Stability and Convergence of Neural Riemann Solvers in Numerical Schemes

Establish rigorous stability and convergence results for finite-volume Godunov-type numerical schemes that employ neural network based Riemann solvers, including the hard-constrained neural Riemann solver proposed in this work, when used to compute interface fluxes for systems of conservation laws such as the shallow water equations and the compressible Euler equations with an ideal-gas equation of state.

Background

The paper introduces a hard-constrained neural Riemann solver (HCNRS) that enforces positivity, consistency, mirror symmetry, Galilean invariance, and scaling invariance by construction, and demonstrates strong empirical agreement with exact Riemann solvers on shallow water and Euler test cases.

While numerical experiments show that HCNRS preserves key properties (e.g., well-balancedness and symmetry) and closely matches exact solutions, the work does not provide formal theoretical guarantees about the behavior of neural Riemann solvers when embedded in finite-volume schemes. Classical analyses for Godunov-type methods rely on properties of exact or specific approximate solvers, and analogous proofs for neural surrogates are not yet established.

The authors explicitly state that theoretical stability and convergence properties for neural Riemann solvers used within numerical schemes are still unresolved, motivating the need for rigorous analysis to complement the empirical evidence presented.