MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913850466500608 |
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| author | Guo, Zi-Hao Li, Hou-Biao |
| author_facet | Guo, Zi-Hao Li, Hou-Biao |
| contents | Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator (MgFNO) that accelerates the training of traditional Fourier neural operators through a novel three-level hierarchical architecture. The key innovation of MgFNO lies in its decoupled training strategy employing three distinct networks at different resolution levels: a coarse-level network first learns low-resolution approximations, an intermediate network refines the solution, and a fine-level network achieves high-resolution accuracy. By combining the frequency principle of deep neural networks with multigrid methodology, MgFNO effectively bridges the complementary learning patterns of neural networks (low-to-high frequency) and multigrid methods (high-to-low frequency error reduction).Experimental results demonstrate that MgFNO achieves relative errors of 0.17%, 0.28%, and 0.22% on the Burgers' equation, Darcy flow, and Navier-Stokes equations, respectively, representing reductions of 89%, 71%, and 83% compared to the conventional FNO. Furthermore, MgFNO supports zero-shot super-resolution prediction, enabling direct application to high-resolution scenarios after training on coarse grids. This study establishes an efficient and high-accuracy new paradigm for solving complex PDEs dominated by high-frequency dynamics. Code and data used are available on https://github.com/guozihao-hub/MgFNO/tree/master. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08615 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations Guo, Zi-Hao Li, Hou-Biao Numerical Analysis 65M55, 65M22 G.1.8; G.1.3 Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator (MgFNO) that accelerates the training of traditional Fourier neural operators through a novel three-level hierarchical architecture. The key innovation of MgFNO lies in its decoupled training strategy employing three distinct networks at different resolution levels: a coarse-level network first learns low-resolution approximations, an intermediate network refines the solution, and a fine-level network achieves high-resolution accuracy. By combining the frequency principle of deep neural networks with multigrid methodology, MgFNO effectively bridges the complementary learning patterns of neural networks (low-to-high frequency) and multigrid methods (high-to-low frequency error reduction).Experimental results demonstrate that MgFNO achieves relative errors of 0.17%, 0.28%, and 0.22% on the Burgers' equation, Darcy flow, and Navier-Stokes equations, respectively, representing reductions of 89%, 71%, and 83% compared to the conventional FNO. Furthermore, MgFNO supports zero-shot super-resolution prediction, enabling direct application to high-resolution scenarios after training on coarse grids. This study establishes an efficient and high-accuracy new paradigm for solving complex PDEs dominated by high-frequency dynamics. Code and data used are available on https://github.com/guozihao-hub/MgFNO/tree/master. |
| title | MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations |
| topic | Numerical Analysis 65M55, 65M22 G.1.8; G.1.3 |
| url | https://arxiv.org/abs/2407.08615 |