A General Framework for Equivariant Neural Networks on Reductive Lie
  Groups
Ilyes Batatia Mario Geiger Jose Munoz Tess Smidt Lior Silberman Christoph Ortner
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).