We explore the feasibility of using machine learning methods to obtain an analytic form of the classical free energy functional for two model fluids, hard rods and Lennard-Jones, in one dimension. The equation learning network proposed by Martius and Lampert [e-print arXiv:1610.02995 (2016)] is suitably modified to construct free energy densities which are functions of a set of weighted densities and which are built from a small number of basis functions with flexible combination rules. This setup considerably enlarges the functional space used in the machine learning optimization as compared to the previous work [S.-C. Lin and M. Oettel, SciPost Phys. 6, 025 (2019)] where the functional is limited to a simple polynomial form. As a result, we find a good approximation for the exact hard rod functional and its direct correlation function. For the Lennard-Jones fluid, we let the network learn (i) the full excess free energy functional and (ii) the excess free energy functional related to interparticle attractions. Both functionals show a good agreement with simulated density profiles for thermodynamic parameters inside and outside the training region.

中文翻译：

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来自方程式学习网络的解析经典密度泛函。

我们探索使用机器学习方法来在一维中对两种模型流体（硬棒和Lennard-Jones）获得经典自由能泛函的解析形式的可行性。适当修改Martius和Lampert提出的方程式学习网络[e-print arXiv：1610.02995（2016）]，以构造自由能量密度，该自由能量密度是一组加权密度的函数，并且由少量具有弹性的基本函数构建组合规则。与以前的工作相比，这种设置大大扩大了机器学习优化中使用的功能空间。Lin和M. Oettel，SciPost物理学。6，025（2019）]中，函数仅限于简单的多项式形式。结果，我们找到了精确的硬杆函数及其直接相关函数的良好近似值。对于Lennard-Jones流体，我们让网络学习（i）完全多余的自由能功能和（ii）与粒子间引力有关的多余自由能功能。两种功能均与训练区域内外的热力学参数的模拟密度分布图显示出良好的一致性。