{
  "abstract": "We apply physics-informed neural networks (PINNs) to recover the time-time component of static, spherically symmetric black-hole exterior metrics from a reduced ordinary differential equation (ODE). Inspired by recent work on solving Einstein field equations with deep learning~\\cite{Li2023}, we encode the vacuum/charged exterior through a residual loss and an asymptotic boundary constraint---not by embedding analytic metric terms such as $2M/r$ directly into the network output. Unlike the distributed PINN (DPINN) strategy of Ref.~\\cite{Li2023}, which partitions the radial domain into subdomains with separate networks, we employ a \\emph{unified} fully connected network over the entire interval $[r_{\\min},r_{\\max}]$, avoiding spurious jumps at subdomain interfaces. Moreover, whereas Ref.~\\cite{Li2023} restricts training to $r\\in(10,300)$---well outside the steep $1/r$ and $1/r^2$ curvature of the inner exterior---we begin at $r_{\\min}=10^{-2}M$, spanning three decades in radius and covering the strongly varying region that DPINNs sidestep by domain truncation. Holding the mass fixed at $M=1$, we sweep electric charge $Q\\in\\{0,0.5,1.0,1.1\\}$ at $Λ=0$ and sweep $Λ\\in\\{0,0.1,-0.1\\}$ at $Q=0.5$. All configurations achieve relative $\\mathcal{L}_2$ errors below $6\\%$ against the analytic reference. For the representative case $Q=0.5$, $Λ=0.1$, three independent trainings with fixed seeds yield relative errors of $3.89\\%$, $1.08\\%$, and $2.76\\%$ (mean $2.58\\%$, standard deviation $1.41\\%$), demonstrating robustness of the mesh-free approach without labeled field data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30332",
  "author_total": 3,
  "authors": [
    {
      "name": "Huan Jin"
    },
    {
      "name": "Fei Wu"
    },
    {
      "name": "Fei Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rzfz9fvkjzcv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30332",
  "fulltext_url": null,
  "id": "0rzfz9fvkjzcv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30332",
  "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",
      "gr-qc"
    ],
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7214588971",
  "pdf_url": "https://arxiv.org/pdf/2609.30332",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rzfz9fvkjzcv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2609.30332",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30332"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps",
  "title_text": "Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:52Z",
  "url": "https://research.draconitic.com/paper/0rzfz9fvkjzcv",
  "venue": "arXiv",
  "versions": []
}