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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Vydáno v:	Computer methods in applied mechanics and engineering Ročník 370; číslo C; s. 113250
Hlavní autoři:	Meng, Xuhui, Li, Zhen, Zhang, Dongkun, Karniadakis, George Em
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.10.2020 Elsevier BV Elsevier
Témata:	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
On-line přístup:	Získat plný text
Tagy:	Přidat tag Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!

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 (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.
AbstractList	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. 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.
ArticleNumber	113250
Author	Li, Zhen Meng, Xuhui Karniadakis, George Em Zhang, Dongkun
Author_xml	– sequence: 1 givenname: Xuhui surname: Meng fullname: Meng, Xuhui organization: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA – sequence: 2 givenname: Zhen surname: Li fullname: Li, Zhen organization: Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA – sequence: 3 givenname: Dongkun surname: Zhang fullname: Zhang, Dongkun organization: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA – sequence: 4 givenname: George Em surname: Karniadakis fullname: Karniadakis, George Em email: george_karniadakis@brown.edu organization: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
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Snippet	Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics...
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SubjectTerms	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
Title	PPINN: Parareal physics-informed neural network for time-dependent PDEs
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