Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions.

将时间并行算法应用于多尺度哈密顿系统以获得稳定的长时间模拟是非常具有挑战性的。在本文中，我们提出了新颖的数据驱动方法，旨在改进 Lions 等人开发的标准副现实算法。 2001 年，针对多尺度哈密顿系统。第一种方法涉及构建一个校正算子，通过使用沿平行真实轨迹在线收集的数据解决 Procrustes 问题来改进给定的不准确粗略求解器。第二种方法涉及通过使用离线生成的数据训练的神经网络构建高效、高保真度的求解器。对于第二种方法，我们解决了基于哈密顿函数的有效数据生成和适当损失函数设计的问题。我们通过将所提出的方法应用于 Fermi-Pasta-Ulam (FPU) 问题来展示概念验证。数值结果表明，与标准准实数方法相比，Procrustes 准实数方法能够产生能量更稳定的解。与具有类似精度的数值求解器相比，神经网络求解器可以实现相当或更好的运行时性能。当与标准副实数算法结合时，改进的神经网络解比改进的数值粗解在能量上稍微稳定一些。