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Encoding physics to learn reaction–diffusion processes

Nature Machine Intelligence volume 5, pages 765–779 (2023)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Modelling complex spatiotemporal dynamical systems, such as reaction–diffusion processes, which can be found in many fundamental dynamical effects in various disciplines, has largely relied on finding the underlying partial differential equations (PDEs). However, predicting the evolution of these systems remains 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. With recent data-driven approaches, it is possible to learn from measurement data while adding prior physics knowledge. However, existing physics-informed machine learning paradigms impose physics laws through soft penalty constraints, and the solution quality largely depends on a trial-and-error proper setting of hyperparameters. Here we propose a deep learning framework that forcibly encodes a given physics structure in a recurrent convolutional neural network to facilitate learning of the spatiotemporal dynamics in sparse data regimes. We show with extensive numerical experiments how the proposed approach can be applied to a variety of problems regarding reaction–diffusion processes and other PDE systems, including forward and inverse analysis, data-driven modelling and discovery of PDEs. We find that our physics-encoding machine learning approach shows high accuracy, robustness, interpretability and generalizability.

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Fig. 1: Schematic architecture of the PeRCNN.
Fig. 2: Error propagation curve and predicted snapshots by PeRCNN, ConvLSTM and PINN on various RD systems.
Fig. 3: Snapshots of the measurement data employed in the experiment and the identified coefficients.
Fig. 4: Error propagation curve of the prediction and the extrapolated snapshots from each data-driven model compared with the reference solution.
Fig. 5: Error propagation curve and the snapshots of the inference result.
Fig. 6: Flowchart of the discovery of governing PDEs.

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Data availability

All the used datasets in this study are available in the Zenodo repository71, the Gitee repository at https://gitee.com/chengzrz/percnn and the GitHub repository at https://github.com/isds-neu/PeRCNN.

Code availability

All the source codes to reproduce the results in this study are available in the Zenodo repository71, the Gitee repository at https://gitee.com/chengzrz/percnn and the GitHub repository at https://github.com/isds-neu/PeRCNN.

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Acknowledgements

The work is supported by the National Natural Science Foundation of China (no. 92270118 and no. 62276269), the Beijing Natural Science Foundation (no. 1232009), the National Key R&D Program of China (no. 2021ZD0110400) and the Beijing Outstanding Young Scientist Program (no. BJJWZYJH012019100020098). We also acknowledge the support by the Huawei MindSpore platform. Y.L. and H.S. acknowledge the support from the Fundamental Research Funds for the Central Universities. C.R. acknowledges the sponsorship of visiting research by H.S. at Renmin University of China.

Author information

Author notes

  1. These authors contributed equally: Chengping Rao, Pu Ren.

Authors and Affiliations

  1. Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China

    Chengping Rao, Qi Wang & Hao Sun

  2. Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Chengping Rao

  3. Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA

    Pu Ren

  4. Department of Civil and Environmental Engineering, MIT, Cambridge, MA, USA

    Oral Buyukozturk

  5. Beijing Key Laboratory of Big Data Management and Analysis Methods, Beijing, China

    Hao Sun

  6. School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China

    Yang Liu

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Contributions

C.R., H.S. and Y.L. contributed to the ideation and design of the research. C.R., P.R. and Q.W. performed the research. C.R., P.R., O.B., H.S. and Y.L. wrote the paper.

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Correspondence to Hao Sun or Yang Liu.

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Nature Machine Intelligence thanks Ilias Bilionis and Lu Lu for their contribution to the peer review of this work.

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Extended data

Extended Data Table 1 Computational parameters for datasets generation
Full size table
Extended Data Table 2 Summary of the coefficient identification results for 2D Gray–Scott reaction–diffusion system
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Extended Data Table 3 Discovered PDEs from the measurement data under various noise levels compared with the ground truth
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Supplementary information

Supplementary Information

Supplementary Figs. 1–24 and Tables 1–19.

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Rao, C., Ren, P., Wang, Q. et al. Encoding physics to learn reaction–diffusion processes. Nat Mach Intell 5, 765–779 (2023). https://doi.org/10.1038/s42256-023-00685-7

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