Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles
This work addresses the problem of automating symmetry discovery in datasets, which is incremental as it applies existing neural network methods to a new mathematical task.
The authors developed a deep learning algorithm to discover and identify continuous symmetry groups in labeled datasets, validated through examples like rotation groups and the Lorentz group, demonstrating its generality for applications in physics and data science.
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.