Group Equivariant Fourier Neural Operators for Partial Differential Equations

Jacob Helwig, Xuan Zhang, Cong Fu, Jerry Kurtin, Stephan Wojtowytsch, Shuiwang Ji
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:12907-12930, 2023.

Abstract

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting $G$-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-helwig23a, title = {Group Equivariant {F}ourier Neural Operators for Partial Differential Equations}, author = {Helwig, Jacob and Zhang, Xuan and Fu, Cong and Kurtin, Jerry and Wojtowytsch, Stephan and Ji, Shuiwang}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12907--12930}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/helwig23a/helwig23a.pdf}, url = {https://proceedings.mlr.press/v202/helwig23a.html}, abstract = {We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting $G$-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).} }
Endnote
%0 Conference Paper %T Group Equivariant Fourier Neural Operators for Partial Differential Equations %A Jacob Helwig %A Xuan Zhang %A Cong Fu %A Jerry Kurtin %A Stephan Wojtowytsch %A Shuiwang Ji %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-helwig23a %I PMLR %P 12907--12930 %U https://proceedings.mlr.press/v202/helwig23a.html %V 202 %X We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting $G$-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).
APA
Helwig, J., Zhang, X., Fu, C., Kurtin, J., Wojtowytsch, S. & Ji, S.. (2023). Group Equivariant Fourier Neural Operators for Partial Differential Equations. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12907-12930 Available from https://proceedings.mlr.press/v202/helwig23a.html.

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