Learning Hamiltonian flows from numerical integrators and examples

TL;DR

A Deep Learning framework that learns the flow maps of Hamiltonian systems to accelerate long-time and ensemble simulations, and achieves substantial speedups while preserving accuracy, enabling scalable simulation of complex Hamiltonian dynamics.

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.

Figures (11)

• Figure 1: (Harmonic oscillator) Residual and phase space plots for 1-step predictions generated by flow maps trained with different residual definitions. Row 1-2: VV residual with $h=0.5$; row 3-4: exact residual. Each column corresponds to a different time collocation mode. Column 1: $N=11$ evenly spaced grid points over $[0,10]$; column 2: $N=21$ grid points; column 3: $N=41$ grid points. In the residual plots, blue curves indicate the residual values while red crosses denote the collocation points. In the phase space plots, the black curve represents the exact flow, the red curve indicates the learned flow, and the green points mark the numerical solution at discrete times ($t=0,h,2h,\dots$).
• Figure 2: (Harmonic oscillator) Phase space plots for 1-step predictions generated by flow maps at different stages during training. In each plot, the black curve represents the exact flow, the red curve indicates the learned flow, and the green points mark the numerical solution at discrete times ($t=0,h,2h,\dots$).
• Figure 3: (NPCOs problem): Trajectories of the slow oscillator for $t\in [0,2000]$ generated by repeatedly applying the neural network-based flow map $\Phi(\cdot,\Delta t)$, $\Delta t=5$, and the reference flow map $\phi_t$. Top row: three trajectories of the slow oscillator with the same energy $H_0=1.13$. Bottom row: trajectories on different energy level sets in the inner region (where the blue curves in the top row occupy).
• Figure 4: (NPCOs problem) 1-step prediction errors, $||\Phi(u_0, t)-\phi_t(u_0)||_2$, of the learned flow maps for a chosen generic $u_0$. In the legend, $\Phi_{[0, T]}$, with $T=20,40,80$, denotes the flow maps learned with time collocation points distributed in the time interval $[0, T]$, respectively.
• Figure 5: (NPCOs) Phase space plots of long-time trajectories over the time interval $[0,2000]$ generated with a fixed step size $t=20$ by the reference flow map (top row) and the network-based flow maps learned up to $T=20,40,80$ respectively.

Theorems & Definitions (3)

• Theorem 2.1: One-step implicit-explicit methods
• Theorem 2.2: Implicit midpoint rule
• Remark 2.1