{
  "abstract": "We construct computable a posteriori error estimators for parametric families of nonlinear elliptic problems by measuring the residual of a variational formulation in the dual norm of a closed subspace of the test space. A single abstract theorem shows that the resulting estimator is bounded above and below by the error in the trial norm, i.e., it is a variationally correct loss function. Two instances are covered. The first considers an $L_2$-like test space, yielding a nonlinear first-order system least-squares loss function that certifies arbitrary, possibly meshless, approximations. The second breaks the test space, yielding a nonlinear discontinuous Petrov-Galerkin loss function computed by element-local problems. All assumptions are verified for quasilinear diffusion, including monotone fluxes and semilinear problems, and a nonlocal Kirchhoff model. Numerical studies train neural networks approximating the parameter-to-solution maps with the latter loss and measure the ratio of loss to squared error.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19296",
  "author_total": 3,
  "authors": [
    {
      "name": "Pablo Cortés Castillo"
    },
    {
      "name": "Wolfgang Dahmen"
    },
    {
      "name": "Jay Gopalakrishnan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/990tvjwgxap0x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19296",
  "fulltext_url": null,
  "id": "990tvjwgxap0x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19296",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7222177003",
  "pdf_url": "https://arxiv.org/pdf/2610.19296",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/990tvjwgxap0x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19296",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19296"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Variationally correct loss functions for nonlinear parametric problems and their neural approximations",
  "title_text": "Variationally correct loss functions for nonlinear parametric problems and their neural approximations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/990tvjwgxap0x",
  "venue": "arXiv",
  "versions": []
}