@Article{CiCP-34-290,
author = {Huang , HaoxiangYang , Vigor and Liu , Yingjie},
title = {Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems},
journal = {Communications in Computational Physics},
year = {2023},
volume = {34},
number = {2},
pages = {290--317},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2022-0285},
url = {http://global-sci.org/intro/article_detail/cicp/21970.html}
}