  Much of the success of deep learning is drawn from building architectures
that properly respect underlying symmetry and structure in the data on which
they operate - a set of considerations that have been united under the banner
of geometric deep learning. Often problems in the physical sciences deal with
relatively small sets of points in two- or three-dimensional space wherein
translation, rotation, and permutation equivariance are important or even vital
for models to be useful in practice. In this work, we present rotation- and
permutation-equivariant architectures for deep learning on these small point
clouds, composed of a set of products of terms from the geometric algebra and
reductions over those products using an attention mechanism. The geometric
algebra provides valuable mathematical structure by which to combine vector,
scalar, and other types of geometric inputs in a systematic way to account for
rotation invariance or covariance, while attention yields a powerful way to
impose permutation equivariance. We demonstrate the usefulness of these
architectures by training models to solve sample problems relevant to physics,
chemistry, and biology.
