  Neural networks that are equivariant to rotations, translations, reflections,
and permutations on n-dimensional geometric space have shown promise in
physical modeling for tasks such as accurately but inexpensively modeling
complex potential energy surfaces to guiding the sampling of complex dynamical
systems or forecasting their time evolution. Current state-of-the-art methods
employ spherical harmonics to encode higher-order interactions among particles,
which are computationally expensive. In this paper, we propose a simple
alternative functional form that uses neurally parametrized linear combinations
of edge vectors to achieve equivariance while still universally approximating
node environments. Incorporating this insight, we design spatial attention
kinetic networks with E(n)-equivariance, or SAKE, which are competitive in
many-body system modeling tasks while being significantly faster.
