{
  "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%.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Saleem Ahmed Naich"
    },
    {
      "affs": [
        0
      ],
      "name": "Sidrah Ahmed",
      "orcid": "0000-0002-6493-7113"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8chx3yw4w431g.bib",
  "cited_by": 0,
  "doi": "10.3390/math14214034",
  "fulltext_url": null,
  "id": "8chx3yw4w431g",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/4034",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/8chx3yw4w431g.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T00:20:49Z",
      "native_id": "10.3390/math14214034",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/4034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study",
  "title_text": "Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study",
  "type": "peer-reviewed",
  "updated_at": "2026-11-06T00:21:06Z",
  "url": "https://research.draconitic.com/paper/8chx3yw4w431g",
  "venue": "Mathematics",
  "versions": []
}