Spherical Metadensity Functional Learning for Inhomogeneous Classical Fluids

We develop classical density functional learning to address fluids with truncated pairwise interparticle interactions in three-dimensional spherical geometry. Simulation data for systems with randomized repulsive pair potentials provide the basis for supervised training of a neural metadensity functional, thereby making efficient use of results for radial distribution functions in the bulk fluid via the test particle route. Specifically, we develop spherical local learning in order to represent the one-body direct correlation functional in terms of a neural network, which captures spatial curvature effects as well as the metadensity functional dependence on the thermally scaled pair potential. The framework yields efficient access to inhomogeneous structuring and related physical phenomena that occur in fluids and general solvents when adsorbed against curved solutes and confined inside of spherical and planar cavities. Test particle setups facilitate accurate prediction of the bulk fluid pair structure and verification of thermodynamic test particle sum rules via functional line integration. Applying the metadensity functional for Henderson inversion allows one to infer accurately the pair potential from the bulk radial distribution function. We address implications of the geometrical setup for two-body quantities and obtain the two-body direct correlation functional from automatic differentiation. For the hard sphere fluid, we confirm metadensity functional predictions against results from a standard neural density functional with fixed pair potential as well as to an analytic functional as given by fundamental measure theory. Simulation results provide further reference and corroborate reliable results of the spherical neural metadensity functional across a broad range of applications.