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 $Δt$, 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.

翻译：模拟哈密顿系统的长时间演化受限于稳定数值积分所需的小时间步长。为克服这一限制，我们引入了一个框架来学习哈密顿流映射，通过预测选定时间跨度 $Δt$ 内的平均相空间演化，从而实现远超经典积分器稳定性极限的稳定大时间步长更新。为此，我们对时间平均的哈密顿动力学施加了平均流一致性条件。与先前方法不同，这允许在独立相空间样本上进行训练，而无需访问未来状态，从而避免了昂贵的轨迹生成。该方法在多种哈密顿系统中得到验证，尤其在使用机器学习力场（MLFF）的分子动力学模拟方面有所改进。我们的模型保持了可比的训练和推理成本，但支持显著更大的积分时间步长，同时可直接在广泛可用的无轨迹MLFF数据集上进行训练。