designed the experiments. J.S. designed, fabricated, and coded the hardware system. D.K., and S.Y. fabricated the devices. J.S., S.Y., and K.C. performed device measurements. Z.Y. bonded the components on PCB. J.S. designed SMTJ-based Ising system. J.S., J.C., Y.C., Y.H. and X.F. developed the optimization algorithm and performed simulations. J.S., S.Y., Y.C., J.Y., X.F. and H.Y. analyzed the data. J.S. and H.Y. wrote the manuscript. H.Y. proposed and supervised this work. All authors discussed the results and revised the manuscript.

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Correspondence to Hyunsoo Yang .

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Si, J., Yang, S., Cen, Y. et al. Energy-efficient superparamagnetic Ising machine and its application to traveling salesman problems. Nat Commun 15 , 3457 (2024). https://doi.org/10.1038/s41467-024-47818-z

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Received : 16 June 2022

Accepted : 11 April 2024

Published : 24 April 2024

DOI : https://doi.org/10.1038/s41467-024-47818-z

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