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This rapid progress in machine learning interatomic potentials has produced a number of new architectures in just the past few years. Particularly notable among these are the atomic cluster expansion, 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 exhibited state-of-the-art accuracy at the time. Here we construct a mathematical framework that unifies these models: atomic cluster expansion is extended and recast as one layer of a multi-layer architecture, while the linearized 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 this unified design space. 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, sheds some light on which design choices are critical to achieving high accuracy. A much-simplified version of NequIP, which we call BOTnet (for body-ordered tensor network), has an interpretable architecture and maintains its accuracy on benchmark datasets.<\/jats:p>","DOI":"10.1038\/s42256-024-00956-x","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T10:05:33Z","timestamp":1736935533000},"page":"56-67","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":158,"title":["The design space of E(3)-equivariant atom-centred interatomic potentials"],"prefix":"10.1038","volume":"7","author":[{"given":"Ilyes","family":"Batatia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8826-2712","authenticated-orcid":false,"given":"Simon","family":"Batzner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D\u00e1vid P\u00e9ter","family":"Kov\u00e1cs","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Albert","family":"Musaelian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gregor N. 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