GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws

TL;DR

GoRINNs are a novel hybrid neural network approach that integrates high-resolution Godunov schemes to accurately solve inverse problems in hyperbolic conservation laws, effectively handling shocks and discontinuities.

Contribution

This work introduces GoRINNs, a new physics-informed neural network framework that learns conservation law closures using numerical-assisted shallow networks, ensuring explainability and conservation.

Findings

High accuracy in smooth and discontinuous regions

Effective handling of shock waves and contact discontinuities

Validated on four benchmark hyperbolic PDE problems

Abstract

We present GoRINNs: numerical analysis-informed (shallow) neural networks for the solution of inverse problems of non-linear systems of conservation laws. GoRINNs is a hybrid/blended machine learning scheme based on high-resolution Godunov schemes for the solution of the Riemann problem in hyperbolic Partial Differential Equations (PDEs). In contrast to other existing machine learning methods that learn the numerical fluxes or just parameters of conservative Finite Volume methods, relying on deep neural networks (that may lead to poor approximations due to the computational complexity involved in their training), GoRINNs learn the closures of the conservation laws per se based on "intelligently" numerical-assisted shallow neural networks. Due to their structure, in particular, GoRINNs provide explainable, conservative schemes, that solve the inverse problem for hyperbolic PDEs, on the basis of approximate Riemann solvers that satisfy the Rankine-Hugoniot condition. The performance of GoRINNs is assessed via four benchmark problems, namely the Burgers', the Shallow Water, the Lighthill-Whitham-Richards and the Payne-Whitham traffic flow models. The solution profiles of these PDEs exhibit shock waves, rarefactions and/or contact discontinuities at finite times. We demonstrate that GoRINNs provide a very high accuracy both in the smooth and discontinuous regions.