Cubic-Equivariant Neural Density Functional Theory for Three-Dimensional Lattice Fluids

What the math gives to ML

The paper provides two transferable mechanisms rather than merely a domain-specific application: exact averaging over the 48-element cubic point group to impose symmetry in three-dimensional convolutions, and learning a thermodynamically meaningful density functional whose functional derivative is the direct-correlation field. The first mechanism can produce symmetry-preserving 3D CNNs or equivariant neural operators without relying on finite data augmentation. The second can replace unconstrained vector-field prediction by a conservative functional-gradient model, making predictions integrable and enabling stable fixed-point inference through an explicit free-energy functional. A stochastic mask over complete profiles is also a useful computational recipe for unbiased dense supervision without materializing overlapping local patches.