@Article{CiCP-28-2002,
author = {D. Jagtap , Ameya and Em Karniadakis , George},
title = {Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations},
journal = {Communications in Computational Physics},
year = {2020},
volume = {28},
number = {5},
pages = {2002--2041},
abstract = {<p style="text-align: justify;">We propose a generalized space-time domain decomposition approach 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 methods, both in terms of applicability as well as domain decomposition approach, which efficiently lends itself to parallelized computation. The XPINN
code is available on $https://github.com/AmeyaJagtap/XPINNs$.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0164},
url = {http://global-sci.org/intro/article_detail/cicp/18403.html}
}