Cubic-equivariant neural density functional theory for three-dimensional lattice fluids

We construct a neural classical density functional that acts directly on unrestricted three-dimensional density profiles. As a computationally tractable test bed, we consider parallel hard cubes of side length three on a simple cubic lattice. A fully convolutional network learns the one-body direct-correlation functional $c^{(1)}[ρ]$ from data obtained with grand-canonical Monte Carlo simulations in randomized external potentials. Complete profiles are used during both training and inference; a stochastic Bernoulli mask on the output sites makes full-profile training effective without explicitly extracting and storing overlapping local density windows. Averaging the first-layer kernels over all 48 rotations and reflections of the cubic point group additionally imposes exact cubic equivariance without data augmentation. We compare the learned functional with independent simulation data and with the lattice fundamental-measure functional of Lafuente and Cuesta. The neural functional markedly improves the homogeneous equation of state and the density profile at a planar hard wall. For the anisotropic pair distribution around a fixed particle, both functionals reproduce the principal packing shells, with their relative accuracy depending on crystallographic direction. These results demonstrate neural density-functional calculations on complete three-dimensional profiles while also identifying accurate full-dimensional training data, thermodynamic consistency, and structural correlations as the central challenges for extensions to continuum fluids.