Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective

TL;DR

This paper develops a mean-field theory for Fourier Neural Operators, analyzing their expressivity and trainability, revealing phase transitions related to mode truncation and gradient behavior, supported by experimental validation.

Contribution

It introduces a mean-field framework for FNO, linking phase transitions to expressivity and trainability, and provides practical insights for stable training.

Findings

Identifies a phase transition in FNO related to mode truncation.

Connects ordered-chaos phases to vanishing/exploding gradients.

Experimental results support theoretical predictions.

Abstract

In this paper, we explores the expressivity and trainability of the Fourier Neural Operator (FNO). We establish a mean-field theory for the FNO, analyzing the behavior of the random FNO from an edge of chaos perspective. Our investigation into the expressivity of a random FNO involves examining the ordered-chaos phase transition of the network based on the weight distribution. This phase transition demonstrates characteristics unique to the FNO, induced by mode truncation, while also showcasing similarities to those of densely connected networks. Furthermore, we identify a connection between expressivity and trainability: the ordered and chaotic phases correspond to regions of vanishing and exploding gradients, respectively. This finding provides a practical prerequisite for the stable training of the FNO. Our experimental results corroborate our theoretical findings.