PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network

Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural netw...

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Veröffentlicht in:Journal of computational physics Jg. 399; S. 108925
Hauptverfasser: Long, Zichao, Lu, Yiping, Dong, Bin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge Elsevier Inc 15.12.2019
Elsevier Science Ltd
Schlagworte:
Artificial neural networks
Computational physics
Convolutional neural network
Dynamic system
Empirical equations
Learning
Mathematical models
Multilayers
Neural networks
Nonlinear response
Operators (mathematics)
Partial differential equations
Response functions
Symbolic neural network
Time dependence
Dynamic system
Partial differential equations
Convolutional neural network
Symbolic neural network
ISSN:0021-9991, 1090-2716
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Zusammenfassung:Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work [1] where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 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. •The proposal of a numeric-symbolic hybrid deep network to recover PDEs from observed dynamic data.•The symbolic network is able to recover concise analytic form of the hidden PDE model.•Our approach only requires minor prior knowledge on the mechanism of the observed dynamic data.•The network can perform accurate long-term prediction without re-training for new initial conditions.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2019.108925