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

• Published in: Journal of computational physics 2019-12, Vol.399, p.108925, Article 108925
• Main Authors: Long, Zichao, Lu, Yiping, Dong, Bin
• Format: Article
• Language: English
• Subjects: 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
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.108925

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.