Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles
Roy T. Forestano, K. Matchev, Katia Matcheva, Alexander Roman, Eyup B. Unlu, Sarunas Verner
Abstract
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. The constructed loss functions ensure that the applied transformations are symmetries and 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> . 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.