Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study
- 1Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan
Abstract
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%.
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BibTeX
@article{naich2026neural,
title = {{Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study}},
author = {Saleem Ahmed Naich and Sidrah Ahmed},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214034},
url = {https://www.mdpi.com/2227-7390/14/21/4034},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study
AU - Saleem Ahmed Naich
AU - Sidrah Ahmed
PY - 2026
DA - 2026/11/04
JO - Mathematics
DO - 10.3390/math14214034
UR - https://www.mdpi.com/2227-7390/14/21/4034
AB - 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%.
SN - 2227-7390
ER -