通过持续同调从局部结构推断超均匀性的特征
Inferring traits of hyperuniformity from local structures via persistent homology
AI总结：
本研究利用持续同调描述点图案的局部结构，通过机器学习实现了有限尺寸系统超均匀性的高精度检测，还可重构超均匀性全局参数、生成指定拓扑性质的图案，为超均匀图案分析与逆向设计提供了新方法。
摘要：
超均匀性指的是大尺度下密度涨落受到抑制的性质。这一性质是有序系统的典型特征，也出现在多种无序物理系统和生物系统中，在这类系统中，超均匀性对于理解图案形成机制、利用其独特属性（如与光的相互作用、输运现象）具有重要意义。尽管超均匀性是一种全局性质，理想情况下是针对无限延伸的系统定义的，但许多无序关联系统的尺寸是有限的。已有研究[Phys. Rev. Research 6, 023107 (2024)]表明，全局超均匀特征与代表局部排布的拓扑性质的分布存在系统性关联。在本工作中，我们基于这一结论，探究并评估了由持续同调描述的点图案中，超均匀性与局部结构之间的逆关系。研究表明，在持续图上训练的标准机器学习算法能够高精度检测周期点图案的超均匀性。由此我们证明，图案局部结构的信息可用于表征有限尺寸排布是否与超均匀图案中存在的排布类似。随后，针对更定量的问题，我们发现通过将目标图案的持续图与参考持续图进行对比，可以重构定义全局超均匀性的参数。我们还探索了具有给定拓扑性质的图案的生成方法。本研究结果为包含局部信息的超均匀图案的高级分析铺平了道路，并为其逆向设计引入了基础概念。
英文摘要：
Hyperuniformity refers to the suppression of density fluctuations at large scales. Typical for ordered systems, this property also emerges in several disordered physical and biological systems, where it is particularly relevant to understand mechanisms of pattern formation and to exploit peculiar attributes, e.g., interaction with light and transport phenomena. While hyperuniformity is a global property, ideally defined for infinitely extended systems, several disordered correlated systems have finite size. It has been shown in [Phys. Rev. Research 6, 023107 (2024)] that global hyperuniform characteristics systematically correlate with distributions of topological properties representative of local arrangements. In this work, building on this information, we explore and assess the inverse relationship between hyperuniformity and local structures in point patterns as described by persistent homology. Standard machine learning algorithms trained on persistence diagrams are shown to detect hyperuniformity of periodic point patterns with high accuracy. Therefore, we demonstrate that the information on patterns' local structures allows for characterizing whether finite size arrangements are analogous to those realized in hyperuniform patterns. Then, addressing more quantitative aspects, we show that parameters defining hyperuniformity globally can be reconstructed by comparing persistence diagrams of targeted patterns with reference ones. We also explore the generation of patterns entailing given topological properties. The results of this study pave the way for advanced analysis of hyperuniform patterns including local information, and introduce basic concepts for their inverse design.
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