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：本文研究利用在频域中操作的傅里叶神经算子（FNO）求解偏微分方程（PDE）。由于物理定律不依赖于描述它们所使用的坐标系，因此将这类对称性编码到神经算子架构中，有助于提升性能并简化学习过程。尽管在物理域中利用群论编码对称性已得到广泛研究，但如何在频域中捕获对称性仍探索不足。在本工作中，我们将群卷积扩展至频域，并利用傅里叶变换的等变性，设计了对旋转、平移和反射等变的傅里叶层。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，并在具有不同对称性水平的环境中表现优异。我们的代码已作为AIRS库（https://github.com/divelab/AIRS）的一部分公开。