Fourier Neural Operators with rank-1 lattice points and hyperbolic cross

2026-06-07

AI Summary

This work addresses the limitations of Fourier neural operators in terms of generalization accuracy and computational efficiency for mappings in function spaces. To overcome these challenges, the authors propose an efficient architecture that uniquely integrates rank-1 lattice sampling with hyperbolic cross truncation in the frequency domain. This combination reduces high-dimensional Fourier transforms to one-dimensional fast Fourier transforms and introduces a structured lattice-based training set in parameter space. The resulting method substantially decreases the number of model parameters, spatial sampling points, and required training samples. Evaluated on elliptic partial differential equations over toroidal domains, the approach achieves higher approximation accuracy while maintaining lower computational complexity compared to existing methods.

Abstract

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.

Problem

• Fourier Neural Operator
• generalization error
• rank-1 lattice
• hyperbolic cross
• function approximation

Innovation

• Fourier Neural Operator
• rank-1 lattice
• hyperbolic cross
• generalization error
• fast Fourier transform