  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.
