Lie Point Symmetry Data Augmentation for Neural PDE Solvers

Johannes Brandstetter, Max Welling, Daniel E Worrall
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:2241-2256, 2022.

Abstract

Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity—Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-brandstetter22a, title = {Lie Point Symmetry Data Augmentation for Neural {PDE} Solvers}, author = {Brandstetter, Johannes and Welling, Max and Worrall, Daniel E}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {2241--2256}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/brandstetter22a/brandstetter22a.pdf}, url = {https://proceedings.mlr.press/v162/brandstetter22a.html}, abstract = {Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity—Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.} }
Endnote
%0 Conference Paper %T Lie Point Symmetry Data Augmentation for Neural PDE Solvers %A Johannes Brandstetter %A Max Welling %A Daniel E Worrall %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-brandstetter22a %I PMLR %P 2241--2256 %U https://proceedings.mlr.press/v162/brandstetter22a.html %V 162 %X Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity—Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
APA
Brandstetter, J., Welling, M. & Worrall, D.E.. (2022). Lie Point Symmetry Data Augmentation for Neural PDE Solvers. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:2241-2256 Available from https://proceedings.mlr.press/v162/brandstetter22a.html.

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