PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs

TL;DR

The paper introduces PPINN, a parallelized physics-informed neural network framework that decomposes long-time PDE problems into short segments, enabling faster and more efficient solutions through parallel training and coarse solver guidance.

Contribution

The paper proposes PPINN, a novel parallel approach combining coarse solvers and PINNs for efficient long-time PDE integration, significantly reducing training time and computational cost.

Findings

PPINN converges within a few iterations for tested PDEs.

Significant speed-ups proportional to the number of time subdomains.

Effective long-time PDE solutions with reduced computational resources.

Abstract

Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree. While effective for relatively short-term time integration, when long time integration of the time-dependent PDEs is sought, the time-space domain may become arbitrarily large and hence training of the neural network may become prohibitively expensive. To this end, we develop a parareal physics-informed neural network (PPINN), hence decomposing a long-time problem into many independent short-time problems supervised by an inexpensive/fast coarse-grained (CG) solver. In particular, the serial CG solver is designed to provide approximate predictions of the solution at discrete times, while initiate many fine PINNs simultaneously to correct the solution iteratively. There is a two-fold benefit from training PINNs with small-data sets rather than working on a large-data set directly, i.e., training of individual PINNs with small-data is much faster, while training the fine PINNs can be readily parallelized. Consequently, compared to the original PINN approach, the proposed PPINN approach may achieve a significant speedup for long-time integration of PDEs, assuming that the CG solver is fast and can provide reasonable predictions of the solution, hence aiding the PPINN solution to converge in just a few iterations. To investigate the PPINN performance on solving time-dependent PDEs, we first apply the PPINN to solve the Burgers equation, and subsequently we apply the PPINN to solve a two-dimensional nonlinear diffusion-reaction equation. Our results demonstrate that PPINNs converge in a couple of iterations with significant speed-ups proportional to the number of time-subdomains employed.