Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective
This work provides theoretical insights into the behavior of FNOs, which is important for researchers in machine learning and scientific computing, but it is incremental as it builds on existing mean-field theories for neural networks.
The paper tackles the problem of understanding the expressivity and trainability of the Fourier Neural Operator (FNO) by establishing a mean-field theory, analyzing it from an edge of chaos perspective, and identifying a connection between expressivity and trainability, with experimental results supporting the theoretical findings.
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.