Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective
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 \emph{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.
Cite
Text
Koshizuka et al. "Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective." Neural Information Processing Systems, 2024. doi:10.52202/079017-0352Markdown
[Koshizuka et al. "Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective." Neural Information Processing Systems, 2024.](https://mlanthology.org/neurips/2024/koshizuka2024neurips-understanding/) doi:10.52202/079017-0352BibTeX
@inproceedings{koshizuka2024neurips-understanding,
title = {{Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective}},
author = {Koshizuka, Takeshi and Fujisawa, Masahiro and Tanaka, Yusuke and Sato, Issei},
booktitle = {Neural Information Processing Systems},
year = {2024},
doi = {10.52202/079017-0352},
url = {https://mlanthology.org/neurips/2024/koshizuka2024neurips-understanding/}
}