Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations...

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Veröffentlicht in:	Advances in computational mathematics Jg. 49; H. 4; S. 62
Hauptverfasser:	Moseley, Ben, Markham, Andrew, Nissen-Meyer, Tarje
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.08.2023 Springer Nature B.V
Schlagworte:	Basis functions Bias Complexity Computational Mathematics and Numerical Analysis Computational Science and Engineering Differential equations Domain decomposition methods Finite element method Mathematical analysis Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Neural networks Numerical methods Optimization Physics Problem solving Visualization 68T01 Parallel computing Forward modelling Differential equations Multi-scale modelling 65M99 Domain decomposition Physics-informed neural networks
ISSN:	1019-7168, 1572-9044
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Abstract	Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs) . FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems.
AbstractList	Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs). FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems. Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs) . FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems.
ArticleNumber	62
Author	Markham, Andrew Nissen-Meyer, Tarje Moseley, Ben
Author_xml	– sequence: 1 givenname: Ben orcidid: 0000-0003-2238-1783 surname: Moseley fullname: Moseley, Ben email: benjamin.moseley@ai.ethz.ch organization: AI Center, ETH Zürich – sequence: 2 givenname: Andrew surname: Markham fullname: Markham, Andrew organization: Department of Computer Science, University of Oxford – sequence: 3 givenname: Tarje surname: Nissen-Meyer fullname: Nissen-Meyer, Tarje organization: Department of Earth Sciences, University of Oxford
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Snippet	Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to...
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SubjectTerms	Basis functions Bias Complexity Computational Mathematics and Numerical Analysis Computational Science and Engineering Differential equations Domain decomposition methods Finite element method Mathematical analysis Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Neural networks Numerical methods Optimization Physics Problem solving Visualization
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Title	Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
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