Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles

TL;DR

This paper introduces a deep learning method to discover and analyze continuous symmetries and their algebraic structures in datasets, with applications in physics and data science.

Contribution

It presents a novel neural network-based approach to identify symmetry groups and their Lie algebras directly from data, bridging machine learning and mathematical symmetry analysis.

Findings

Successfully identified symmetries in various groups including SO(2), SO(3), SO(4), and Lorentz groups.

Demonstrated the method's generality with applications to different datasets and symmetry types.

Opened new avenues for using machine learning in mathematical studies of Lie groups.

Abstract

We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. We construct loss functions that ensure that the applied transformations are symmetries and that the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups $SO(2)$, $SO(3)$, and $SO(4)$, and of the Lorentz group $SO(1,3)$. Other examples include squeeze mapping, piecewise discontinuous labels, and $SO(10)$, demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.