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).

翻译：我们研究利用在频域中运行的傅里叶神经算子（FNO）求解偏微分方程（PDE）。由于物理定律不依赖于用于描述它们的坐标系，因此将此类对称性编码到神经算子架构中，以期获得更优性能并简化学习过程，是值得期待的。尽管利用群论在物理域中编码对称性已被广泛研究，但如何在频域中捕获对称性仍探索不足。在本工作中，我们将群卷积扩展到频域，并利用傅里叶变换的等变性质，设计了对于旋转、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均表现出良好的泛化能力，并在具有不同对称性程度的场景中性能优异。我们的代码已作为AIRS库（https://github.com/divelab/AIRS）的一部分公开。