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

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.

The R-mAtrIx Net

“…Remarkably, machine learning is already proving helpful in the analysis of symmetry in physical systems. In particular, one may verify the presence of a conjectured symmetry or even automate its search using machine learning [19,[46][47][48][49][50]87]. It would be very interesting to explicate the interplay of our program in this broader line of investigation.…”

“…This has made it possible to reliably train increasingly deeper networks which are optimized to carry out increasingly sophisticated tasks such as the direct computation of Ricci flat metrics on Calabi Yau manifolds [12][13][14][15] and the solution of differential equations without necessarily providing the neural network data obtained from explicitly sampling the solution. Further, in recent interesting developments, deep learning has been applied to analyze various aspects of symmetry in physical systems ranging from their classification to their automated detection [19,[46][47][48][49][50].…”