GoRINNs: Godunov-Riemann informed neural networks for learning hyperbolic conservation laws

Abstract

We introduce Godunov-Riemann Informed Neural Networks (GoRINNs), a hybrid framework that combines shallow neural networks with high-resolution finite volume (FV) Godunov-type schemes to solve inverse problems in nonlinear conservation laws. In contrast to other proposed - based on deep neural networks - schemes, that learn numerical fluxes of conservative FV methods or model parameters, GoRINNs directly learn physical flux functions using numerical analysis- informed shallow neural networks, preserving conservation laws while reducing computational complexity. Using second-order accurate schemes with flux limiters and approximate Riemann solvers (satisfying the Rankine-Hugoniot condition), GoRINNs demonstrate high accuracy across benchmark problems such as Burgers', Shallow Water, Lighthill-Whitham-Richards, and Payne-Whitham traffic flow models, showcasing a robust and interpretable approach to integrating machine learning with classical numerical methods. An uncertainty quantification was also conducted by evaluating training performance variability across multiple realizations, each with different randomly sampled datasets and initial parameter values.