Neural density functionals: Local learning and pair-correlation matching

Recently Dijkman et al. (arxiv:2403.15007) proposed training classical neural density functionals via bulk pair-correlation matching. We show their method to be an efficient regularizer for neural functionals based on local learning of inhomogeneous one-body direct correlations [Samm\"uller et al., Proc. Natl. Acad. Sci. 120, e2312484120 (2023)]. We demonstrate that local one-body learning allows flexible neural modelling of the full Mermin-Evans density functional map, but that bulk pair-correlation matching alone does not. Using spatial localization gives access to accurate neural free energy functionals, including convolutional neural networks, that transcend the training box.