Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected n...
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| Vydáno v: | Computer methods in applied mechanics and engineering Ročník 390; s. 114502 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier B.V
15.02.2022
Elsevier BV |
| Témata: | |
| ISSN: | 0045-7825, 1879-2138 |
| On-line přístup: | Získat plný text |
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| Abstract | Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs. |
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| AbstractList | Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs. |
| ArticleNumber | 114502 |
| Author | Gao, Han Zahr, Matthew J. Wang, Jian-Xun |
| Author_xml | – sequence: 1 givenname: Han surname: Gao fullname: Gao, Han – sequence: 2 givenname: Matthew J. orcidid: 0000-0003-4066-981X surname: Zahr fullname: Zahr, Matthew J. – sequence: 3 givenname: Jian-Xun orcidid: 0000-0002-9030-1733 surname: Wang fullname: Wang, Jian-Xun email: jwang33@nd.edu |
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| Keywords | Physics-informed machine learning Mechanics Graph convolutional neural networks Partial differential equations Inverse problem |
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| Snippet | Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as... |
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| StartPage | 114502 |
| SubjectTerms | Artificial neural networks Boundary conditions Computational geometry Continuity (mathematics) Convexity Graph convolutional neural networks Inverse problem Inverse problems Mechanics Neural networks Optimization Partial differential equations Physics-informed machine learning Polynomials Training |
| Title | Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems |
| URI | https://dx.doi.org/10.1016/j.cma.2021.114502 https://www.proquest.com/docview/2635268634 |
| Volume | 390 |
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