Bridging electronic and classical density-functional theory using universal machine-learned functional approximations

The accuracy of density-functional theory (DFT) is determined by the quality of the approximate functionals, such as exchange-correlation in electronic DFT and the excess functional in the classical DFT formalism of fluids. The exact functional is highly nonlocal for both electrons and fluids, yet most approximate functionals are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations are promising in both electronic and classical DFT, but have so far employed disparate approaches with limited generality. Here, we formulate a universal approximation framework and training protocol for nonlocal ML functionals, combining features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this approach for several 1D and quasi-1D problems and demonstrate that a functional with exactly the same hyperparameters achieves excellent accuracy for the hard-rod fluid, the inhomogeneous Ising model, the exact exchange functional for electrons, the electron kinetic energy functional for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to exact 3D functionals spanning electronic and classical DFT.