Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study

We compare three neural approaches to the Riemann problem of the isentropic Cargo–LeRoux model with an exact solver and with classical Godunov-type schemes. The model has a linearly degenerate contact field and a strictly convex closure that is singular at aρ=1. We pose the star-state system as constrained residual minimisation. Without entropy constraints, the minimiser contains an expansion shock on 78.6% of well-conditioned cases; with entropy bounds on each wave, it is correct on all 1237 solvable cases and identifies all 163 vacuum cases. Eliminating the contact relation reduces the problem to a monotone scalar equation whose root exists, uniquely, exactly when no vacuum forms; a bracketed solve gives the same reliability at about 1/116 of the cost. Godunov schemes built on the two solvers agree to 8 × 10−10, need no fallback, and are 1.4–2.8 times more accurate than Rusanov; without entropy constraints, the scheme forms an expansion shock at the sonic point. At a contact, cell averaging breaks the equilibrium of p+q; a contact-consistent update restores it to machine precision on an isolated contact. Over five wave patterns and three seeds, a Mixture-of-Experts physics-informed network matches a first-order Rusanov scheme. A neural finite volume (NFV) scheme fitted to each target is 6–34 times more accurate than Rusanov, but its flux is inconsistent, its error grows 6–19-fold on held-out data, and it produces negative densities near vacuum in one of three seeds. A hybrid scheme’s detector flags every vacuum case, yet its surrogate star states violate the Riemann relations by 18–25%.