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 Lion, Maday, and Turinici 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-Ulum (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.

翻译：将并行时间算法应用于多尺度哈密顿系统以实现稳定的长时间模拟极具挑战性。本文提出新颖的数据驱动方法，旨在改进Lion、Maday和Turinici于2001年提出的标准Parareal算法在多尺度哈密顿系统中的应用。第一种方法通过沿Parareal轨迹在线收集数据求解Procrustes问题，构建校正算子以改善给定不精确的粗粒度求解器。第二种方法采用离线生成数据训练的神经网络构建高效高保真求解器。针对第二种方法，我们解决了有效数据生成问题，并基于哈密顿函数设计了合适的损失函数。通过将所提方法应用于Fermi-Pasta-Ulum（FPU）问题验证其概念可行性。数值结果表明，Procrustes Parareal方法相比标准Parareal能生成能量更稳定的解。神经网络求解器在运行时性能方面可与同等精度的数值求解器相当或更优。当与标准Parareal算法结合时，改进的神经网络解在能量稳定性上略优于改进的数值粗解。