Overview
A novel framework has been developed for learning Hamiltonian flow maps directly through the application of neural networks. This method aims to address the computational expense associated with long-time simulations of Hamiltonian systems characterized by widely separated timescales, particularly for large ensembles of trajectories. The approach leverages either the residual of a convergent numerical scheme or existing reference trajectory data for training. It incorporates specific techniques for variable-timestep maps and stiff oscillatory systems, demonstrating long-time accuracy and identifying conditions under which it can decrease the cost of multiscale ensemble simulations.
Research Context
Hamiltonian systems that exhibit widely separated timescales are prevalent in various scientific domains, including molecular dynamics, classical mechanics, and plasma physics. Simulating these systems over extended durations poses a significant computational challenge. Standard direct integrators typically necessitate resolving the fastest dynamics within these systems, even when the primary quantities of interest evolve at much slower rates. This requirement leads to substantial computational costs, which become particularly severe when simulating large ensembles of trajectories.
Approach
The developed framework centers on learning Hamiltonian flow maps via neural networks. Two primary training mechanisms are employed:
- Training from the residual of a convergent numerical scheme.
- Training from reference trajectory data.
For variable-timestep maps, the methodology involves enforcing correct short-time behavior through truncated Taylor expansions, with a neural network subsequently representing the remainder of the map. For stiff oscillatory systems, the training process utilizes an energy-balanced norm for measuring losses. This norm specifically weights position errors according to their associated frequencies. Additionally, an HMC-$H_0$ procedure is introduced for generating training samples from microcanonical energy surfaces.
The analytical component of the framework clarifies the mechanisms underlying residual training and elucidates the propagation of local flow-map errors. Specifically, it establishes that for a one-step scheme with a nonsingularity condition on its implicit part, any $C^1$ critical point of the unrestricted function-space residual functional will exhibit zero residual, thereby reproducing the numerical scheme. For a class of multiscale oscillatory Hamiltonians, the amplification rate of recursively applied flow-map errors is shown to be independent of the stiff frequencies, provided the local error is controlled using the energy-balanced norm. This condition leads to a linear accumulation of errors over the corresponding pre-asymptotic time window.
Findings
Numerical experiments were conducted on diverse system types, including separable, nonseparable, and noncanonical Hamiltonian systems. These experiments indicated:
- The learned flow maps achieve long-time accuracy.
- The approach identifies specific regimes where learned flow maps can reduce the computational cost associated with multiscale ensemble simulations.
Why This Matters
The computational expense of simulating Hamiltonian systems with disparate timescales is a known impediment to long-duration studies, especially for large trajectory ensembles. The demonstrated long-time accuracy and cost reduction capabilities in multiscale ensemble simulation suggest a potential advancement in addressing these computational bottlenecks, which are relevant across molecular dynamics, classical mechanics, and plasma physics.