{
  "abstract": "Residual-trained neural surrogates carry no error guarantee, a limitation for free-boundary problems such as phase change and contact. We derive guaranteed error bounds for neural approximations of parabolic obstacle problems in first-order state--flux--multiplier form. The main result bounds the flux error of the network and the energy error of its feasible projection $\\max\\{v,g\\}$ by the constitutive residual, the initial mismatch, the complementarity defect, and the PDE residual weighted by $c_F^2/α$, without the coercivity and norm-equivalence constants of the first-order formulation. It extends the Apushkinskaya--Repin majorant for the heat operator to general coefficients, free multipliers and a flux bound. A penetration-consistency result shows that neural triples with arbitrarily small bound exist. Outward-rounded cellwise Taylor-model enclosures turn the bound into a finite computation valid on any adaptive partition. The flux and gradient errors of the projected output are certified below $0.27$ and $0.33$ of the solution scale for a one-dimensional Stefan problem and, after adaptive refinement, below one half for all three trained networks on a two-dimensional expanding disk. On an expanding ellipse the gradient bound stays above one half; substitution diagnostics trace the floor of the trained networks to the projected state near the free boundary, and the remaining gap to enclosure overhead. Without a closed-form solution, a validated lower bound replaces the solution norm; there the enclosure is nearly exact, but the trained networks have errors above the solution scale and no useful accuracy is certified. The bounds control a state--flux norm, not the multiplier or the free-boundary location.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24202",
  "author_total": 1,
  "authors": [
    {
      "name": "Davood Damircheli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpddfj3mv4r7d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24202",
  "fulltext_url": null,
  "id": "fpddfj3mv4r7d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24202",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223055883",
  "pdf_url": "https://arxiv.org/pdf/2610.24202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpddfj3mv4r7d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Certified neural approximations for parabolic obstacle problems",
  "title_text": "Certified neural approximations for parabolic obstacle problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fpddfj3mv4r7d",
  "venue": "arXiv",
  "versions": []
}