  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.
