{
  "abstract": "We develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles, which are solutions of $$ Q^{\\prime\\prime}=bQ-γQ^{p},\\quad Q>0,\\quad p>1, \\quad b, γ>0, $$ the stationary profile equation for the nonlinear Schrödinger, generalized Korteweg-de Vries, and nonlinear Klein-Gordon equations. Its explicit sech-type solutions serve as exact reference for error computation and benchmark hyperparameter selection. We improve on the standard Adam-trained physics-informed neural network (PINN) baseline by introducing higher-order optimization via Gauss-Newton/Levenberg-Marquardt and L-BFGS refinements, as well as Fourier-feature PINNs, and constrained Deep Ritz (DR) formulations. In DR approaches we introduce two new neural-network formulations, which implement two classical variational characterizations: a Nehari-constrained Deep Ritz method, in which the height is learned by projecting onto the Nehari/Pokhozhaev manifold, and a Weinstein Deep Ritz method, in which a normalized profile minimizes the Weinstein quotient. The Deep Ritz formulations perform best overall, in particular, the Weinstein variant gives the best error-runtime balance. The neural-network based solvers remain less accurate and efficient than classical Petviashvili iteration in 1D, however, the current study provides a proof of concept: the mesh-free formulation, non-reliance on exact profiles or the related features make them promising in higher dimensions and for more complicated nonlinearities and dispersive operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27500",
  "author_total": 3,
  "authors": [
    {
      "name": "Chandler Haight"
    },
    {
      "name": "Alex D Rodriguez"
    },
    {
      "name": "Svetlana Roudenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dre7hjx4tj8qe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27500",
  "fulltext_url": null,
  "id": "dre7hjx4tj8qe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27500",
  "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.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "nlin.PS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.27500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dre7hjx4tj8qe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.27500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27500"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.PS",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods",
  "title_text": "Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/dre7hjx4tj8qe",
  "venue": "arXiv",
  "versions": []
}