{
  "abstract": "We study the numerical resolution of uniformly elliptic Dirichlet boundary value problems (BVPs) by minimization of the PDE residual over a hypothesis space of Neural Networks made to satisfy exactly the boundary conditions through multiplication by a user-defined \\emph{distance-to-boundary approximation} $ρ$. While an $L^2(\\partialΩ)$ boundary penalty cannot generally guarantee $H^2(Ω)$ error control, we show that exact enforcement can, provided $ρ$ is chosen appropriately. More precisely, we identify a sharp sufficient condition on $ρ$ preserving the regularity needed for $H^2(Ω)$ error control, provide practical constructions of such $ρ$ for general smooth domains $Ω$, and refer to the resulting networks as \\emph{boundary-adapted}. For boundary-adapted Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we then derive explicit high-probability $H^2(Ω)$ convergence rates by combining approximation-theoretic and statistical-learning arguments, making explicit the dependence on $ρ$. Numerical experiments support our theoretical results, showing that appropriate choices of $ρ$ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.19167",
  "author_total": 4,
  "authors": [
    {
      "name": "Nathanael Tepakbong"
    },
    {
      "name": "Jun Fan"
    },
    {
      "name": "Xiang Zhou"
    },
    {
      "name": "Ding-Xuan Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ctt5qqhh05rx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.19167",
  "fulltext_url": null,
  "id": "0ctt5qqhh05rx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.19167",
  "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.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7170073162",
  "pdf_url": "https://arxiv.org/pdf/2607.19167",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ctt5qqhh05rx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2607.19167",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.19167"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "$H^2(Ω)$ A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems",
  "title_text": "H^2(Ω) A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/0ctt5qqhh05rx",
  "venue": "arXiv",
  "versions": []
}