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。