Mixing things up: Optimizing fluid mixing with machine learning

Can an optimized mixing be achieved for laminar flows instead? To answer this question, a team of researchers from Japan, in a new study, turned to machine learning. In their study published in Scientific Reports, the team resorted to an approach called "reinforcement learning" (RL), in which intelligent agents take actions in an environment to maximize the cumulative reward (as opposed to an instantaneous reward).

"Since RL maximizes the cumulative reward, which is global-in-time, it can be expected to be suitable fortackling the problem of efficient fluid mixing, which is also a global-in-time optimization problem," explains Associate Professor Masanobu Inubushi, the corresponding author of the study. "Personally, I have a conviction that it is important to find the right algorithm for the right problem rather than blindly apply a machine learning algorithm. Luckily, in this study, we managed to connect the two fields (fluid mixing and reinforcement learning) after considering their physical and mathematical characteristics." The work included contributions from Mr. Mikito Konishi, a graduate student, and Prof. Susumu Goto, both from Osaka University.

One major roadblock awaited the team, however. While RL is suitable for global optimization problems, it is not particularly well-suited for systems involving high-dimensional state spaces, i.e., systems requiring a large number of variables for their description. Unfortunately, fluid mixing was just such a system.

To address this issue, the team adopted an approach used in the formulation of another optimization problem, which enabled them to reduce the state space dimension for fluid flow to one. Put simply, the fluid motion could now be described using only a single parameter!

The RL algorithm is usually formulated in terms of a "Markov decision process" (MDP), a mathematical framework for decision making in situations where the outcomes are part random and part controlled by the decision maker. Using this approach, the team showed that RL was effective in optimizing fluid mixing.

"We testedour RL-based algorithm for the two-dimensional fluid mixing problem and found that the algorithm identified an effective flow control, which culminated in an exponentially fast mixing without any prior knowledge," says Dr. Inubushi. "The mechanism underlying this efficient mixing was explained by looking at the flow around the fixed points from a dynamical system theory perspective."

Another significant advantage of the RL method was an effective transfer learning (applying the knowledge gained to a different but related problem) of the trained "mixer." In the context of fluid mixing, this implied that a mixer trained at a certain Péclet number (the ratio of the rate of advection to the rate of diffusion in the mixing process) could be used to solve a mixing problem at another Péclet number. This greatly reduced the time and cost of training the RL algorithm.

While these results are encouraging, Dr. Inubishi points out that this is still the first step. "There are still many issues to be solved, such as the method's application to more realistic fluid mixing problems and improvement of RL algorithms and their implementation methods," he remarks.

While it is certainly true that two-dimensional fluid mixing is not representative of the actual mixing problems in the real world, this study provides a useful starting point. Moreover, while it focuses on mixing in laminar flows, the method is extendable to turbulent mixing as well. It is, therefore, versatile and has potential for major applications across various industries employing fluid mixing.

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Reference DOI: https://doi.org/10.1038/s41598-022-18037-7

About The Tokyo University of Science Tokyo University of Science (TUS) is a well-known and respected university, and the largest science-specialized private research university in Japan, with four campuses in central Tokyo and its suburbs and in Hokkaido. Established in 1881, the university has continually contributed to Japan's development in science through inculcating the love for science in researchers, technicians, and educators.

With a mission of "Creating science and technology for the harmonious development of nature, human beings, and society," TUS has undertaken a wide range of research from basic to applied science. TUS has embraced a multidisciplinary approach to research and undertaken intensive study in some of today's most vital fields. TUS is a meritocracy where the best in science is recognized and nurtured. It is the only private university in Japan that has produced a Nobel Prize winner and the only private university in Asia to produce Nobel Prize winners within the natural sciences field.

Website: https://www.tus.ac.jp/en/mediarelations/

About Associate Professor Masanobu Inubushi from Tokyo University of Science Masanobu Inubushi is currently an Associate Professor at the Tokyo University of Science, Japan. He obtained his undergraduate degree in 2008 from the Tokyo Institute of Technology, Japan. He then obtained his PhD in Mathematics from the Research Institute for Mathematical Sciences (RIMS) at Kyoto University Graduate School in 2013. After working at NTT, Communication Science Laboratories from 2013-2018, he joined Osaka University as Assistant Professor in 2018. Dr. Inubushi has over 25 published research works that have been cited over 400 times. His research interests include fluid mechanics, chaos theory, and mathematical physics, and machine learning.

Funding information This work was partially supported by JSPS Grant-in-Aid for Early-Career Scientists No. 19K14591and JSPS Grants-in-Aid for Scientific Research Nos. 19KK0067, 20H02068, 20K20973, and 22K03420.