  Machine-learning interatomic potentials (MLIPs) have made a significant
contribution to the recent progress in the fields of computational materials
and chemistry due to MLIPs' ability of accurately approximating energy
landscapes of quantum-mechanical models while being orders of magnitude more
computationally efficient. However, the computational cost and number of
parameters of many state-of-the-art MLIPs increases exponentially with the
number of atomic features. Tensor (non-neural) networks, based on low-rank
representations of high-dimensional tensors, have been a way to reduce the
number of parameters in approximating multidimensional functions, however, it
is often not easy to encode the model symmetries into them.
  In this work we develop a formalism for rank-efficient equivariant tensor
networks (ETNs), i.e., tensor networks that remain invariant under actions of
SO(3) upon contraction. All the key algorithms of tensor networks like
orthogonalization of cores and DMRG carry over to our equivariant case.
Moreover, we show that many elements of modern neural network architectures
like message passing, pulling, or attention mechanisms, can in some form be
implemented into the ETNs. Based on ETNs, we develop a new class of
polynomial-based MLIPs that demonstrate superior performance over existing
MLIPs for multicomponent systems.
