{
  "abstract": "We approximate the solution operator $(a,f)\\mapsto u$ of $-\\operatorname{div}(a\\nabla u)=f$ by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry--Émery bound $\\mathrm{BE}(-K,\\infty)$: an explicit scheme reaches $\\dot H^1$-error $\\varepsilon$ uniformly over coefficients with bounded carré du champ and bounded generator image and over $L^2$ data, with parameters depending only on the coefficient bounds, the accuracy and $K$; on the torus this gives a $C^2$-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity $A^{1+θ}$, with an explicit rate attained by the scheme's product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of $\\dot H^1$ without Galerkin truncation, and synthesises the coupling $\\nabla a\\cdot\\nabla u$ from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20106",
  "author_total": 1,
  "authors": [
    {
      "name": "Dawid Hanrahan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jkf7eqr57saq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20106",
  "fulltext_url": null,
  "id": "6jkf7eqr57saq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20106",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222088946",
  "pdf_url": "https://arxiv.org/pdf/2610.20106",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jkf7eqr57saq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20106",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20106"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products",
  "title_text": "Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/6jkf7eqr57saq",
  "venue": "arXiv",
  "versions": []
}