, where the {} brackets for the G functions denote a list over all permutations of the parameters ζ R , R R s , η A , R A s , θ s and ζ . {A i } is the set of the four attributes for atom i, namely, q i , ε i , σ i and λ i . The parameter values used to construct the features are given in Table . In total, there are 24 radial features and 128 angular features, along with another 24 radial features using the charged cutoff function f cq , and the four atomic attributes, for a total of 180 features per atom. We denote the list of the ghost particle feature vectors as F GP . Note that while F GP is invariant to translation and rotation, it is not invariant to the re-indexing of atoms. However, the mapped set of features, M(F GP ) is symmetric to re-indexing. In this way, our loss function, energies and forces are invariant to all three symmetries.