Implicit Form Neural Network for Learning Scalar Hyperbolic Conservation Laws

Xiaoping Zhang, Tao Cheng, Lili Ju
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:1082-1098, 2022.

Abstract

Conservation laws are critical to our understanding of the physical world. Numerical solution of hyperbolic conservation laws has been a very important and challenging task with some difficul- ties such as nonphysical solutions, solution discontinuities and shock wave capturing, and a large amount of conventional numerical solvers of finite difference, finite volume and discontinuous Galerkin types have been developed in the past decades. Along with the booming and great suc- cess of deep learning in computer vision and natural language processing, many excellent works also have emerged for their application to PDE related scientific problems in recent years. In this paper, we propose a deep learning method for solving scalar hyperbolic conservation laws. More specifically, we design a neural network model in the unsupervised fashion, called “IFNN”, based on a special implicit form for the solution of the problem. The proposed IFNN does not directly process the target PDEs, instead it deals with the nonlinear equations characterizing implicitly the solution of the PDEs. Numerical experiments and performance comparisons of our IFNN with the well-known PINN are performed on two well-known problems under various boundary conditions, the inviscid Burgers’ equation and the Lighthill-Whitham-Richards (LWR) model for traffic flow, and the results show that IFNN can significantly outperform PINN in capturing the shock waves.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-zhang22a, title = {Implicit Form Neural Network for Learning Scalar Hyperbolic Conservation Laws}, author = {Zhang, Xiaoping and Cheng, Tao and Ju, Lili}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {1082--1098}, year = {2022}, editor = {Bruna, Joan and Hesthaven, Jan and Zdeborova, Lenka}, volume = {145}, series = {Proceedings of Machine Learning Research}, month = {16--19 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v145/zhang22a/zhang22a.pdf}, url = {https://proceedings.mlr.press/v145/zhang22a.html}, abstract = { Conservation laws are critical to our understanding of the physical world. Numerical solution of hyperbolic conservation laws has been a very important and challenging task with some difficul- ties such as nonphysical solutions, solution discontinuities and shock wave capturing, and a large amount of conventional numerical solvers of finite difference, finite volume and discontinuous Galerkin types have been developed in the past decades. Along with the booming and great suc- cess of deep learning in computer vision and natural language processing, many excellent works also have emerged for their application to PDE related scientific problems in recent years. In this paper, we propose a deep learning method for solving scalar hyperbolic conservation laws. More specifically, we design a neural network model in the unsupervised fashion, called “IFNN”, based on a special implicit form for the solution of the problem. The proposed IFNN does not directly process the target PDEs, instead it deals with the nonlinear equations characterizing implicitly the solution of the PDEs. Numerical experiments and performance comparisons of our IFNN with the well-known PINN are performed on two well-known problems under various boundary conditions, the inviscid Burgers’ equation and the Lighthill-Whitham-Richards (LWR) model for traffic flow, and the results show that IFNN can significantly outperform PINN in capturing the shock waves. } }
Endnote
%0 Conference Paper %T Implicit Form Neural Network for Learning Scalar Hyperbolic Conservation Laws %A Xiaoping Zhang %A Tao Cheng %A Lili Ju %B Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2022 %E Joan Bruna %E Jan Hesthaven %E Lenka Zdeborova %F pmlr-v145-zhang22a %I PMLR %P 1082--1098 %U https://proceedings.mlr.press/v145/zhang22a.html %V 145 %X Conservation laws are critical to our understanding of the physical world. Numerical solution of hyperbolic conservation laws has been a very important and challenging task with some difficul- ties such as nonphysical solutions, solution discontinuities and shock wave capturing, and a large amount of conventional numerical solvers of finite difference, finite volume and discontinuous Galerkin types have been developed in the past decades. Along with the booming and great suc- cess of deep learning in computer vision and natural language processing, many excellent works also have emerged for their application to PDE related scientific problems in recent years. In this paper, we propose a deep learning method for solving scalar hyperbolic conservation laws. More specifically, we design a neural network model in the unsupervised fashion, called “IFNN”, based on a special implicit form for the solution of the problem. The proposed IFNN does not directly process the target PDEs, instead it deals with the nonlinear equations characterizing implicitly the solution of the PDEs. Numerical experiments and performance comparisons of our IFNN with the well-known PINN are performed on two well-known problems under various boundary conditions, the inviscid Burgers’ equation and the Lighthill-Whitham-Richards (LWR) model for traffic flow, and the results show that IFNN can significantly outperform PINN in capturing the shock waves.
APA
Zhang, X., Cheng, T. & Ju, L.. (2022). Implicit Form Neural Network for Learning Scalar Hyperbolic Conservation Laws. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:1082-1098 Available from https://proceedings.mlr.press/v145/zhang22a.html.

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