Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

2026-01-29

AI Summary

This work addresses the challenge of efficiently simulating long-term dynamics in Hamiltonian systems, where traditional numerical integrators are constrained by stability requirements that necessitate extremely small time steps. The authors propose a machine learning–based approach that models the average evolution of phase space over a fixed time interval Δt, enabling stable updates with significantly larger step sizes. The key innovation lies in the introduction of a mean-flow consistency condition, which allows the model to be trained using only independent phase-space samples—without requiring trajectory data or future state information. This is the first method capable of directly leveraging the abundant trajectory-free data from modern machine-learned force fields (MLFFs). Experiments across diverse Hamiltonian systems demonstrate substantial increases in allowable time steps while maintaining computational costs for training and inference comparable to existing methods.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Multiagent Systems: Multiagent Learning
• Humans and AI: Human-in-the-loop Machine Learning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.

Problem

• Hamiltonian systems
• molecular dynamics
• large-timestep simulation
• numerical integration stability
• phase-space evolution

Innovation

• Hamiltonian Flow Maps
• Mean Flow Consistency
• Large-Timestep Integration
• Machine-Learned Force Fields
• Trajectory-Free Training