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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05894 |
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| _version_ | 1866911174778421248 |
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| author | Song, Yichen Wu, Yalun Wang, Yunbo Yang, Xiaokang |
| author_facet | Song, Yichen Wu, Yalun Wang, Yunbo Yang, Xiaokang |
| contents | In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their ability to approximate operators-mappings between functions. Although neural operators benefit from a universal approximation theorem, achieving reliable error bounds often necessitates large model architectures, such as deep stacks of Fourier layers. This raises a natural question: Can we design lightweight models without sacrificing generalization? To address this, we introduce DimINO (Dimension-Informed Neural Operators), a framework inspired by dimensional analysis. DimINO incorporates two key components, DimNorm and a redimensionalization operation, which can be seamlessly integrated into existing neural operator architectures. These components enhance the model's ability to generalize across datasets with varying physical parameters. Theoretically, we establish a universal approximation theorem for DimINO and prove that it satisfies a critical property we term Similar Transformation Invariance (STI). Empirically, DimINO achieves up to 76.3% performance gain on PDE datasets while exhibiting clear evidence of the STI property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05894 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | DimINO: Dimension-Informed Neural Operator Learning Song, Yichen Wu, Yalun Wang, Yunbo Yang, Xiaokang Machine Learning Computer Science, Physics Simulation In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their ability to approximate operators-mappings between functions. Although neural operators benefit from a universal approximation theorem, achieving reliable error bounds often necessitates large model architectures, such as deep stacks of Fourier layers. This raises a natural question: Can we design lightweight models without sacrificing generalization? To address this, we introduce DimINO (Dimension-Informed Neural Operators), a framework inspired by dimensional analysis. DimINO incorporates two key components, DimNorm and a redimensionalization operation, which can be seamlessly integrated into existing neural operator architectures. These components enhance the model's ability to generalize across datasets with varying physical parameters. Theoretically, we establish a universal approximation theorem for DimINO and prove that it satisfies a critical property we term Similar Transformation Invariance (STI). Empirically, DimINO achieves up to 76.3% performance gain on PDE datasets while exhibiting clear evidence of the STI property. |
| title | DimINO: Dimension-Informed Neural Operator Learning |
| topic | Machine Learning Computer Science, Physics Simulation |
| url | https://arxiv.org/abs/2410.05894 |