The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speedup, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of learned coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn model, and viscous Burgers model, show that learning can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.

翻译：Parareal算法是求解演化方程的一类重要并行时空算法，已在实践中得到广泛应用。为实现有效加速，算法中粗传播子的选择至关重要。本文研究了基于学习的粗传播子构建方法。在误差估计框架的基础上，我们提出了一套系统的粗传播子构造流程，使其兼具理想的稳定性和一致性阶数。此外，我们为由此得到的Parareal算法提供了初步的数学保证。在线性扩散模型、Allen-Cahn模型和粘性Burgers模型等多类场景下的数值实验表明，与一些传统且广泛使用的经验性粗传播子相比，学习策略能显著提升并行效率。