Group Equivariant Fourier Neural Operators for Partial Differential Equations
Group Equivariant Fourier Neural Operators for Partial Differential Equations
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DOI：
10.48550/arxiv.2306.05697
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发表时间：
2023-06
期刊：
影响因子：
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通讯作者：
Jacob Helwig;Xuan Zhang;Cong Fu;Jerry Kurtin;Stephan Wojtowytsch;Shuiwang Ji
中科院分区：
文献类型：
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作者：
Jacob Helwig;Xuan Zhang;Cong Fu;Jerry Kurtin;Stephan Wojtowytsch;Shuiwang Ji
我们考虑用傅里叶神经算子（FNO）求解偏微分方程（PDE），傅里叶神经算子在频域中运行。由于物理定律不依赖于用于描述它们的坐标系，因此期望在神经算子架构中对这种对称性进行编码，以获得更好的性能和更易的学习。虽然使用群论在物理域中对对称性进行编码已被广泛研究，但如何在频域中捕捉对称性却研究不足。在这项工作中，我们将群卷积扩展到频域，并通过利用傅里叶变换的等变性性质设计了对旋转、平移和反射具有等变性的傅里叶层。由此产生的$G$-FNO架构在不同输入分辨率下具有良好的泛化能力，并且在具有不同对称程度的设置中表现良好。我们的代码作为AIRS库（https://github.com/divelab/AIRS）的一部分公开可用。
We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting $G$-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).