Designing Neural Networks for Hyperbolic Conservation Laws

TL;DR

This paper introduces a conservative form neural network (CFN) that effectively learns hyperbolic conservation law dynamics, accurately capturing shock speeds and maintaining robustness in noisy, sparse data scenarios.

Contribution

The paper develops a novel conservative form neural network that explicitly learns flux functions, improving prediction accuracy and physical fidelity over standard neural networks.

Findings

CFN outperforms non-conservative networks in accuracy.

CFN accurately captures shock propagation speeds.

CFN remains robust with noisy and sparse data.

Abstract

We propose a new data-driven method to learn the dynamics of an unknown hyperbolic system of conservation laws using deep neural networks. Inspired by classical methods in numerical conservation laws, we develop a new conservative form network (CFN) in which the network learns the flux function of the unknown system. Our numerical examples demonstrate that the CFN yields significantly better prediction accuracy than what is obtained using a standard non-conservative form network, even when it is enhanced with constraints to promote conservation. In particular, solutions obtained using the CFN consistently capture the correct shock propagation speed without introducing non-physical oscillations into the solution. They are furthermore robust to noisy and sparse observation environments.