Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations

By Ameya Dilip Jagtap, G. Em Karniadakis. 2020

1,077 citations recorded by the source corpus.

We propose a generalized space-time domain decomposition framework for the physics-informed neural networks (PINNs) to solve nonlinear partial differential equations (PDEs) on arbitrary complex-geometry domains. The proposed framework, named eXtended PINNs ( XPINNs ), further pushes the boundaries of both PINNs as well as conservative PINNs (cPINNs), which is a recently proposed domain decomposition approach in the PINN framework tailored to conservation laws. Compared to PINN, the XPINN method has large representation and parallelization capacity due to the inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, the domain can be decomposed in any arbitrary way (in space and time), which is not possible in cPINN. Thus, XPINN offers both space and time parallelization, thereby reducing the training cost more effectively. In each subdomain, a separate neural network is employed with optimally selected hyperparameters, e.g., depth/width of the network, number and location of residual points, activation function, optimization method, etc. A deep network can be employed in a subdomain with complex solution, whereas a shallow neural network can be used in a subdomain with relatively simple and smooth solutions. We demonstrate the versatility of XPINN by solving both forward and inverse PDE problems, ranging from one-dimensional to three-dimensional problems, from time-dependent to time-independent problems, and from continuous to discontinuous problems, which clearly shows that the XPINN method is promising in many practical problems. The proposed XPINN method is the generalization of PINN and cPINN approaches, both in terms of applicability as well as domain decomposition approach, which efﬁciently lends itself to parallelized computation. The XPINN code will be available on https://github.com/AmeyaJagtap/XPINNs .

Subfields: Statistical and Nonlinear Physics

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Citations in the corpus

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 spatio-temporal domain accelerates the training of PINNs and improves their accuracy [19, 20, 21].”
GNOT: A General Neural Operator Transformer for Operator Learning (2023, methodology): “Inspired by the domain decomposition methods that are widely used to handle multi-scale problems (Jagtap & Karniadakis, 2021; Hu et al., 2022), we propose to use the geometric coordinates of input points for the gating network and we found that this could be viewed as a soft domain decomposition.”
Physics-informed neural networks for inverse problems in supersonic flows (2022, methodology): “Here 1q denotes the indicator function; see [5] for more details.”
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, background): “Physics-informed neural networks (PINNs) were introduced first 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”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “In addition, the variational hp-VPINN considered domain decomposition, and similar pointwise versions were also studied in CPINN [25], and XPINN [26].”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, methodology): “A key consideration in any domain decomposition approach is to ensure that individual subdomain solutions are communicated and match across the subdomain interfaces, and Jagtap and Karniadakis [2020] rely upon additional interfaces terms in their PINN loss function to do so.”
Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions (2021, methodology): “The implementation of PINNs with a domain decomposition in space and space-time domain is successfully studied by Jagtap et al. (2020) and Jagtap and Karniadakis (2020), which is further extended to a multi-GPU platform by Shukla et al. (2021).”