{
  "abstract": "For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\\pm 0.450)\\times 10^{-4}), ((2.868\\pm 0.947)\\times 10^{-5}), and ((1.756 \\pm 0.712)\\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35196",
  "author_total": 3,
  "authors": [
    {
      "name": "Yanxin Zhang"
    },
    {
      "name": "Yong Zhang"
    },
    {
      "name": "Houbiao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f81jjpmvajtz7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35196",
  "fulltext_url": null,
  "id": "f81jjpmvajtz7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35196",
  "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.AI"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7214708119",
  "pdf_url": "https://arxiv.org/pdf/2609.35196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f81jjpmvajtz7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "GAC-PINN: Geometry-Adaptive and Constraint-Enhanced Physics-Informed Neural Networks",
  "title_text": "GAC-PINN: Geometry-Adaptive and Constraint-Enhanced Physics-Informed Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/f81jjpmvajtz7",
  "venue": "arXiv",
  "versions": []
}