PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network

By Zichao Long, Yiping Lu, Bin Dong. Journal of Computational Physics, 2018

645 citations recorded by the source corpus.

Abstract 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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3ad8760d9e86a7bed1ad230eaa7bae3ab799457c

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Citations in the corpus

Data-driven Discovery of Partial Differential Equations for Multiple-Physics Electromagnetic Problem (2019, methodology): “Long et al. (2018) [8] upgraded their network by imposing appropriate constraints on ﬁlters and using a newly designed symbolic neural network to express the analytical form of function clearly.”
Disentangling Physical Dynamics From Unknown Factors for Unsupervised Video Prediction (2020, methodology): “PDE-Net [35, 34] discretizes a broad class of PDEs by approximating partial derivatives with convolutions.”
Learning to Optimize: A Primer and A Benchmark (2021, methodology): “Beyond first-order algorithms, unrolling is also applied to second-order (Newton or Quasi-Newton) methods [236; 120], Differential Equations [131; 132; 259] and Interior Point method [58; 23], and Frank-Wolfe algorithm [125; 168].”
Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing (2019, background): “[74] impose more constraints on the learnable filters and introduce a symbolic neural network to approximate the unknown response function.”