Learning the Exact Flux: Neural Riemann Solvers with Hard Constraints

TL;DR

This paper introduces a neural Riemann solver with hard constraints to accurately and efficiently solve Riemann problems in fluid dynamics, maintaining physical properties and symmetry.

Contribution

It proposes a novel hard-constrained neural Riemann solver that enforces key physical and symmetry constraints, improving accuracy over existing neural and approximate solvers.

Findings

HCNRS accurately reproduces exact Riemann solver solutions.

Unconstrained neural models violate symmetry and conservation.

HCNRS outperforms traditional approximate solvers in benchmark tests.

Abstract

Godunov-type methods, which obtain numerical fluxes through local Riemann problems at cell interfaces, are among the most fundamental and widely used numerical methods in computational fluid dynamics. Exact Riemann solvers faithfully solve the underlying equations, but can be computationally expensive due to the iterative root-finding procedures they often require. Consequently, most practical computations rely on classical approximate Riemann solvers, such as Rusanov and Roe, which trade accuracy for computational speed. Neural networks have recently shown promise as an alternative for approximating exact Riemann solvers, but most existing approaches are data-driven or impose weak constraints. This may result in problems with maintaining balanced states, symmetry breaking, and conservation errors when integrated into a Godunov-type scheme. To address these issues, we propose a hard-constrained neural Riemann solver (HCNRS) and enforce five constraints: positivity, consistency, mirror symmetry, Galilean invariance, and scaling invariance. Numerical experiments are carried out for the shallow water and ideal-gas Euler equations on standard benchmark problems. In the absence of hard constraints, violations of the well-balanced property, mass conservation, and symmetry are observed. Notably, in the Euler implosion problem, the exact Riemann solver with MUSCL-Hancock captures the jet structure well, whereas the Rusanov flux is too diffusive and smears it out. HCNRS accurately reproduces the solution obtained by the exact Riemann solver. In contrast, an unconstrained neural formulation lacks mirror symmetry, which makes the solution depend on the choice of flux normal direction. As a result, the jet is either shifted or lost, along with diagonal symmetry.