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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2021-11-01
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| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-021-00773-x |
| _version_ | 1818823179547705344 |
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| 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 |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-18T23:35:52Z |
| format | Article |
| id | doaj.art-8ec4005490a346b3a3bcdf7273108d82 |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-12-18T23:35:52Z |
| publishDate | 2021-11-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-8ec4005490a346b3a3bcdf7273108d822022-12-21T20:47:32ZengNature PortfolioScientific Reports2045-23222021-11-0111111410.1038/s41598-021-00773-xDeepGreen: deep learning of Green’s functions for nonlinear boundary value problemsCraig R. Gin0Daniel E. Shea1Steven L. Brunton2J. Nathan Kutz3Department of Population Health and Pathobiology, North Carolina State UniversityDepartment of Materials Science and Engineering, University of WashingtonDepartment of Mechanical Engineering, University of WashingtonDepartment of Applied Mathematics, University of WashingtonAbstract 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.https://doi.org/10.1038/s41598-021-00773-x |
| spellingShingle | Craig R. Gin Daniel E. Shea Steven L. Brunton J. Nathan Kutz DeepGreen: deep learning of Green’s functions for nonlinear boundary value problems Scientific Reports |
| title | 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_short | 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 |
| url | https://doi.org/10.1038/s41598-021-00773-x |
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