From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

2026-06-30

AI Summary

This work establishes rigorous approximation and sample complexity guarantees for Fourier Neural Operators (FNOs) in learning solution operators of dissipative evolutionary partial differential equations. By constructing a class of evolution operators grounded in spectral methods, it bridges classical spectral approximation theory with modern operator learning within a unified analytical framework. The study proves that FNOs achieve polynomial sample complexity when approximating solution operators for equations such as Navier–Stokes, Allen–Cahn, and Cahn–Hilliard. The derived learning rates explicitly depend on the smoothness of the input functions, spatial dimensionality, regularity of the nonlinear terms, and the strength of dissipation, thereby providing a systematic theoretical explanation for the empirical efficiency of FNOs across a range of dissipative dynamical systems.

Abstract

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Problem

• Fourier neural operators
• sample complexity
• dissipative evolution equations
• solution operators
• spectral methods

Innovation

• Fourier Neural Operators
• spectral methods
• sample complexity
• operator learning
• dissipative evolution equations