Metadensity Functional Learning for Classical Fluids: Regularizing with Pair Correlations

We investigate and exploit consequences of the recent neural metadensity functional theory by [Kampa, S. M. Phys. Rev. Lett. 2025, 134, 107301] for describing the physics of inhomogeneous fluids. The metadensity dependence on the pair potential is relevant for soft matter design and Henderson inversion, and it allows one to change the pair potential on the fly at the prediction stage. Here we consider one-dimensional systems with short-ranged (truncated) interparticle forces and draw on the functional pair potential dependence to investigate "metadirect" routes toward the bulk fluid pair correlation structure. Classical density functional theory provides the required functional relationships. Efficient variational calculus is implemented by neural functional line integration and automatic differentiation. We regularize local learning of neural functionals by comparing the pair structure from different routes. Thereby, results from metadirect functional differentiation are matched against accurate test particle data from an initial locally trained metadensity functional. Accessing the pair structure via the metadensity functional dependence circumvents Ornstein-Zernike inversion, and it is based on first principles.