Analysis of Fourier Neural Operators via Effective Field Theory

Abstract: Fourier Neural Operators (FNOs) have emerged as leading surrogates for high-dimensional partial-differential equations, yet their stability, generalization and frequency behavior lack a principled explanation. We present the first systematic effective-field-theory analysis of FNOs in an infinite-dimensional function space, deriving closed recursion relations for the layer kernel and four-point vertex and then examining three practically important settings-analytic activations, scale-invariant cases and architectures with residual connections. The theory shows that nonlinear activations inevitably couple frequency inputs to high-frequency modes that are otherwise discarded by spectral truncation, and experiments confirm this frequency transfer. For wide networks we obtain explicit criticality conditions on the weight-initialization ensemble that keep small input perturbations to have uniform scale across depth, and empirical tests validate these predictions. Taken together, our results quantify how nonlinearity enables neural operators to capture non-trivial features, supply criteria for hyper-parameter selection via criticality analysis, and explain why scale-invariant activations and residual connections enhance feature learning in FNOs.