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Many learning tasks, including learning potential energy surfaces from abinitio calculations, involve global spatial symmetries and permutationalsymmetry between atoms or general particles. Equivariant graph neural networksare a standard approach to such problems, with one of the most successfulmethods employing tensor products between various tensors that transform underthe spatial group. However, as the number of different tensors and thecomplexity of relationships between them increase, maintaining parsimony andequivariance becomes increasingly challenging. In this paper, we propose usingfusion diagrams, a technique widely employed in simulating SU($2$)-symmetricquantum many-body problems, to design new equivariant components forequivariant neural networks. This results in a diagrammatic approach toconstructing novel neural network architectures. When applied to particleswithin a given local neighborhood, the resulting components, which we term"fusion blocks," serve as universal approximators of any continuous equivariantfunction defined in the neighborhood. We incorporate a fusion block intopre-existing equivariant architectures (Cormorant and MACE), leading toimproved performance with fewer parameters on a range of challenging chemicalproblems. Furthermore, we apply group-equivariant neural networks to studynon-adiabatic molecular dynamics of stilbene cis-trans isomerization. Ourapproach, which combines tensor networks with equivariant neural networks,suggests a potentially fruitful direction for designing more expressiveequivariant neural networks.

Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism