{
  "abstract": "Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: IEEE Conference on Decision and Control (CDC), 2026",
  "arxiv": "2609.18894",
  "author_total": 4,
  "authors": [
    {
      "name": "Amartya Mukherjee"
    },
    {
      "name": "Maxwell Fitzsimmons"
    },
    {
      "name": "David C. Del Rey Fernández"
    },
    {
      "name": "Jun Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ak9g9dcjhzx59.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18894",
  "fulltext_url": null,
  "id": "ak9g9dcjhzx59",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18894",
  "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.LG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7213506047",
  "pdf_url": "https://arxiv.org/pdf/2609.18894",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ak9g9dcjhzx59.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18894",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18894"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Learning Lyapunov Operators for Nonlinear Systems",
  "title_text": "Learning Lyapunov Operators for Nonlinear Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/ak9g9dcjhzx59",
  "venue": "arXiv",
  "versions": []
}