Reductive Lie Groups, such as the orthogonal groups, the Lorentz group, or the unitary groups, play essential roles across scientific fields as diverse as high energy physics, quantum mechanics, quantum chromodynamics, molecular dynamics, computer vision, and imaging. In this paper, we present a general Equivariant Neural Network architecture capable of respecting the symmetries of the finite-dimensional representations of any reductive Lie Group G. Our approach generalizes the successful ACE and MACE architectures for atomistic point clouds to any data equivariant to a reductive Lie group action. We also introduce the lie-nn software library, which provides all the necessary tools to develop and implement such general G-equivariant neural networks. It implements routines for the reduction of generic tensor products of representations into irreducible representations, making it easy to apply our architecture to a wide range of problems and groups. The generality and performance of our approach are demonstrated by applying it to the tasks of top quark decay tagging (Lorentz group) and shape recognition (orthogonal group).

翻译：约化李群，例如正交群、洛伦兹群或酉群，在高能物理、量子力学、量子色动力学、分子动力学、计算机视觉和成像等不同科学领域发挥着关键作用。在本文中，我们提出了一种通用的等变神经网络架构，能够尊重任何约化李群G的有限维表示的对称性。我们的方法将成功的ACE和MACE原子点云架构推广到任何对约化李群作用等变的数据上。我们还介绍了lie-nn软件库，它提供了开发和实现此类通用的G-等变神经网络所需的所有必要工具。该库实现了将表示的通用张量积约化为不可约表示的程序，使得我们的架构易于应用于广泛的问题和群。通过将其应用于顶夸克衰变标记（洛伦兹群）和形状识别（正交群）任务，我们证明了该方法的通用性和性能。