PPINN: Parareal physics-informed neural network for time-dependent PDEs

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 (Raissi et al., 2019). While effective for rel...

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Vydané v:Computer methods in applied mechanics and engineering Ročník 370; číslo C; s. 113250
Hlavní autori: Meng, Xuhui, Li, Zhen, Zhang, Dongkun, Karniadakis, George Em
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Amsterdam Elsevier B.V 01.10.2020
Elsevier BV
Elsevier
Predmet:
Burgers equation
Conservation laws
Convergence
Datasets
Deep neural network
Long-time integration
Machine learning
Multiscale
Neural networks
Parallel-in-time
Physics
PINN
Reaction-diffusion equations
Time dependence
Time integration
Training
Deep neural network
Long-time integration
PINN
Multiscale
Machine learning
Parallel-in-time
ISSN:0045-7825, 1879-2138
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Popis
Shrnutí: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 (Raissi et al., 2019). 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 speed-up 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 few iterations with significant speed-ups proportional to the number of time-subdomains employed.
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USDOE
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2020.113250