Characterizing the entropy of a system is a crucial, and often computationally costly, step in understanding its thermodynamics. It plays a key role in the study of phase transitions, pattern formation, protein folding, and more. Current methods for entropy estimation suffer from a high computational cost, lack of generality, or inaccuracy and inability to treat complex, strongly interacting systems. In this paper, we present a method, termed machine-learning iterative calculation of entropy (MICE), for calculating the entropy by iteratively dividing the system into smaller subsystems and estimating the mutual information between each pair of halves. The estimation is performed with a recently proposed machine-learning algorithm which works with arbitrary network architectures that can be chosen to fit the structure and symmetries of the system at hand. We show that our method can calculate the entropy of various systems, both thermal and athermal, with state-of-the-art accuracy. Specifically, we study various classical spin systems and identify the jamming point of a bidisperse mixture of soft disks. Finally, we suggest that besides its role in estimating the entropy, the mutual information itself can provide an insightful diagnostic tool in the study of physical systems.

表征系统的熵是理解其热力学的关键步骤，并且通常在计算上昂贵。它在相变，模式形成，蛋白质折叠等研究中起着关键作用。当前用于熵估计的方法遭受高计算成本，缺乏通用性，或不精确且不能处理复杂的，相互作用强烈的系统的困扰。在本文中，我们提出了一种称为机器学习熵的迭代计算（MICE）的方法，该方法用于通过将系统迭代地划分为较小的子系统并估计每对两半之间的互信息来计算熵。估计是使用最近提出的机器学习算法执行的，该算法可与任意网络体系结构一起使用，可以选择这些体系结构以适合当前系统的结构和对称性。我们证明了我们的方法可以以最先进的精度计算各种系统的热和非热的熵。具体来说，我们研究各种经典的自旋系统，并确定软盘双分散混合物的阻塞点。最后，我们建议，互信息除了在估计熵中发挥作用外，还可以为物理系统的研究提供有见地的诊断工具。我们研究了各种经典的自旋系统，并确定了软盘双分散混合物的干扰点。最后，我们建议，互信息除了在估计熵中发挥作用外，还可以为物理系统的研究提供有见地的诊断工具。我们研究了各种经典的自旋系统，并确定了软盘双分散混合物的干扰点。最后，我们建议，互信息除了可以在估计熵中发挥作用外，还可以为物理系统的研究提供有见地的诊断工具。