A Unified Framework to Enforce, Discover, and Promote Symmetry in
  Machine Learning
Samuel E. Otto Nicholas Zolman J. Nathan Kutz Steven L. Brunton
Symmetry is present throughout nature and continues to play an increasingly
central role in physics and machine learning. Fundamental symmetries, such as
Poincar\'{e} invariance, allow physical laws discovered in laboratories on
Earth to be extrapolated to the farthest reaches of the universe. Symmetry is
essential to achieving this extrapolatory power in machine learning
applications. For example, translation invariance in image classification
allows models with fewer parameters, such as convolutional neural networks, to
be trained on smaller data sets and achieve state-of-the-art performance. In
this paper, we provide a unifying theoretical and methodological framework for
incorporating symmetry into machine learning models in three ways: 1. enforcing
known symmetry when training a model; 2. discovering unknown symmetries of a
given model or data set; and 3. promoting symmetry during training by learning
a model that breaks symmetries within a user-specified group of candidates when
there is sufficient evidence in the data. We show that these tasks can be cast
within a common mathematical framework whose central object is the Lie
derivative associated with fiber-linear Lie group actions on vector bundles. We
extend and unify several existing results by showing that enforcing and
discovering symmetry are linear-algebraic tasks that are dual with respect to
the bilinear structure of the Lie derivative. We also propose a novel way to
promote symmetry by introducing a class of convex regularization functions
based on the Lie derivative and nuclear norm relaxation to penalize symmetry
breaking during training of machine learning models. We explain how these ideas
can be applied to a wide range of machine learning models including basis
function regression, dynamical systems discovery, multilayer perceptrons, and
neural networks acting on spatial fields such as images.