{
  "abstract": "We develop a framework for constructing and rigorously validating controls for semilinear wave equations with localized interior control. We analyze the continuous energy-space problem and certify exact control for finite-dimensional Galerkin systems. Under explicit smallness and neighborhood conditions, our a posteriori theorem proves the existence of an exact Galerkin control near a computed candidate. It combines a validated terminal-state error, a bounded right inverse of the derivative of the control-to-terminal-state map, and a Lipschitz bound for that derivative. Under additional uniform right-inverse and neighborhood conditions, we also show that Newton correction converges with quadratic decay of terminal-state errors. We use residual estimates and interval arithmetic to enclose the nonlinear trajectory and its linearized controllability Gramian. For the specified eight-mode tanh problem, we rigorously bound a source-response operator to prove contraction of the exact source-frozen iteration. We compare deterministic controls based on the Hilbert Uniqueness Method with neural approximations of temporal modal coefficients. Two Newton corrections reduce the computed terminal-state errors of all six tested neural candidates below $10^{-8}$. For the three certified candidates, we obtain smaller bounds on the distance to an exact Galerkin steering control by refining the validation mesh while keeping the computed controls fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.28029",
  "author_total": 3,
  "authors": [
    {
      "name": "Tan-Phuc Nguyen"
    },
    {
      "name": "Minh-Binh Tran"
    },
    {
      "name": "Son Tu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/es391zphxch6n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.28029",
  "fulltext_url": null,
  "id": "es391zphxch6n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.28029",
  "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",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7166574869",
  "pdf_url": "https://arxiv.org/pdf/2606.28029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/es391zphxch6n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.28029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.28029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations",
  "title_text": "Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/es391zphxch6n",
  "venue": "arXiv",
  "versions": []
}