让神经算子更符合物理对称性，提升求解效率并大幅减少参数量。
Isotropic Fourier Neural Operators
- 引入各向同性傅里叶层，强制保持空间对称性
- 2D场景参数量减少16倍，3D场景减少96倍
- 特别适合追求高效与物理一致性建模的研究者
傅里叶神经算子是学习函数空间映射的深度学习模型，可用于快速求解偏微分方程（PDE），在某些情况下显著快于传统求解器。其核心是傅里叶层，直接对傅里叶模态进行线性变换，参数依赖波数。然而，大多数物理系统具有各向同性，结果不随坐标系变化，而现有线性变换未必保持此对称性。本文提出改进的线性变换方式，确保空间对称性被尊重，称为各向同性傅里叶神经算子。该方法不仅提升模型性能，还使2D场景参数量减少至1/16，3D场景减少至1/96。
原文摘要 · Abstract (English)
Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.
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