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 $\dot x=f(x)$ 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 $F_θ$.