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 28 articles , 1 standard article )

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  1. Kang, Sung Ha; Liao, Wenjing; Liu, Yingjie: IDENT: identifying differential equations with numerical time evolution (2021)
  2. Keller, Rachael T.; Du, Qiang: Discovery of dynamics using linear multistep methods (2021)
  3. Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em: DeepXDE: a deep learning library for solving differential equations (2021)
  4. Su, Wei-Hung; Chou, Ching-Shan; Xiu, Dongbin: Deep learning of biological models from data: applications to ODE models (2021)
  5. Arridge, S.; Hauptmann, A.: Networks for nonlinear diffusion problems in imaging (2020)
  6. Chen, Zhen; Wu, Kailiang; Xiu, Dongbin: Methods to recover unknown processes in partial differential equations using data (2020)
  7. Darbon, Jérôme; Langlois, Gabriel P.; Meng, Tingwei: Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures (2020)
  8. Fan, Yuwei; Ying, Lexing: Solving electrical impedance tomography with deep learning (2020)
  9. Li, Yingzhou; Lu, Jianfeng; Mao, Anqi: Variational training of neural network approximations of solution maps for physical models (2020)
  10. Magiera, Jim; Ray, Deep; Hesthaven, Jan S.; Rohde, Christian: Constraint-aware neural networks for Riemann problems (2020)
  11. Puligilla, Shivakanth Chary; Jayaraman, Balaji: Assessment of end-to-end and sequential data-driven learning for non-intrusive modeling of fluid flows (2020)
  12. Schaeffer, Hayden; Tran, Giang; Ward, Rachel; Zhang, Linan: Extracting structured dynamical systems using sparse optimization with very few samples (2020)
  13. Uhlmann, Gunther; Wang, Yiran: Convolutional neural networks in phase space and inverse problems (2020)
  14. Wu, Kailiang; Qin, Tong; Xiu, Dongbin: Structure-preserving method for reconstructing unknown Hamiltonian systems from trajectory data (2020)
  15. Yao, Houpu; Gao, Yi; Liu, Yongming: FEA-Net: a physics-guided data-driven model for efficient mechanical response prediction (2020)
  16. Zhang, Hai-Miao; Dong, Bin: A review on deep learning in medical image reconstruction (2020)
  17. Arridge, Simon; Maass, Peter; Öktem, Ozan; Schönlieb, Carola-Bibiane: Solving inverse problems using data-driven models (2019)
  18. Cotter, Andrew; Jiang, Heinrich; Gupta, Maya; Wang, Serena; Narayan, Taman; You, Seungil; Sridharan, Karthik: Optimization with non-differentiable constraints with applications to fairness, recall, churn, and other goals (2019)
  19. Gulian, Mamikon; Raissi, Maziar; Perdikaris, Paris; Karniadakis, George: Machine learning of space-fractional differential equations (2019)
  20. He, Juncai; Xu, Jinchao: Mgnet: A unified framework of multigrid and convolutional neural network (2019)

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