Inferring traits of hyperuniformity from local structures via persistent homology

TL;DR

This work connects finite-size hyperuniform patterns to local geometric structure by leveraging persistent homology (PH) and machine learning. By transforming persistence diagrams into robust features (algebraic, binning, and Wasserstein-distance-based) and training diverse ML models, the authors achieve high accuracy in detecting HU and in quantifying Hu parameters via reference diagrams. They further show that Wasserstein distances between PH diagrams can estimate structure-factor parameters ($oldsymbol{ abla}=( ext{e.g., }oldsymbol{ abla} oldsymbol{ ext{a}})$, $H$, $K$) and even enable approximate inverse design by constructing patterns whose $h_1$ diagrams match a target, thereby linking local topology to global HU characteristics. The approach provides a practical framework for analyzing HU in finite domains and for topology-driven pattern design with potential applications to transport and optical properties.

Abstract

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.

Figures (13)

• Figure 1: Examples of point patterns with a structure factor as in equation \ref{['eq:formulaofstruc']}. First row: Point patterns for different structure factor parameters reported above the panels. Second row: Structure factors, illustrated by both the $\mathcal{S}_0(|\mathbf{k}|)$ imposed in the generation procedure (solid black lines) and discrete values computed for the patterns in the first rows for $\mathcal{S}(|\mathbf{k}|)$ imposed at appropriate wavenumbers $|\mathbf{k}|=2\pi n$ with $n\in \mathbb{N}_{>0}$ (red dots). We remark that $\widetilde{H}$ in equation \ref{['eq:htilde']} corresponds to the value of $\mathcal{S}(|\mathbf{k}|)$ for the $|\mathbf{k}|=2\pi$, i.e. the leftmost red point in the structure factor plots and may deviate significantly from the parameter $H$ for relatively small $\alpha$. See, for instance, the case with $\alpha=0.5$ (first column).
• Figure 2: Illustration of the Čech persistent homology and the persistence diagram for a point pattern of four points.
• Figure 3: Persistence diagram for the point pattern reported in figure \ref{['fig:points_Sk']} with $\alpha\,$=$\,0.5$, $K\,$=$\,40\pi$ and $H\,$=$\,10^{-3}$. $h_0$ points $(b_j^0,d_j^0)$ (blue circles) denote the birth and death time of 0-dimensional holes (i.e., connected components). $h_1$ points $(b_j^1,d_j^1)$ (red squares) denote the birth and death time of 1-dimensional holes.
• Figure 4: Binning of the persistence diagram in figure \ref{['fig:pattern_and_diagram']}. (a) Modified persistence diagram neglecting the $b > d$ region by replacing the death times ($d_j^k$) with the life times ($l_j^k=d_j^k-b_j^k$), i.e. applying the mapping $(b_j^k,d_j^k)\mapsto (b_j^k, l_j^k)$. (b) Histogram of $h_0$ points in panel (a) for a suitable size of the (uniform) bins, here for $n=15$, with $n$ the number of bins. (c) A color map presenting the binning of $h_1$, here for $n=15$ with $n$ the binning of the axis (see further details in the text). The color indicates the number of $h_1$ points in each bin.
• Figure 5: Examples of three distance maps $D_i(\alpha,K)$ used to determine effective structure factor parameters of a given pattern. Parameters of the corresponding reference arrangements ($\alpha_{\rm R}$, $H_{\rm R}$, $K_{\rm R}$) are reported above the panels.