Discovering Symmetry Groups with Flow Matching

Yuxuan Chen, Jung Yeon Park, Floor Eijkelboom, Jianke Yang, Jan-Willem Van De Meent, Lawson L.S. Wong, Robin Walters
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:16295-16323, 2026.

Abstract

Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data, yet discovering these symmetries automatically is challenging. We propose LieFlow, a novel framework that reframes symmetry discovery as a distribution learning problem on Lie groups. Instead of searching for the symmetry generators, our approach operates directly in group space, modeling a symmetry distribution over a large hypothesis group $G$. The support of the learned distribution reveals the underlying symmetry group $H \subseteq G$. Unlike previous works, LieFlow can discover both continuous and discrete symmetries within a unified framework, without assuming a fixed Lie algebra basis or a specific distribution over the group elements. Experiments on synthetic 2D and 3D point clouds, ModelNet10, and a real-world MI-Motion dataset show that LieFlow accurately discovers continuous and discrete subgroups, significantly outperforming a state-of-the-art baseline, LieGAN, in identifying discrete symmetries.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-chen26dg, title = {Discovering Symmetry Groups with Flow Matching}, author = {Chen, Yuxuan and Park, Jung Yeon and Eijkelboom, Floor and Yang, Jianke and Van De Meent, Jan-Willem and Wong, Lawson L.S. and Walters, Robin}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {16295--16323}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/chen26dg/chen26dg.pdf}, url = {https://proceedings.mlr.press/v306/chen26dg.html}, abstract = {Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data, yet discovering these symmetries automatically is challenging. We propose LieFlow, a novel framework that reframes symmetry discovery as a distribution learning problem on Lie groups. Instead of searching for the symmetry generators, our approach operates directly in group space, modeling a symmetry distribution over a large hypothesis group $G$. The support of the learned distribution reveals the underlying symmetry group $H \subseteq G$. Unlike previous works, LieFlow can discover both continuous and discrete symmetries within a unified framework, without assuming a fixed Lie algebra basis or a specific distribution over the group elements. Experiments on synthetic 2D and 3D point clouds, ModelNet10, and a real-world MI-Motion dataset show that LieFlow accurately discovers continuous and discrete subgroups, significantly outperforming a state-of-the-art baseline, LieGAN, in identifying discrete symmetries.} }
Endnote
%0 Conference Paper %T Discovering Symmetry Groups with Flow Matching %A Yuxuan Chen %A Jung Yeon Park %A Floor Eijkelboom %A Jianke Yang %A Jan-Willem Van De Meent %A Lawson L.S. Wong %A Robin Walters %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-chen26dg %I PMLR %P 16295--16323 %U https://proceedings.mlr.press/v306/chen26dg.html %V 306 %X Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data, yet discovering these symmetries automatically is challenging. We propose LieFlow, a novel framework that reframes symmetry discovery as a distribution learning problem on Lie groups. Instead of searching for the symmetry generators, our approach operates directly in group space, modeling a symmetry distribution over a large hypothesis group $G$. The support of the learned distribution reveals the underlying symmetry group $H \subseteq G$. Unlike previous works, LieFlow can discover both continuous and discrete symmetries within a unified framework, without assuming a fixed Lie algebra basis or a specific distribution over the group elements. Experiments on synthetic 2D and 3D point clouds, ModelNet10, and a real-world MI-Motion dataset show that LieFlow accurately discovers continuous and discrete subgroups, significantly outperforming a state-of-the-art baseline, LieGAN, in identifying discrete symmetries.
APA
Chen, Y., Park, J.Y., Eijkelboom, F., Yang, J., Van De Meent, J., Wong, L.L. & Walters, R.. (2026). Discovering Symmetry Groups with Flow Matching. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:16295-16323 Available from https://proceedings.mlr.press/v306/chen26dg.html.

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