Analysis of Fourier Neural Operators via Effective Field Theory
This provides theoretical foundations for understanding and improving neural operators, which are important for solving functional problems in scientific computing.
The researchers analyzed Fourier Neural Operators (FNOs) using effective field theory to explain their stability, generalization, and frequency behavior, showing that nonlinear activations couple frequency inputs to high-frequency modes and deriving criticality conditions for weight initialization that ensure uniform scale across depth.
Fourier Neural Operators (FNOs) have emerged as leading surrogates for solver operators for various functional problems, yet their stability, generalization and frequency behavior lack a principled explanation. We present a 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 derive explicit criticality conditions on the weight initialization ensemble that ensure small input perturbations maintain a uniform scale across depth, and we confirm experimentally that the theoretically predicted ratio of kernel perturbations matches the measurements. Taken together, our results quantify how nonlinearity enables neural operators to capture non-trivial features, supply criteria for hyperparameter selection via criticality analysis, and explain why scale invariant activations and residual connections enhance feature learning in FNOs.