{
  "abstract": "We propose a time-stepping minimum-residual (MinRes) framework for transient coercive variational problems, applicable to finite-element and neural-network trial approximations. For the Backward Euler scheme, we measure the residual at each time level in the dual norm induced by the steady test-space norm, scaled by the time step. This choice yields stability estimates controlling a discrete parabolic energy quantity, up to the time-discretization defect. We develop conforming and broken-test formulations and specialize the analysis to diffusion-advection-reaction problems. For finite-dimensional test spaces, we derive fully discrete reliability estimates by supplementing the computable discrete residual with a complementary contribution that accounts for residual components the test space does not resolve. We then extend the framework to neural-network trial classes and obtain computable residual decompositions for conforming and broken polynomial test spaces. Numerical experiments confirm the expected finite-element convergence rates, examine the effect of test-space refinement for a global space-time neural approximation, and demonstrate residual-driven spatial refinement for a transient problem with a moving localized feature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27548",
  "author_total": 2,
  "authors": [
    {
      "name": "Judit Muñoz-Matute"
    },
    {
      "name": "Sergio Rojas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e332fjk6mjtdw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27548",
  "fulltext_url": null,
  "id": "e332fjk6mjtdw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27548",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214150304",
  "pdf_url": "https://arxiv.org/pdf/2609.27548",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e332fjk6mjtdw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27548",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27548"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems",
  "title_text": "Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/e332fjk6mjtdw",
  "venue": "arXiv",
  "versions": []
}