{
  "abstract": "We represent the flow map of an autonomous ODE $\\dot x=f(x)$ as a conjugate $Φ_t=F_θ^{-1}\\circΨ_t\\circ F_θ$ of an explicitly integrable reference flow $Ψ_t$ by an invertible neural network $F_θ$, the neural vector field transformation. The representation is an exact flow for every parameter value and inherits the phase portrait of the reference. A $C^1$ conjugacy preserves the eigenvalues of equilibria, the periods and Floquet multipliers of periodic orbits and the frequency-amplitude relation of centers, so these invariants must be learnable parameters of $Ψ$. We give parametrized reference flows with explicit integral curves, show that an action-angle reference is Hamiltonian only if its learnable frequency map is a gradient, and show that an orientation-preserving $F_θ$ cannot reverse the rotation sense of the reference, so it must be fixed before training. The main contribution is a training recipe: a push-forward equation residual made relative to $|f|$ and sampled log-uniformly in $|x|$ around an equilibrium, with the anchors $F_θ(0)=0$, $DF_θ(0)=I$; a symplectic transformation with the gradient-form frequency map; and invariants initialized from the field or data. With it the full spectrum of a three- and a six-dimensional nonlinear sink is recovered (eigenvalue deviation $8.3\\times10^{-3}$ and relative error 2.3%, three seeds each), and the action-angle variables of a four-dimensional integrable Hamiltonian system with frequency maps and actions correct to a median of 0.09% and 0.22%, and an exactly symplectic flow. On a dispersive system a time-input network is eight times more accurate inside its horizon, and the data-only identification of the Van der Pol oscillator yields a conjugacy on the data region with the Floquet exponent 17% off; an oracle run with the true invariants frozen attributes this to the expressivity of $F_θ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29039",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew Qing He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5qqt5eswtd1bj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29039",
  "fulltext_url": null,
  "id": "5qqt5eswtd1bj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29039",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223243927",
  "pdf_url": "https://arxiv.org/pdf/2610.29039",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5qqt5eswtd1bj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29039",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29039"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows",
  "title_text": "Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/5qqt5eswtd1bj",
  "venue": "arXiv",
  "versions": []
}