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Multiscale geometrical and topological learning in the analysis of soft matter collective dynamics

Tetiana Orlova1,*,†, Amaranta Membrillo Solis2,‡,§, Hayley R. O. Sohn3,4,5, Tristan Madeleine2, Giampaolo D'Alessandro2, Ivan I. Smalyukh3,4,5, Malgosia Kaczmarek1,∥, and Jacek Brodzki2

  • *Contact author: tetiana.orlova@ysu.am
  • †Present address: Institute of Physics, Yerevan State University, 1 Alex Manoogian, Yerevan 0025, Armenia.
  • ‡Contact author: i.a.membrillosolis@qmul.ac.uk
  • §Present address: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
  • ∥Contact author: mfk@soton.ac.uk
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Phys. Rev. Materials 10, 015602 – Published 9 January, 2026

DOI: https://doi.org/10.1103/dht2-w1gz

Abstract

Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and nonliving self-assembled systems. The arrays of moving liquid crystal skyrmions studied here are a representative example of hierarchically organized materials that exhibit complex spatiotemporal dynamics driven by multiscale processes. Joint geometric and topological data analysis (TDA) offers a powerful framework for investigating such systems by capturing the underlying structure of the data at multiple scales. In the TDA approach, we introduce the Ψ function, a robust numerical topological descriptor related to both the spatiotemporal changes in the size and shape of individual topological solitons and the emergence of regions with their different spatial organization. The geometric method based on the analysis of vector fields generated from images of skyrmion ensembles offers insights into the nonlinear physical mechanisms of the system's response to external stimuli and provides a basis for comparison with theoretical predictions. The methodology presented here is very general and can provide a characterization of system behavior both at the level of individual pattern-forming agents and as a whole, allowing one to relate the results of image data analysis to processes occurring in a physical, chemical, or biological system in the real world.

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Phys. Rev. Materials 10, 015602– Published 9 January, 2026
  • Received 30 July 2025
  • Accepted 24 November 2025
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