Stabilization of parareal algorithms for long time computation of a class of highly oscillatory Hamiltonian flows using data

TL;DR

This paper introduces data-driven enhancements to the parareal algorithm for long-term, stable simulation of highly oscillatory Hamiltonian systems, using correction operators and neural networks trained on offline data.

Contribution

It presents two novel data-driven methods—Procrustes correction and neural network solvers—to improve the stability and efficiency of parareal algorithms for multiscale Hamiltonian systems.

Findings

Procrustes parareal improves energy stability.

Neural network solver achieves comparable or better runtime.

Combined methods enhance long-term stability.

Abstract

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.