U-WNO: U-Net Enhanced Wavelet Neural Operator for Solving Parametric Partial Differential Equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916804314529792 |
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| author | Lei, Wei-Min Li, Hou-Biao |
| author_facet | Lei, Wei-Min Li, Hou-Biao |
| contents | High-frequency features are critical in multiscale phenomena such as turbulent flows and phase transitions, since they encode essential physical information. The recently proposed Wavelet Neural Operator (WNO) utilizes wavelets' time-frequency localization to capture spatial manifolds effectively. While its factorization strategy improves noise robustness, it suffers from high-frequency information loss caused by finite-scale wavelet decomposition. In this study, a new U-WNO network architecture is proposed. It incorporates the U-Net path and residual shortcut into the wavelet layer to enhance the extraction of high-frequency features and improve the learning of spatial manifolds. Furthermore, we introduce an adaptive activation mechanism to mitigate spectral bias through trainable slope parameters. Extensive benchmarks across seven PDE families (Burgers, Darcy flow, Navier-Stokes, etc.) show that U-WNO achieves 45--83% error reduction compared to baseline WNO, with mean $L_2$ relative errors ranging from 0.043% to 1.56%. This architecture establishes a framework combining multiresolution analysis with deep feature learning, addressing the spectral-spatial tradeoff in operator learning. Code and data used are available on https://github.com/WeiminLei/U-WNO.git. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | U-WNO: U-Net Enhanced Wavelet Neural Operator for Solving Parametric Partial Differential Equations Lei, Wei-Min Li, Hou-Biao Numerical Analysis 65M55, 65M22 G.1.8 High-frequency features are critical in multiscale phenomena such as turbulent flows and phase transitions, since they encode essential physical information. The recently proposed Wavelet Neural Operator (WNO) utilizes wavelets' time-frequency localization to capture spatial manifolds effectively. While its factorization strategy improves noise robustness, it suffers from high-frequency information loss caused by finite-scale wavelet decomposition. In this study, a new U-WNO network architecture is proposed. It incorporates the U-Net path and residual shortcut into the wavelet layer to enhance the extraction of high-frequency features and improve the learning of spatial manifolds. Furthermore, we introduce an adaptive activation mechanism to mitigate spectral bias through trainable slope parameters. Extensive benchmarks across seven PDE families (Burgers, Darcy flow, Navier-Stokes, etc.) show that U-WNO achieves 45--83% error reduction compared to baseline WNO, with mean $L_2$ relative errors ranging from 0.043% to 1.56%. This architecture establishes a framework combining multiresolution analysis with deep feature learning, addressing the spectral-spatial tradeoff in operator learning. Code and data used are available on https://github.com/WeiminLei/U-WNO.git. |
| title | U-WNO: U-Net Enhanced Wavelet Neural Operator for Solving Parametric Partial Differential Equations |
| topic | Numerical Analysis 65M55, 65M22 G.1.8 |
| url | https://arxiv.org/abs/2408.08190 |