  The rapid progress of machine learning interatomic potentials over the past
couple of years produced a number of new architectures. Particularly notable
among these are the Atomic Cluster Expansion (ACE), which unified many of the
earlier ideas around atom density-based descriptors, and Neural Equivariant
Interatomic Potentials (NequIP), a message passing neural network with
equivariant features that showed state of the art accuracy. In this work, we
construct a mathematical framework that unifies these models: ACE is
generalised so that it can be recast as one layer of a multi-layer
architecture. From another point of view, the linearised version of NequIP is
understood as a particular sparsification of a much larger polynomial model.
Our framework also provides a practical tool for systematically probing
different choices in the unified design space. We demonstrate this by an
ablation study of NequIP via a set of experiments looking at in- and
out-of-domain accuracy and smooth extrapolation very far from the training
data, and shed some light on which design choices are critical for achieving
high accuracy. Finally, we present BOTNet (Body-Ordered-Tensor-Network), a
much-simplified version of NequIP, which has an interpretable architecture and
maintains accuracy on benchmark datasets.
