Recently,Prof.Sun Hao's teampublished a paperentitledEncoding physics to learn reaction-diffusion processon Nature Machine Intelligenceunder the title ofEncoding physics to learn reaction-diffusion process.The journal ofNature Machine Intelligencepublishes high-quality original research and reviews in a wide range of topics in machine learning, robotics and AIisinterested in the best research from across the fields of artificial intelligence, machine learning and robotics.The journalalso explores and discusses the significant impact of theaboveacademicfields on other scientific disciplines, as well as on many aspects of our society and industry. This is the first time thatour faculty atGaoling School of Artificial Intelligence has published a paper on this top-notch journal.

The paper proposesa novel network, namely, the physics-encodedrecurrent convolutional neural network (PeRCNN), for modelling and discovery of nonlinear spatiotemporal dynamical systems based on sparse and noisy data.The proposed approach can be applied to a variety of problems regardingsuch asreaction–diffusion processes and other PDE systems, including forward and inverse analysis, data-driven modelling and discovery of PDEs. The prior physics knowledge is forcibly ‘encoded’, which makes the network possess interpretability.In particular, tThe paperalso proposesa deep learning framework whichforcibly encodes a given physics structure in a recurrent convolutional neural network to facilitate learning of the spatiotemporal dynamics in sparse data regimes. Thestudy findsthat thephysics-encoding machine learning approachsuch a computational paradigmshows high accuracy, robustness, interpretability and generalizability.In this paper, Sun’s team demonstrated the capabilities of the proposed network architecture by applying it to various tasks in scientific modelling of spatiotemporal dynamics such as reaction–diffusion processes.

Generally speaking, predicting the evolution of Modelling complex spatiotemporal dynamical systemsis a challenging task for many cases owing to insufficient prior knowledge and a lack of explicit PDE formulation for describing the nonlinear process of the system variables.Common machine learning methodsneed to rely on a large amount of training data, a method that always leads topossessingproblems such as poor interpretability, weak generalization, and uncontrollablemodelingerrors. WithThanks to therecentdevelopment ofdata-driven approaches, it is possible to learnspatiotemporal dynamicsfrom measurement data while adding prior physics knowledge. However, existing physics-informed machine learning paradigms impose physicals lawsor governing equationsthrough soft penalty constraints, and the solution quality largely depends on a trial-and-error proper setting of hyperparameters.Therefore, it is of great necessity to develop new knowledge-embedded learning model to learn the underlying spatiotemporal dynamics from data.

To this end, the paper proposes a novel network, namely, the physics-encoded recurrent convolutional neural network (PeRCNNmodel), as shown in Fig. 1(see the paper).One major advantage of the PeRCNN is that the prior physics knowledge can be encoded into the network, which guarantees that the resulting network strictly obeys given physics (for example, ICs and BCs, general PDE structure, and known terms in PDEs). This brings distinct benefits for improving the convergence of training and accuracy of the model.By encoding given physical structure withintotherecurrent convolutional neural network, the performance ofmodelingspatiotemporal dynamical systems based on sparse and noisy data was improved.

The paper shows with extensive numericalexperiments how the proposed approach can be applied tomodel and discovera variety ofproblems regarding reaction–diffusion processes and other PDE systems. By comparingthe proposed approach with some existing methods (oraka.,baselinemodelss)for governing PDE discovery, including PDE-FIND 16, sparse regression coupled with an FCNN or PDE-Net 5, the team found that the approach outperforms (if not performs as good as) the considered baselines consistently under different noise levels and data richness.

In addition, the paper integrates the sparse regression technique with ourthePeRCNNmodel fordiscoveryofingtheexplicit form ofPDEs. The proposed framework for the PDEdiscovery is presented in Fig. 5(see the paper)with the example of 2D GS RD equation. The entire procedure consists of three steps, data reconstruction, sparse regression and fine-tuning of coefficients.

Although the paper demonstrates the effectiveness of the PeRCNN on various RD systems, the model is in theory applicable to other types of spatiotemporal PDEs(for example, the 2D Burgers’ equation with the convection term shown in Supplementary Note F.4, and the Kolmogorov turbulent flows at Reynolds number 1,000, discussed in Supplementary Note J).

This research isexpected to boost the development of data-driven modeling of complex spatiotemporal dynamical systems, providing scientists and engineers with more powerful tools to understand and predict natural and engineering phenomena. This approach, which combines deep learning and aourprior physics knowledge, is expected tobe applicabletedin multiple disciplines and play an important role, including fluid mechanics, biochemistry, environmental science, engineering, materials science, and more. Let's wait andWe look forward toseeingthe further development and application of this new approach, which willmayreveal more mysteries about complex spatiotemporal dynamical systems and bring new breakthroughs for future scientific and technological development.

About the author:

Hao Sun is an Associate Professor with Tenure in the Gaoling School of Artificial Intelligence at Renmin University of China (RUC). He is also a Research Affiliate at MIT and an Affiliate Professor at Northeastern University (Boston, MA). He received his Ph.D. in Engineering Mechanics from Columbia University in 2014 and did his Postdoc training at MIT during 2014-2017. He was a Tenure-Track Assistant Professor at the University of Pittsburgh (2017-2018) and at Northeastern University (2018-2021) before joining RUC.His research interests lie inAI for Science as well as AI-enabled Scientific Computing. He has published over 60 peer-reviewed articles in top-tiered journals (e.g., Nature Machine Intelligence, Nature Communications) and top computer science conferences (e.g., ICLR, NeurIPS).His research interests lie in "mathematical and physical foundation of AI and its interdisciplinary applications".Professor.Sun has publishedover 60 peer-reviewed articles in top-tiered journals, including Nature Machine Intelligence、Nature Communications.Dr. Sun has been a PI or a major Co-PI for dozens of research projects (with a total funding level of 30 Million RMB) sponsored by National Natural Science Foundation of China, National Science Foundation (US), Beijing Natural Science Foundation, etc.Thanks to his academic contributions,Dr. Sun was named to the prestigious 2018 Forbes“30 Under 30”: Science, a list of the world’s most inspiring young innovators, bright rising stars and the leaders of tomorrow who are transforming the world. He also received the 2019 Top Ten Outstanding Chinese American Youth Award and the 2022 DeepTech Intelligent Computing Innovator Award of China.Sun was named to the prestigious 2018 Forbes "30 Under 30": Science, a list of the world’smost inspiring young innovators, bright rising stars and the leaders of tomorrow who are transforming the world, in recognition of his accomplishment in applied machine learning and infrastructure informatics. He also received the 2019 Top Ten Outstanding Chinese American Youth Award.DeepTech. Sun wasalsoselected as one of DeepTech's "2022 China Intelligent Computing Technology Innovators”.