Learning Hamiltonian Flow Maps from Numerical-Scheme Residuals for Long-Time Multiscale Simulation

Abstract

Hamiltonian systems with widely separated timescales arise in molecular dynamics, classical mechanics, and plasma physics. Long-time simulation of such systems is expensive because standard direct integrators generally need to resolve the fastest dynamics even when the quantities of interest evolve on much slower scales. The cost is particularly severe for large ensembles of trajectories. We develop a framework for learning Hamiltonian flow maps directly with neural networks. The map is trained either from the residual of a convergent numerical scheme or from reference trajectory data. For variable-timestep maps, truncated Taylor expansions enforce the correct short-time behavior while a neural network represents the remainder. For stiff oscillatory systems, the losses are measured in an energy-balanced norm that weights position errors according to their associated frequencies. We also introduce an HMC-$H_0$ procedure for generating training samples from microcanonical energy surfaces. The analysis clarifies what the residual training learns and how local flow-map errors propagate. Under a nonsingularity condition on the implicit part of a one-step scheme, every $C^1$ critical point of the unrestricted function-space residual functional has zero residual and therefore reproduces that numerical scheme. For a class of multiscale oscillatory Hamiltonians, the amplification rate of recursively applied flow-map errors is independent of the stiff frequencies when the local error is controlled in the energy-balanced norm; linear accumulation follows on the corresponding pre-asymptotic time window. Numerical experiments on separable, nonseparable, and noncanonical systems demonstrate long-time accuracy and identify regimes in which learned flow maps can reduce the cost of multiscale ensemble simulation.