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 [Sammüller et al., Proc. Natl. Acad. Sci. USA 120, e2312484120 (2023)]. While Dijkman et al. demonstrated pair-correlation matching of a global neural free-energy functional, we argue in favor of local one-body learning for flexible neural modeling of the full Mermin-Evans density-functional map. Using spatial localization gives access to accurate neural free-energy functionals, including convolutional neural networks, that transcend the training box.