The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems ar...

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Autor principal:	Stefano Markidis
Formato:	article
Lenguaje:	EN
Publicado:	Frontiers Media S.A. 2021
Materias:	physics-informed deep-learning PINN scientific computing Poisson solvers deep-learning Information technology T58.5-58.64
Acceso en línea:	https://doaj.org/article/fe27750990e74469aaee567ae2e486e3
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spelling	oai:doaj.org-article:fe27750990e74469aaee567ae2e486e32021-11-19T06:04:34ZThe Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?2624-909X10.3389/fdata.2021.669097https://doaj.org/article/fe27750990e74469aaee567ae2e486e32021-11-01T00:00:00Zhttps://www.frontiersin.org/articles/10.3389/fdata.2021.669097/fullhttps://doaj.org/toc/2624-909XPhysics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.Stefano Markidis Frontiers Media S.A.articlephysics-informed deep-learningPINNscientific computingPoisson solversdeep-learningInformation technologyT58.5-58.64ENFrontiers in Big Data, Vol 4 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	physics-informed deep-learning PINN scientific computing Poisson solvers deep-learning Information technology T58.5-58.64
spellingShingle	physics-informed deep-learning PINN scientific computing Poisson solvers deep-learning Information technology T58.5-58.64 Stefano Markidis The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
description	Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.
format	article
author	Stefano Markidis
author_facet	Stefano Markidis
author_sort	Stefano Markidis
title	The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
title_short	The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
title_full	The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
title_fullStr	The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
title_full_unstemmed	The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
title_sort	old and the new: can physics-informed deep-learning replace traditional linear solvers?
publisher	Frontiers Media S.A.
publishDate	2021
url	https://doaj.org/article/fe27750990e74469aaee567ae2e486e3
work_keys_str_mv	AT stefanomarkidis theoldandthenewcanphysicsinformeddeeplearningreplacetraditionallinearsolvers AT stefanomarkidis oldandthenewcanphysicsinformeddeeplearningreplacetraditionallinearsolvers
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The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

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