Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows
Abstract
We represent the flow map of an autonomous ODE ẋ=f(x) as a conjugate Φt=Fθ−1∘Ψt∘ Fθ of an explicitly integrable reference flow Ψt by an invertible neural network Fθ, the neural vector field transformation. The representation is an exact flow for every parameter value and inherits the phase portrait of the reference. A C1 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 Fθ 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 Fθ(0)=0, DFθ(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 8.3×10−3 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 Fθ.
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BibTeX
@misc{he2026neural,
title = {{Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows}},
author = {Andrew Qing He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29039},
eprint = {2610.29039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29039},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows
AU - Andrew Qing He
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29039
UR - https://arxiv.org/abs/2610.29039
AB - We represent the flow map of an autonomous ODE ẋ=f(x) as a conjugate Φ_t=F_θ^−1∘Ψ_t∘ 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 8.3×10^−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.
ER -