Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part II: 2D Problems

TL;DR

This paper extends neural network methods with local converging inputs to 2D conservation laws, demonstrating accurate predictions of shocks, contacts, and smooth regions in two-dimensional Euler systems.

Contribution

It introduces a 2D extension of NNLCI methods for conservation laws, showing their effectiveness and ease of training in higher dimensions.

Findings

Methods accurately predict shocks and contacts in 2D

Neural networks perform well despite smeared local inputs

Approach is efficient and easy to train

Abstract

In our prior work [arXiv:2109.09316], neural network methods with inputs based on domain of dependence and a converging sequence were introduced for solving one dimensional conservation laws, in particular the Euler systems. To predict a high-fidelity solution at a given space-time location, two solutions of a conservation law from a converging sequence, computed from low-cost numerical schemes, and in a local domain of dependence of the space-time location, serve as the input of a neural network. In the present work, we extend the methods to two dimensional Euler systems and introduce variations. Numerical results demonstrate that the methods not only work very well in one dimension [arXiv:2109.09316], but also perform well in two dimensions. Despite smeared local input data, the neural network methods are able to predict shocks, contacts, and smooth regions of the solution accurately. The neural network methods are efficient and relatively easy to train because they are local solvers.