What is this paper about? The authors teach a machine to write a reusable rulebook for liquids. In classical density functional theory (cDFT) the key ingredient is an “excess free‑energy functional” that encodes how particles interact. That functional is usually unknown for realistic three‑dimensional fluids. The paper introduces Equi‑cDFT, a method that learns this missing functional from three‑dimensional equilibrium density fields while preserving spatial symmetry and the variational structure of cDFT.

What the researchers did. They represent the fluid by its one‑body density on a regular 3D grid. For each grid point the method encodes the local neighborhood with symmetry‑adapted features built from Cartesian moments. A shared neural readout turns those features into a local excess free energy per particle. Summing these local contributions recovers a global excess free‑energy functional F_exc[ρ,T]. The team trains the model without ever giving it free energies or chemical‑potential labels. Instead they differentiate the learned functional automatically to get the one‑body direct correlation, form a local chemical‑potential field, and minimize the spatial variance of that local chemical potential. For canonical training data they remove the unknown constant chemical potential by subtracting the predicted spatial mean.

How it works at a high level. The model enforces the correct transformation of the functional under rotations and reflections of the Cartesian grid (equivariance) by using features that are invariant under the cubic point‑group operations. Learning the functional “energy‑first” lets the same model deliver both equilibrium densities and the derivative fields that control response. The authors test this approach on a standard model fluid, the Lennard–Jones truncated and shifted at 2.5σ (LJTS) system, using equilibrium molecular dynamics (MD) density fields collected under varying external fields.