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Inferring traits of hyperuniformity from local structures via persistent homology.

Abel H G Milor¹, Marco Salvalaglio^1,2

¹Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany.

Journal of Physics. Condensed Matter : an Institute of Physics Journal

|January 31, 2025

Summary

This study reveals that local structural information, analyzed using persistent homology and machine learning, can accurately detect and quantify hyperuniformity (HU) in finite point patterns. This enables the inverse design of patterns with specific topological and global properties.

Keywords:

Wasserstein distance correlated disorder hyperuniformity machine learning persistent homology point patterns

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Area of Science:

Physics
Materials Science
Data Science

Background:

Hyperuniformity (HU) describes suppressed large-scale density fluctuations, observed in both ordered and disordered systems.
HU is crucial for understanding pattern formation and phenomena like light interaction and transport.
Finite-sized disordered systems present challenges for characterizing global HU properties.

Purpose of the Study:

To explore the inverse relationship between hyperuniformity and local structural properties in point patterns.
To assess the use of persistent homology and machine learning for detecting HU in finite systems.
To enable the reconstruction of global HU parameters from local topological data.

Main Methods:

Utilizing persistent homology to analyze local arrangements within point patterns.
Training standard machine learning algorithms on persistence diagrams.
Comparing persistence diagrams of targeted patterns with reference HU patterns.

Main Results:

Machine learning accurately detects hyperuniformity in periodic point patterns using topological data.
Local structural information effectively characterizes finite arrangements analogous to HU patterns.
Global hyperuniformity parameters can be reconstructed from local topological features.

Conclusions:

Persistent homology and machine learning provide powerful tools for analyzing hyperuniformity in finite systems.
Local structural analysis is key to understanding and characterizing global hyperuniform properties.
This work lays the foundation for inverse design of hyperuniform patterns with desired topological characteristics.