{
  "abstract": "An efficient shallow Ritz method was developed in [8, 9] for one-dimensional linear diffusion and diffusion-reaction problems. In this paper, the method is extended to a nonlinear diffusion-reaction equation with a cubic reaction term: (1) a basic error estimate of the shallow Ritz discretization is established and (2) a reduced block Newton (rBN) method for solving the resulting non-convex optimization problem is developed. Again, the computational cost of the rBN method is $\\mathcal{O}(n)$ for one-dimensional problems, where $n$ is the number of neurons. Combining a convex splitting scheme for time discretization with the efficient shallow Ritz method, we develop a time-stepping neural network method for solving the Allen-Cahn equation. The unconditional energy stability of the time discretization and an error estimate in the $L^2$ norm are established. Numerical experiments demonstrate that the neural network approach yields accurate solutions for singularly perturbed nonlinear diffusion-reaction equations and the Allen-Cahn equation. The method effectively resolves interior and boundary layers without oscillations and significantly outperforms the finite element method on fixed uniform meshes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17553",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiqiang Cai"
    },
    {
      "name": "César Herrera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/av6x4xbnkda1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17553",
  "fulltext_url": null,
  "id": "av6x4xbnkda1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17553",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7221762091",
  "pdf_url": "https://arxiv.org/pdf/2610.17553",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/av6x4xbnkda1c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17553",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17553"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations",
  "title_text": "Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/av6x4xbnkda1c",
  "venue": "arXiv",
  "versions": []
}