这篇论文用等变学习搞定了三维经典密度泛函，一个模型就能跨温度、尺寸用，还能算液汽共存和胶体力，挺有意思。
研究者提出直接从三维平衡密度场学习经典密度泛函，无需自由能或化学势标签，保持空间对称性与变分一致性。单个学习到的泛函可跨温度、系统尺寸和统计系综转移，能恢复结构因子、状态方程、液-汽共存和界面展宽，这些均非训练目标。在复杂三维几何中，该泛函预测了胶体间溶剂耗尽桥形成与断裂的非单调力，以及互连螺旋孔中的吸附。
Equivariant learning of a transferable three-dimensional classical density functional
Liquids exhibit collective behavior that depends sensitively on thermodynamic conditions, interfaces and confinement, yet predicting each new state commonly requires a separate atomistic simulation. Classical density functional theory offers a reusable variational description, but its central excess free-energy functional is generally unknown, and learned approximations have largely remained restricted to planar or lower-dimensional settings. Here we show that this functional can be learned directly from fully three-dimensional equilibrium density fields while preserving spatial symmetry and variational consistency, without free-energy or chemical-potential labels. A single learned functional transfers across temperatures, system sizes and statistical ensembles, and recovers structure factors, the equation of state, liquid--vapor coexistence and interfacial broadening, none of which are used as training targets. Applied to complex three-dimensional geometries, it predicts the non-monotonic force associated with formation and rupture of a solvent-depleted bridge between colloids and adsorption in an interconnected gyroid pore. These results demonstrate that equilibrium density data can be converted into a transferable thermodynamic generator connecting microscopic liquid structure to response, phase behavior and collective phenomena.