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

By Xuhui Meng, Zhen Li, Dongkun Zhang, G. Em Karniadakis. 2019

601 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Numerical Methods & Scientific Computing, AI for Physical Sciences

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, background): “PINNs have been extended to convolutional (Zhu et al., ∂x2 with initial condition u(x, 0) = f(x) and Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0.”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, unknown): “5) [226] [251] [301] [303] [70] [196] [228] [248] [318] [82] [102] [144] [292] [216] [195] [23] [90] [320] [154] [58] [236] [53] [70] [192] [252] [26] [61] [149] [88] [299] [216] [195] [225] [145] [198] [171] [44] [188]”
A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks (2022, background): “For problems in a large domain, decomposition of the spatiotemporal domain accelerates the training of PINNs and improves their accuracy [19, 20, 21].”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, unknown): “Solve PDEs [147] [65] [140] [9] [215] [42] [264] [106] [235] [107] [131] [209] [234] [279] [281] [60] [177] [208] [231] [301] [71] [89] [129] [268] [194] [176] [16] [48] [218] [44] [60] [172] [235] [19] [51] [131] [76] [278] [194] [176] [206] [166]”
Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2020, methodology): “To enhance the robustness and generality of PINN, multiple variations of PINN have also been developed, such as Variational PINNs [18], Parareal PINNs [19], and nonlocal PINN [20].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “In Meng et al [109] the authors study a two-dimensional diffusion-reaction equation that involves long-time integration and they use a parareal PINN (PPINN) to divide the time interval into equal-length sub-domains.”
B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data (2020, background): “On the other hand, PINNs have shown eﬀectiveness in both forward and inverse problems for a wide range of PDEs [9, 10, 11, 12, 13].”
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, result): “…in 2017 [7, 8] for forward, inverse and hybrid problems, and since then there have also been rapid developments in this area [9, 10, 11, 12, 13, 14, 15, 16, 17], although the achievements so far in scientiﬁc and engineering applications have been modest compared to applications in imaging,…”