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.

该研究介绍了一种名为PPINN的新型神经网络结构，可在短时间内解决时间依赖性偏微分方程问题，通过将一个长时间问题分解成许多由粗粒度求解器监督的独立短时间问题，PPINN可以在几个迭代中实现收敛并获得显著加速。

PPINN:时变偏微分方程的Parareal物理信息神经网络