{
  "abstract": "We give checkable positive-definiteness criteria for the limiting neural tangent kernel (NTK) and high-probability finite-width gradient-descent guarantees for scalar- or vector-valued physics-informed neural networks (PINNs) with linear constraints. The constraints may be strong-form differential rows of any fixed finite order, including coupled systems, with any linear initial or boundary conditions; weak-form residual and boundary functionals; or finite-measure nonlocal observations such as integral, nonlocal-diffusion, and Dirac-data rows, all in any dimension. For each class, positive definiteness of the limiting NTK is equivalent to a rank condition on a finite coefficient, functional, or moment matrix of the fixed design: a certificate computed before training that detects structural zero modes. The model hypotheses are those of an ordinary two-layer PINN, a smooth nonpolynomial activation with bounded symmetric initialization, and are met by standard choices such as $\\tanh$ with uniform initialization. Given a certificate, explicit width and step-size conditions ensure, with high probability, that the empirical NTK retains at least half the limiting gap and that full-batch gradient descent decreases the training loss geometrically at every iteration. Controlled experiments check the certificates and the finite-width mechanism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04357",
  "author_total": 1,
  "authors": [
    {
      "name": "Zifan Lyu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7176xeagpd2ka.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04357",
  "fulltext_url": null,
  "id": "7176xeagpd2ka",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04357",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7220661582",
  "pdf_url": "https://arxiv.org/pdf/2610.04357",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7176xeagpd2ka.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04357",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04357"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Checkable NTK Positivity and Finite-Width Gradient Descent for Scalar- and Vector-Valued PINNs with Strong-Form, Weak-Form, and Nonlocal Linear Constraints",
  "title_text": "Checkable NTK Positivity and Finite-Width Gradient Descent for Scalar- and Vector-Valued PINNs with Strong-Form, Weak-Form, and Nonlocal Linear Constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7176xeagpd2ka",
  "venue": "arXiv",
  "versions": []
}