An approximate Riemann solver approach in Physics-Informed Neural Networks for hyperbolic conservation laws

TL;DR

This paper introduces a novel approach combining Riemann solvers with Physics-Informed Neural Networks to better model discontinuities in hyperbolic conservation laws, improving shock detection and resolution.

Contribution

We develop a generalized LLPINN framework that dynamically computes shock speeds and incorporate an approximate Roe Riemann solver within PINNs for enhanced shock capturing.

Findings

Sharper shock transitions compared to traditional methods

Smoother solutions in vortex regions

Effective shock detection in 2D Riemann problems

Abstract

This study enhances the application of Physics-Informed Neural Networks (PINNs) for modeling discontinuous solutions in both hydrodynamics and relativistic hydrodynamics. Conventional PINNs, trained with partial differential equation residuals, frequently face convergence issues and lower accuracy near discontinuities. To address these issues, we build on the recently proposed locally linearized PINNs (LLPINNs), which improve shock detection by modifying the Jacobian matrix resulting from the linearization of the equations, only in regions where the velocity field exhibits strong compression. However, the original LLPINN framework required a priori knowledge of shock velocities, limiting its practical utility. We present a generalized LLPINN method that dynamically computes shock speeds using neighboring states and applies jump conditions through entropy constraints. Additionally, we introduce locally Roe PINNs (LRPINNs), which incorporate an approximate Roe Riemann solver to improve shock resolution and conservation properties across discontinuities. These methods are adapted to two-dimensional Riemann problems by using a divergence-based shock detection combined with dimensional splitting, delivering precise solutions. Compared to a high-order weighted essentially non-oscillatory solver, our method produces sharper shock transitions but smoother solutions in areas with small-scale vortex structures. Future research will aim to improve the resolution of these small-scale features without compromising the model's ability to accurately capture shocks.