Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems
Year: 2023
Author: Haoxiang Huang, Vigor Yang, Yingjie Liu
Communications in Computational Physics, Vol. 34 (2023), Iss. 2 : pp. 290–317
Abstract
A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state, including shock waves and contact discontinuities. In conventional approaches, a low-cost solution patch is usually used as the input to a neural network for predicting the high-fidelity solution patch. With that technique, however, there is no way to distinguish a smeared discontinuity from a smooth solution with large gradient in the input, and the two almost identical inputs correspond to two fundamentally different high-fidelity solution patches in training and predicting. To circumvent this difficulty, we use local patches of two low-cost numerical solutions of the conservation laws in a converging sequence as the input to a neural network. The neural network then makes a correct prediction by identifying whether the solution contains discontinuities or just smooth variations with large gradients, because the former becomes increasingly steep in a converging sequence in the input, and the latter does not. The inputs can be computed from low-cost numerical schemes with coarse resolution, in a local domain of dependence of a space-time location where the prediction is to be made. Despite smeared input solutions, the output provides sharp approximations of solutions containing shock waves and contact discontinuities. The method works effectively not only for regions with discontinuities, but also for smooth regions of the solution. It is efficient to implement, once trained, and has broader applications for different types of differential equations.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2022-0285
Communications in Computational Physics, Vol. 34 (2023), Iss. 2 : pp. 290–317
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Neural network neural networks with local converging inputs physics informed machine learning conservation laws differential equation multi-fidelity optimization.