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The parareal algorithm represents an important class of parallel-in-timealgorithms for solving evolution equations and has been widely applied inpractice. To achieve effective speedup, the choice of the coarse propagator inthe algorithm is vital. In this work, we investigate the use of learned coarsepropagators. Building upon the error estimation framework, we present asystematic procedure for constructing coarse propagators that enjoy desirablestability and consistent order. Additionally, we provide preliminarymathematical guarantees for the resulting parareal algorithm. Numericalexperiments on a variety of settings, e.g., linear diffusion model, Allen-Cahnmodel, and viscous Burgers model, show that learning can significantly improveparallel efficiency when compared with the more ad hoc choice of someconventional and widely used coarse propagators.

Learning Coarse Propagators in Parareal Algorithm