{
  "abstract": "This paper presents a unified error analysis of randomized neural networks (RaNNs) for elliptic interface problems. We first derive an integral representation of Barron functions using the standard $\\tanh$ activation, which yields a $C^2$-approximation bound valid for any hidden-parameter distribution supported on a cube with strictly positive density. We then establish a quantitative $\\mathcal{O}(n^{-1})$ generalization bound for the composite loss that enforces the partial differential equation, boundary, and interface conditions simultaneously, where $n$ denotes the number of training samples. Combining these results with the optimization guarantees for RaNNs, we obtain an a priori $L^2$ error estimate for the neural network solution in the presence of interface discontinuities. Numerical experiments are presented to validate the theoretical results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26534",
  "author_total": 2,
  "authors": [
    {
      "name": "Jun Hu"
    },
    {
      "name": "Sidi Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70k5x0mnmm8m2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26534",
  "fulltext_url": null,
  "id": "70k5x0mnmm8m2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26534",
  "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",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7214106969",
  "pdf_url": "https://arxiv.org/pdf/2609.26534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70k5x0mnmm8m2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks",
  "title_text": "Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/70k5x0mnmm8m2",
  "venue": "arXiv",
  "versions": []
}