{
  "abstract": "A scalar potential landscape provides an intuitive description of the stability, transitions, and global organization of dynamical systems. For non-gradient dynamics, however, constructing global Lyapunov-type functions for nonlinear flows with recurrent structures remains a formidable challenge. Here, we introduce the generalized Lyapunov function (GLF), a scalar potential that is monotonically non-increasing along deterministic trajectories, as a unifying framework for nonequilibrium potentials. Conventional Lyapunov functions, Freidlin--Wentzell quasi-potentials, and potentials arising from Ao-type decompositions naturally emerge as special cases within this framework. Furthermore, we develop a data-free physics-informed neural network (PINN) approach in which the Lyapunov inequality and a weak divergence-scale compatibility condition are directly embedded into the loss function. The framework is demonstrated on diverse systems, including linear dynamics, the Hopf normal form, the van der Pol oscillator, a three-dimensional Hopf-link flow, and symmetric and asymmetric competitive Lotka--Volterra systems. The learned landscapes agree closely with analytical results where available and, more generally, successfully reveal invariant sets as low- or constant-potential structures. In particular, the van der Pol oscillator, Hopf-link flow, and asymmetric Lotka--Volterra system demonstrate that coherent GLFs can be constructed without relying on known closed-form expressions, establishing a versatile route to potential-landscape construction for complex nonlinear non-gradient systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Commun. Theor. Phys. 78 (2026) 125601",
  "arxiv": "2606.14174",
  "author_total": 1,
  "authors": [
    {
      "name": "Z. C. Tu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1qvccbmzmg55.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14174",
  "fulltext_url": null,
  "id": "a1qvccbmzmg55",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14174",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "nlin.CD"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.14174",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1qvccbmzmg55.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:58Z",
      "native_id": "2606.14174",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14174"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Universal Construction of Generalized Lyapunov Functions for Nonlinear Dynamical Systems Using Physics-Informed Neural Networks",
  "title_text": "Universal Construction of Generalized Lyapunov Functions for Nonlinear Dynamical Systems Using Physics-Informed Neural Networks",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:17Z",
  "url": "https://research.draconitic.com/paper/a1qvccbmzmg55",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1088/1572-9494/aeaad5",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}