The swift progression of machine learning (ML) have not gone unnoticed in the realm of statistical mechanics. ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to compute the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption and capillarity.

翻译：机器学习（ML）的迅猛发展在统计力学领域并未被忽视。经典密度泛函理论（DFT）社区对ML技术产生了兴趣，因为这些技术能够发现自由能泛函，从而确定多粒子系统的平衡密度分布。在DFT中，外势描述了多粒子系统与外部场的相互作用，进而影响密度分布。在此背景下，我们引入了一个统计学习框架来推断施加于多粒子系统的外势。我们将贝叶斯推断方法与经典DFT框架相结合以重构外势，从而得到外势函数形式的概率化描述，并具备内在的不确定性量化。该框架通过一个正则系综的受限几何一维粒子系综（具有排除体积相互作用）进行了示例验证。所需的训练数据集通过蒙特卡洛（MC）模拟生成，其中外势被施加于该正则系综。MC模拟产生的粒子坐标被输入学习框架，以揭示外势。最终，我们能够利用DFT工具计算系统的平衡密度分布。我们的方法将推断得到的密度与通过DFT公式使用真实外势计算得到的精确密度进行了基准对比。所提出的贝叶斯方法能够准确推断外势和密度分布。我们还重点分析了基于可用模拟数据量的外势不确定性量化。本文介绍的看似简单的案例研究，或可作为研究包括吸附和毛细现象在内广泛应用的范例。