From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators
2026-07-01 • Machine Learning
Machine Learning
AI summaryⓘ
The authors studied how Fourier neural operators (FNOs), a type of machine learning model, can predict the behavior of certain time-dependent physical systems described by dissipative equations. They showed that FNOs work well when the underlying problems can be accurately represented using spectral methods and provided guarantees on how many examples are needed to learn these operators. Their results apply broadly to many important equations like Navier-Stokes and Allen-Cahn and account for different types of nonlinearities. Essentially, the authors linked traditional mathematical methods with modern neural networks to explain when FNOs can efficiently learn complex dynamic systems.
Fourier neural operatorsDissipative evolution equationsSpectral methodsPolynomial sample complexityNavier-Stokes equationsAllen-Cahn equationCahn-Hilliard equationNonlinear operatorsOperator learningApproximation theory
Authors
Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek
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.