Volume 29, Issue 3
DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data

Hao Xu, Haibin ChangDongxiao Zhang

Commun. Comput. Phys., 29 (2021), pp. 698-728.

Published online: 2021-01

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  • Abstract

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is to discover unknown physics and corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by the diffusion equation, the convection-diffusion equation, the Burgers equation, and the Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

  • Keywords

Data-driven discovery, machine learning, deep neural network, sparse regression, noisy data.

  • AMS Subject Headings

35E99

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COPYRIGHT: © Global Science Press

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@Article{CiCP-29-698, author = {Xu , Hao and Chang , Haibin and Zhang , Dongxiao}, title = {DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {3}, pages = {698--728}, abstract = {

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is to discover unknown physics and corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by the diffusion equation, the convection-diffusion equation, the Burgers equation, and the Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0142}, url = {http://global-sci.org/intro/article_detail/cicp/18563.html} }
TY - JOUR T1 - DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data AU - Xu , Hao AU - Chang , Haibin AU - Zhang , Dongxiao JO - Communications in Computational Physics VL - 3 SP - 698 EP - 728 PY - 2021 DA - 2021/01 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0142 UR - https://global-sci.org/intro/article_detail/cicp/18563.html KW - Data-driven discovery, machine learning, deep neural network, sparse regression, noisy data. AB -

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is to discover unknown physics and corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by the diffusion equation, the convection-diffusion equation, the Burgers equation, and the Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

Hao Xu, Haibin Chang & Dongxiao Zhang. (2021). DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data. Communications in Computational Physics. 29 (3). 698-728. doi:10.4208/cicp.OA-2020-0142
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