Learning Neural Free-Energy Functionals with Pair-Correlation Matching

The intrinsic Helmholtz free-energy functional, the centerpiece of classical density functional theory, is at best only known approximately for 3D systems. Here we introduce a method for learning a neural-network approximation of this functional by exclusively training on a dataset of radial distribution functions, circumventing the need to sample costly heterogeneous density profiles in a wide variety of external potentials. For a supercritical Lennard-Jones system with planar symmetry, we demonstrate that the learned neural free-energy functional accurately predicts inhomogeneous density profiles under various complex external potentials obtained from simulations.