{
  "abstract": "We present LatentLieODE, a novel neural ODE method to discover linear Lie symmetries and first integrals of the learned dynamics from timeseries observations such as video inputs. It starts with an autoencoder, which learns a low dimensional representation of the observed system and then maps trajectories to decorrelated latent coordinates $z \\in \\mathbb{R}^d$. A bank of linear infinitesimal generators $\\mathbf{A}_k \\in \\mathbb{R}^{d \\times d}$ is fitted jointly against a differentiable Lie-bracket residual $\\mathbf{J}_f(z)\\mathbf{A}z - \\mathbf{A} f(z)$, while an MLP optimises the latent ODE $f(z) = \\mathrm{d}z/\\mathrm{d}t$ to reconstruct the observation-space timeseries. After suitable generators and $f$ are found, we fit a third network $H_ϕ$ to generate the first integral of the system. Across six benchmark systems (harmonic oscillator, simple and twin pendulums, Lotka-Volterra, pixel pendulum, sphere geodesics) we find each system's relevant symmetries, and the discovered invariants match ground-truth conserved quantities at $|ρ| \\geq 0.978$. On the sphere, the pipeline recovers a non-abelian $\\mathfrak{so}(3)$, certified by the negative-definite Killing form of its recovered structure constants. This work presents significant advances in data-driven discovery of symmetries and provides a new pathway for neural surrogates of dynamical systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35967",
  "author_total": 2,
  "authors": [
    {
      "name": "Matt L. Wiemman"
    },
    {
      "name": "Peter Melchior"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3d6y19ennd8na.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35967",
  "fulltext_url": null,
  "id": "3d6y19ennd8na",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35967",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7224082297",
  "pdf_url": "https://arxiv.org/pdf/2610.35967",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3d6y19ennd8na.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:59Z",
      "native_id": "2610.35967",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35967"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs",
  "title_text": "Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:59Z",
  "url": "https://research.draconitic.com/paper/3d6y19ennd8na",
  "venue": "arXiv",
  "versions": []
}