On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investi...
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| Vydané v: | Computer methods in applied mechanics and engineering Ročník 384; číslo C; s. 113938 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier B.V
01.10.2021
Elsevier BV Elsevier |
| Predmet: | |
| ISSN: | 0045-7825, 1879-2138 |
| On-line prístup: | Získať plný text |
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| Abstract | Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction–diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at https://github.com/PredictiveIntelligenceLab/MultiscalePINNs.
•We argue that spectral bias in deep neural networks in fact corresponds to “NTK eigenvector bias”.•We show that Fourier feature mappings can modulate the frequency of the NTK eigenvectors.•By analyzing the NTK eigenspace, we engineer new effective architectures for multi-scale problems.•We put forth a collection of challenging benchmarks for PINNs. |
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| AbstractList | Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction–diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at https://github.com/PredictiveIntelligenceLab/MultiscalePINNs.
•We argue that spectral bias in deep neural networks in fact corresponds to “NTK eigenvector bias”.•We show that Fourier feature mappings can modulate the frequency of the NTK eigenvectors.•By analyzing the NTK eigenspace, we engineer new effective architectures for multi-scale problems.•We put forth a collection of challenging benchmarks for PINNs. Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction–diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at https://github.com/PredictiveIntelligenceLab/MultiscalePINNs. |
| ArticleNumber | 113938 |
| Author | Perdikaris, Paris Wang, Hanwen Wang, Sifan |
| Author_xml | – sequence: 1 givenname: Sifan surname: Wang fullname: Wang, Sifan email: sifanw@sas.upenn.edu organization: Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA 19104, United States of America – sequence: 2 givenname: Hanwen orcidid: 0000-0002-4990-6810 surname: Wang fullname: Wang, Hanwen email: wangh19@sas.upenn.edu organization: Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA 19104, United States of America – sequence: 3 givenname: Paris orcidid: 0000-0002-2816-3229 surname: Perdikaris fullname: Perdikaris, Paris email: pgp@seas.upenn.edu organization: Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, United States of America |
| BackLink | https://www.osti.gov/biblio/1786215$$D View this record in Osti.gov |
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| Snippet | Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they... |
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| SubjectTerms | Deep learning Eigenvectors Inverse problems Mathematical analysis Neural networks Neural Tangent Kernel Partial differential equations Robustness (mathematics) Scientific machine learning Spectral bias Wave propagation |
| Title | On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks |
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