## Introduction Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning. However, both pursuits require knowledge of the underlying symmetries in data. To address this, we propose learning symmetries directly from data via flow matching on Lie groups. We formulate symmetry discovery as learning a distribution over a larger hypothesis group, such that the learned distribution matches the symmetries observed in data. Our method, \lieflow, is more flexible in terms of the types of groups it can discover and requires fewer assumptions. Experiments on 2D and 3D point clouds demonstrate the successful discovery of discrete groups, including reflections by flow matching over a complex domain. We identify a key challenge where the symmetric arrangement of the target modes causes 'last-minute convergence,' where samples remain stationary until relatively late in the flow, and introduce a novel interpolation scheme for flow matching for symmetry discovery. ## Contextual Technicality Lie groups are a fundamental concept in mathematics and physics. A Lie group is a collection of transformations that satisfy certain mathematical properties, such as unity and commutativity. The discovery of symmetries on data uses these groups to understand the geometric structures present in the data. Flow matching is a statistical technique that uses probability flows to calculate dependence between variables. In the context of this problem, flow matching on Lie groups serves to discover symmetries directly from data. ## Conclusion Discovering symmetries on data is a fundamental task in understanding physical systems and improving performance of machine learning models. The \lieflow method proposes a new approach for tackling this challenge, using flow matching on Lie groups to explore symmetries directly from data. ## Future Prospects Discovering symmetries on data offers many future prospects in understanding physical systems and improving performance of machine learning models. It is likely that machine learning technologies will be used to discover symmetries in complex data, such as images or genetic data.