{
  "abstract": "We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains $Ω\\subset\\mathbb{R}^d$, $d\\ge2$. For each $s>0$, explicit deterministic rules prescribe all hidden parameters independently of the target $u\\in H^s(Ω)$. At frequency resolution $M$, the $N_M\\asymp M^d$ prescribed features span a linear space $V_M$. For every sufficiently large integer $N$, we choose $M\\asymp N^{1/d}$ with $N_M\\le N$ and construct a bounded linear operator $A_N:H^s(Ω)\\to V_M$ such that \\[ \\|u-A_Nu\\|_{H^m(Ω)} \\le C N^{-(s-m)/d}\\|u\\|_{H^s(Ω)} \\] for every $u\\in H^s(Ω)$ and every integer $0\\le m<s$, with $C$ independent of $u$, $N$, and $m$. Sobolev width lower bounds show that this rate is optimal among linear spaces of dimension at most $N$ chosen independently of the target. We use explicit spherical cubature nodes for the directions and Gauss-Legendre nodes or equispaced midpoints for the offsets. We show that the exterior offset margin can shrink to zero as $M\\to\\infty$ while preserving the approximation rate. A margin beyond $\\sup_{x\\inΩ}|x|$ of order $M^{-1}\\log M$ for the Gaussian and $M^{-1}(\\log M)^2$ for the tanh difference suffices to truncate the ridge integral and remove exterior midpoint nodes. Numerical experiments in two and three dimensions examine $L^2$ and $H^1$ approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09338",
  "author_total": 2,
  "authors": [
    {
      "name": "Ran Bi"
    },
    {
      "name": "Weibing Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84jeecp6fdp3t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09338",
  "fulltext_url": null,
  "id": "84jeecp6fdp3t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09338",
  "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",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7221055948",
  "pdf_url": "https://arxiv.org/pdf/2610.09338",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84jeecp6fdp3t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09338",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09338"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation",
  "title_text": "Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/84jeecp6fdp3t",
  "venue": "arXiv",
  "versions": []
}