Where the "'" indicates that the sum only considers terms for which at least two of , , and are non-zero, ensuring a true three-body interaction. Previous work has shown that this functional form is well suited for describing variety of systems, including inorganic materials, covalent materials, condensed phase reacting systems, and molecular systems and materials. Models are fit by force-, energy-, and/or stress matching to gas-and/or condensed-phase atomic configurations labeled by a ground-truth method such as DFT; this is achieved by minimizing an objective function of the form: