  Physical theories grounded in mathematical symmetries are an essential
component of our understanding of a wide range of properties of the universe.
Similarly, in the domain of machine learning, an awareness of symmetries such
as rotation or permutation invariance has driven impressive performance
breakthroughs in computer vision, natural language processing, and other
important applications. In this report, we argue that both the physics
community and the broader machine learning community have much to understand
and potentially to gain from a deeper investment in research concerning
symmetry group equivariant machine learning architectures. For some
applications, the introduction of symmetries into the fundamental structural
design can yield models that are more economical (i.e. contain fewer, but more
expressive, learned parameters), interpretable (i.e. more explainable or
directly mappable to physical quantities), and/or trainable (i.e. more
efficient in both data and computational requirements). We discuss various
figures of merit for evaluating these models as well as some potential benefits
and limitations of these methods for a variety of physics applications.
Research and investment into these approaches will lay the foundation for
future architectures that are potentially more robust under new computational
paradigms and will provide a richer description of the physical systems to
which they are applied.
