DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems

Abstract Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, qua...

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Autores principales:	Craig R. Gin, Daniel E. Shea, Steven L. Brunton, J. Nathan Kutz
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2021
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/8ec4005490a346b3a3bcdf7273108d82
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spelling	oai:doaj.org-article:8ec4005490a346b3a3bcdf7273108d822021-11-08T10:47:42ZDeepGreen: deep learning of Green’s functions for nonlinear boundary value problems10.1038/s41598-021-00773-x2045-2322https://doaj.org/article/8ec4005490a346b3a3bcdf7273108d822021-11-01T00:00:00Zhttps://doi.org/10.1038/s41598-021-00773-xhttps://doaj.org/toc/2045-2322Abstract Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green’s function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green’s function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator L and Green’s function G which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm–Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation and can solve nonlinear BVPs at orders of magnitude faster than traditional methods without the need for an initial guess. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green’s functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.Craig R. GinDaniel E. SheaSteven L. BruntonJ. Nathan KutzNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 11, Iss 1, Pp 1-14 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Craig R. Gin Daniel E. Shea Steven L. Brunton J. Nathan Kutz DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
description	Abstract Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green’s function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green’s function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator L and Green’s function G which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm–Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation and can solve nonlinear BVPs at orders of magnitude faster than traditional methods without the need for an initial guess. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green’s functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.
format	article
author	Craig R. Gin Daniel E. Shea Steven L. Brunton J. Nathan Kutz
author_facet	Craig R. Gin Daniel E. Shea Steven L. Brunton J. Nathan Kutz
author_sort	Craig R. Gin
title	DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
title_short	DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
title_full	DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
title_fullStr	DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
title_full_unstemmed	DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems
title_sort	deepgreen: deep learning of green’s functions for nonlinear boundary value problems
publisher	Nature Portfolio
publishDate	2021
url	https://doaj.org/article/8ec4005490a346b3a3bcdf7273108d82
work_keys_str_mv	AT craigrgin deepgreendeeplearningofgreensfunctionsfornonlinearboundaryvalueproblems AT danieleshea deepgreendeeplearningofgreensfunctionsfornonlinearboundaryvalueproblems AT stevenlbrunton deepgreendeeplearningofgreensfunctionsfornonlinearboundaryvalueproblems AT jnathankutz deepgreendeeplearningofgreensfunctionsfornonlinearboundaryvalueproblems
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DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems

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