{
  "abstract": "Accurate learning-based solvers for coupled reaction–diffusion systems should preserve mathematical structure rather than merely reduce pointwise residuals. We develop a structure-preserving deep Galerkin method that combines weak residual minimization with positivity, mass-balance, and energy-dissipation constraints. Adaptive quadrature concentrates test points near fronts and structural defects, while staged optimization stabilizes stiff training. Under coercivity, one-sided reaction monotonicity, and entropy compatibility, we establish an energy stability estimate, a weak-residual a posteriori error bound, and a unified structural reliability inequality. Numerical studies on pattern formation, nonlinear diffusion, and heterogeneous media show lower reported aggregate solution errors and structural drift than conventional deep Galerkin, variational, and physics-informed neural solvers; no inferential significance claim is made because seed-resolved records are not available in the retained archive. The results support invariant-aware variational learning as a reliable framework for nonlinear PDE approximation.",
  "abstract_shown": true,
  "affiliations": [
    "Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119991, Russia",
    "Key Laboratory of Internet Information Retrieval of Hainan Province, Hainan University, Haikou 570228, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Xinrui Li"
    },
    {
      "affs": [
        1
      ],
      "name": "Yuheng Li",
      "orcid": "0009-0003-1849-2728"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/895t9k93q469m.bib",
  "cited_by": 0,
  "doi": "10.3390/math14213890",
  "fulltext_url": null,
  "id": "895t9k93q469m",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/3890",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223708182",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/895t9k93q469m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14213890",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/3890"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3109|Model Reduction and Neural Networks",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems",
  "title_text": "A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/895t9k93q469m",
  "venue": "Mathematics",
  "versions": []
}