[Paper Review] Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

Winfried Ripken, Michael Plainer+6 more|arXiv (Cornell University)|Jan 29, 2026

Machine Learning in Materials Science0 citations

TL;DR

Introduces Hamiltonian Flow Maps trained with a trajectory-free mean-flow consistency objective to enable stable large-timestep molecular dynamics using ab-initio data without trajectory generation.

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.

Motivation & Objective

• Motivate reducing computational cost in long-time Hamiltonian dynamics by enabling large timestep simulations.
• Propose a trajectory-free training objective based on mean flow consistency for learning Hamiltonian flow maps.
• Enable training directly on standard MLFF datasets without trajectory generation or teacher models.
• Demonstrate stable, accurate large-timestep MD rollouts across diverse systems while preserving physical constraints.

Proposed method

• Model a Hamiltonian flow map that advances phase-space state over a time interval Δt.
• Define a startpoint-conditioned mean displacement field as the time-averaged velocity and force over [t, t+Δt].
• Derive a trajectory-free consistency equation by differentiating the mean displacement and enforcing it to match instantaneous dynamics.
• Train a neural network to predict the mean displacement using a loss that combines instantaneous forces with a time-consistency term (Mean Flow).
• Recover the full Hamiltonian flow map by multiplying the learned mean displacement by Δt and adding the current state.
• Use Jacobian-vector products for efficient loss evaluation and employ stop-gradient targets to reduce computational overhead.
• Apply inference-time filters (energy/angular momentum conservation, translation removal, random rotation) to stabilize rollouts.

Figure 1 : Hamiltonian Flow Maps (HFMs) for large timesteps in phase space. Top row : Existing approaches rely on trajectory data, typically generated using a teacher MLFF through sequential simulation with small timesteps. While this enables training large-timestep models via direct regression, the

Experimental results

Research questions

• RQ1 Can a trajectory-free, mean-flow consistency objective learn accurate large-timestep Hamiltonian flow maps from instantaneous force and position data?
• RQ2 Does training on decorrelated MLFF datasets without trajectories enable stable, large-timestep MD rollouts across diverse systems while preserving physical observables?
• RQ3 How does the learned flow map perform compared to classical integrators (e.g., Velocity Verlet) at large timesteps on single-particle, N-body, and molecular systems?
• RQ4 Can the method recover both instantaneous forces and time-averaged dynamics within a single model?
• RQ5 What data efficiency and stability limits characterize the approach across different molecular systems?

Key findings

• HFMs enable stable rollouts at timesteps far beyond stability limits of standard MLFFs.
• HFMs trained on decorrelated instantaneous data perform competitively with MLFF baselines at small timesteps and surpass them at larger timesteps.
• In 4 molecular cases (Aspirin, Ethanol, Naphthalene, Salicylic Acid), HFMs maintain accuracy up to Δt of 9 fs, with MAEs close to or better than MLFFs at 1 fs and gradually increasing yet stable up to 9 fs.
• NVE simulations with HFMs explore conformational space faster than VV-based baselines, evidenced by Ramachandran plots for paracetamol.
• HFMs accurately reproduce structural statistics (h(r) MAE) in NVT simulations and yield reasonable temporal observables (power spectra).
• For alanine dipeptide, HFMs with Δtmax ≈ 15 fs produce accurate free-energy surfaces at Δt ≈ 12 fs, demonstrating scalability to complex metastable systems.

Figure 2 : HFMs vs. classical integration: Symplectic integrators such as Velocity Verlet (VV) advance the system through many local half-steps (left). HFMs instead predict the phase-space displacement over the interval directly (right). By modeling the mean velocity and force $(\bar{{{\bm{v}}}},\ba