AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-22
Neural Vector Field Transformations: Learning the Flow of Autonomous Systems by Conjugation to Explicitly Integrable Reference Flows
A method represents the flow map of an autonomous ODE as a conjugate of an explicitly integrable reference flow under an invertible neural network, so invariants such as equilibrium eigenvalues, periods and Floquet multipliers become learnable parameters of the reference. A training recipe built on a relative push-forward equation residual, a symplectic transformation and initialized invariants recovers the full spectrum of three- and six-dimensional nonlinear sinks with eigenvalue deviation of 8.3×10⁻³ and relative error of 2.3%. It also obtains action-angle variables for a four-dimensional integrable Hamiltonian system with frequency and action maps correct to a median of 0.09% and 0.22%, along with an exactly symplectic flow.