用特殊点网格提升神经算子效率，更少参数更准解方程
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
- 用秩-1格点替代传统网格，降低计算复杂度
- 在椭圆方程上实现更少参数、更少采样点的高精度求解
- 结合双曲截断，仅需一维快速傅里叶变换，适合大规模问题
傅里叶神经算子（FNO）是一种学习函数空间映射的神经网络架构，其高效实现依赖于多维傅里叶变换。通过推导关于空间和参数变量的一般正则性界，我们证明：将空间张量积网格替换为定制的秩-1格点，并在参数空间中使用精心构造的第二个格点作为训练点，可提升FNO的泛化误差性能。该方法在更少网络参数、更少空间点和更少训练样本下实现更精确高效的近似。此外，架构被简化，因为基于秩-1格点的高维傅里叶变换只需一维快速傅里叶变换，且可结合双曲截断频率索引集。我们在环面上的椭圆型偏微分方程上验证了基于格点的双曲截断FNO的优势。
原文摘要 · Abstract (English)
The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.
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