Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs

TL;DR

This paper introduces a neural network approach for solving scalar hyperbolic conservation laws, replacing traditional flux methods with trainable networks, applicable in supervised and unsupervised settings, and demonstrating superior performance in experiments.

Contribution

The paper proposes a novel neural network-based scheme that preserves conservation laws and can be trained with synthetic data or weak formulations, outperforming classical numerical methods.

Findings

Outperforms Godunov, WENO, and Discontinuous Galerkin schemes in experiments.

Theoretically guarantees arbitrarily good performance with bounds on network size.

Successfully applied to traffic prediction using highway data.

Abstract

We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative structure of the scheme. The model can be trained both in a supervised setting with efficiently generated synthetic data or in an unsupervised manner, leveraging the weak formulation of the partial differential equation. We provide theoretical results that our model can perform arbitrarily well, and provide associated upper bounds on neural network size. Extensive experiments demonstrate that our method often outperforms efficient schemes such as Godunov's scheme, WENO, and Discontinuous Galerkin for comparable computational budgets. Finally, we demonstrate the effectiveness of our method on a traffic prediction task, leveraging field experimental highway data from the Berkeley DeepDrive drone dataset.