Learning Hamiltonian flows from numerical integrators and examples

Abstract: Hamiltonian systems with multiple timescales arise in molecular dynamics, classical mechanics, and theoretical physics. Long-time numerical integration of such systems requires resolving fast dynamics with very small time steps, which incurs a high computational cost - especially in ensemble simulations for uncertainty quantification, sensitivity analysis, or varying initial conditions. We present a Deep Learning framework that learns the flow maps of Hamiltonian systems to accelerate long-time and ensemble simulations. Neural networks are trained, according to a chosen numerical scheme, either entirely without data to approximate flows over large time intervals or with data to learn flows in intervals far from the initial time. For the latter, we propose a Hamiltonian Monte Carlo-based data generator. The architecture consists of simple feedforward networks that incorporate truncated Taylor expansions of the flow map, with a neural network remainder capturing unresolved effects. Applied to benchmark non-integrable and non-canonical systems, the method achieves substantial speedups while preserving accuracy, enabling scalable simulation of complex Hamiltonian dynamics.