PDE-Net: Learning PDEs from Data. In this paper, we present an initial attempt to learn evolution PDEs from data. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. The basic idea of the proposed PDE-Net is to learn differential operators by learning convolution kernels (filters), and apply neural networks or other machine learning methods to approximate the unknown nonlinear responses. Comparing with existing approaches, which either assume the form of the nonlinear response is known or fix certain finite difference approximations of differential operators, our approach has the most flexibility by learning both differential operators and the nonlinear responses. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). We also discuss relations of the PDE-Net with some existing networks in computer vision such as Network-In-Network (NIN) and Residual Neural Network (ResNet). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

References in zbMATH (referenced in 63 articles , 1 standard article )

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  1. Cui, Tao; Wang, Ziming; Xiang, Xueshuang: An efficient neural network method with plane wave activation functions for solving Helmholtz equation (2022)
  2. Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas: Optimization with learning-informed differential equation constraints and its applications (2022)
  3. Li, Yixin; Hu, Xianliang: Artificial neural network approximations of Cauchy inverse problem for linear PDEs (2022)
  4. Ren, Pu; Rao, Chengping; Liu, Yang; Wang, Jian-Xun; Sun, Hao: PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs (2022)
  5. Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel: Deep splitting method for parabolic PDEs (2021)
  6. Bezgin, Deniz A.; Schmidt, Steffen J.; Adams, Nikolaus A.: A data-driven physics-informed finite-volume scheme for nonclassical undercompressive shocks (2021)
  7. Bocquet, Marc; Farchi, Alban; Malartic, Quentin: Online learning of both state and dynamics using ensemble Kalman filters (2021)
  8. Chen, Zhen; Xiu, Dongbin: On generalized residual network for deep learning of unknown dynamical systems (2021)
  9. Darbon, Jérôme; Meng, Tingwei: On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations (2021)
  10. Forgione, Marco; Piga, Dario: Continuous-time system identification with neural networks: model structures and fitting criteria (2021)
  11. Glasner, Karl: Optimization algorithms for parameter identification in parabolic partial differential equations (2021)
  12. Kang, Sung Ha; Liao, Wenjing; Liu, Yingjie: IDENT: identifying differential equations with numerical time evolution (2021)
  13. Keller, Rachael T.; Du, Qiang: Discovery of dynamics using linear multistep methods (2021)
  14. Khoo, Yuehaw; Lu, Jianfeng; Ying, Lexing: Solving parametric PDE problems with artificial neural networks (2021)
  15. Li, Wei; Bazant, Martin Z.; Zhu, Juner: A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches (2021)
  16. Lu, Fei; Maggioni, Mauro; Tang, Sui: Learning interaction kernels in heterogeneous systems of agents from multiple trajectories (2021)
  17. Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em: DeepXDE: a deep learning library for solving differential equations (2021)
  18. Lu, Peter; Lermusiaux, Pierre F. J.: Bayesian learning of stochastic dynamical models (2021)
  19. Maulik, Romit; Botsas, Themistoklis; Ramachandra, Nesar; Mason, Lachlan R.; Pan, Indranil: Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation (2021)
  20. Meidani, Kazem; Barati Farimani, Amir: Data-driven identification of 2D partial differential equations using extracted physical features (2021)

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