Discrete-time generalized canonical transformations for non-autonomous systems
arXiv:2607.12914
2026
Dynamics
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper constructs discrete flows for non-autonomous Hamiltonian systems by lifting time to an extended cotangent phase space and requiring the lifted map to be symplectic. Its transferable mechanism is a structure-preserving optimizer whose updates conserve the extended symplectic form, Poisson bracket, and phase-space volume instead of introducing arbitrary numerical damping. Neural-network parameters, momenta, training time, and conjugate time-energy can be treated as the extended state. A symmetric composition of canonical subflows gives a concrete optimizer with a measurable stability boundary on quadratic losses.
Ideas from this paper
✗ Failed on benchmark
2026
Replace a dissipative optimizer update with a canonical discrete flow on the extended state $(\theta,p,t,e)$, where $\theta$ are network parameters, $p$ is momentum, $t$ is training time, and $e$ is its conjugate energy variable. Use a symmetric composition of exact Hamiltonian subflows for kinetic energy, loss, and time translation; this preserves the extended symplectic form and avoids artificial phase-volume collapse. Weak restarts or occasional damping can be added separately if convergence…
Useful8/10
Difficulty5/10
Novelty6/10