Models of collective motion usually reduce social interactions to pairwise forces, despite empirical evidence for many-body effects. We introduce the directional atomic cluster expansion (dACE), which adapts the atomic cluster expansion (ACE) of interatomic potentials to self-propelled agents. dACE provides a systematically improvable, exactly equivariant expansion of an agent's acceleration that splits uniquely into 1-, 2-, 3-body and higher contributions, along with a self term for the agent's own past motion that is fitted so that it does not absorb the interactions. On simulated collectives in two and three dimensions with known rules, dACE recovers the interaction law body order by body order from little data, while the neural model SocialNet, the current closest competitor, needs more data, has much larger errors on many-body or chiral rules and is restricted to 2D. On real data sets of tracked zebrafish and golden shiners, the laws learned by dACE agree better between disjoint halves of the data than SocialNet's. The learned laws are interpretable, and in the shiners the 3-body term opposes the pairwise response as neighbourhoods get dense. Finally, we show that next-step prediction accuracy mostly measures self-propulsion and noise rather than the interactions, and suggest more robust combinations of error measures for interaction learning.

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