用三维平衡密度数据学习可迁移的液体自由能泛函，无需标签即可预测相变和界面行为。
Equivariant learning of a transferable three-dimensional classical density functional
- 从三维平衡密度场直接学习保持对称性的自由能泛函
- 单个泛函跨温度、尺寸、系综转移，准确复现结构因子与气液共存
- 适用于复杂几何，预测胶体间溶剂桥形成与断裂等非单调力
液体的集体行为对热力学条件、界面和受限环境敏感，但每次新状态通常需独立原子模拟。经典密度泛函理论提供可重用的变分描述，但其核心过剩自由能泛函通常未知，已有学习方法多限于平面或低维情形。本文展示可直接从全三维平衡密度场中学习该泛函，同时保持空间对称性与变分一致性，且无需自由能或化学位标签。单一学习泛函可在不同温度、系统尺寸及统计系综间迁移，复现结构因子、物态方程、气液共存及界面展宽，这些均未作为训练目标。应用于复杂三维几何时，成功预测胶体间溶剂贫乏桥的形成与破裂所产生的非单调作用力，以及互连螺旋孔中的吸附行为。结果表明，平衡密度数据可转化为可迁移的热力学生成器，连接微观液体结构与响应、相变及集体现象。
原文摘要 · Abstract (English)
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.
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