Microscopic information, such as the pair distribution and direct correlation functions, can be obtained from experimental data. From these correlation functions, thermodynamical quantities and the potential interaction function can be recovered. Derivations of Ornstein-Zernike equation and Hopfield Neural Network method are given first, as a theoretical background to follow the present work. From these two frameworks, structural information, such as the radial distribution (g(r)) and direct correlation (C(r)) functions, were retrieved from neutron scattering experimental data. The problem was solved considering simple initial conditions, which does not require any previous information about the system, making it clear the robustness of the Hopfield Neural Network method. The pair interaction potential was estimated in the Percus-Yevick (PY) and hypernetted chain (HNC) approximations and a poor agreement, compared with the Lennard-Jones 6-12 potential, was observed for both cases, suggesting the necessity of a more accurate closure relation to describe the system. In this sense, the Hopfield Neural Network together with experimental information provides an alternative approach to solve the Ornstein-Zernike equations, avoiding the limitations imposed by the closure relation.

中文翻译：

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使用 Hopfield 神经网络方法间接求解 Ornstein-Zernike 方程

微观信息，如对分布和直接相关函数，可以从实验数据中获得。从这些相关函数中，可以恢复热力学量和势相互作用函数。首先给出 Ornstein-Zernike 方程和 Hopfield 神经网络方法的推导，作为遵循当前工作的理论背景。从这两个框架中，从中子散射实验数据中检索结构信息，例如径向分布 (g(r)) 和直接相关 (C(r)) 函数。该问题在考虑简单初始条件的情况下得到解决，不需要任何有关系统的先前信息，从而明确了 Hopfield 神经网络方法的鲁棒性。在 Percus-Yevick (PY) 和超网状链 (HNC) 近似中估计了配对相互作用势，并且与 Lennard-Jones 6-12 势相比，在两种情况下都观察到了较差的一致性，这表明需要更准确的描述系统的闭包关系。从这个意义上说，Hopfield 神经网络与实验信息一起提供了一种解决 Ornstein-Zernike 方程的替代方法，避免了闭合关系带来的限制。