aiqbench analysis
This research introduces LieFlow, a framework that treats symmetry discovery as a distribution learning problem on Lie groups to automatically identify both continuous and discrete symmetries in data. By operating directly within group space rather than searching for generators, the method significantly outperforms existing baselines like LieGAN across synthetic point clouds and real-world motion datasets. The approach advances machine learning by leveraging mathematical structures from physics to improve sample efficiency without requiring fixed algebraic assumptions.
Discovering Symmetry Groups with Flow Matching
arXiv:2512.20043v3 Announce Type: replace 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.
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.
Artificial Intelligence (cs.AI)
arXiv:2512.20043 [cs.AI]
(or arXiv:2512.20043v3 [cs.AI] for this version)
Full text under the publisher's open licence · 164 words
Related coverage
LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery
arXiv · Sep 30, 2026
Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs
arXiv · Oct 28, 2026
Horizon Simulation’s Quantum Dressing Shares 1.5% in Graviton Channel
Quantum Zeitgeist · Aug 3, 2026
LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems
arXiv · Aug 4, 2026