Machine Learning Lie Structures & Applications to Physics

Abstract

Classical and exceptional Lie algebras and their representations are among the most important tools in the analysis of symmetry in physical systems. In this letter we show how the computation of tensor products and branching rules of irreducible representations are machine-learnable, and can achieve relative speed-ups of orders of magnitude in comparison to the non-ML algorithms.

Machine Learning Symmetry

“…Classifying which representations of su(5) contain the matter content of the standard model is an important problem for particle physics model building. Such computations are seen to run on exponential time scales [23] when carried out on LieART, a Mathematica software for computations in Lie algebra representation theory [40].…”

“…The dimensions of these representations range from 1, for (0, 0, 0, 0) to 9765625 for (4,4,4,4). This data is again fed to a relu activated MLP [23].…”

“…Doing so might unveil new structures and organizing principles for the given parameter space, and eventually to new insight. In the following, we will report on some initial progress in using neural networks for these purposes [22,23].…”

“…The works [22,23] are in a sense part of a more general program of applying methods from data science and machine learning to study problems in mathematical physics, see [24,25] for a pedagogical treatment. By now, machine learning and data science have been used to learn various aspects of number theory [26,27,28], quiver gauge theories and cluster algebras [29], knot theory [30,31,32] as well as graph theory [33] and commutative algebra [34].…”

“…By now, machine learning and data science have been used to learn various aspects of number theory [26,27,28], quiver gauge theories and cluster algebras [29], knot theory [30,31,32] as well as graph theory [33] and commutative algebra [34]. Importantly, and in hindsight somewhat as a precursor of [23], support vector machines and neural networks were used to learn the algebraic structures of discrete groups and rings [35]. It was explicitly shown there that these machine learning frameworks are able to solve difficult problems in the representation theory of discrete groups without having to appeal to computationally expensive methods like Sylow theorems.…”