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

2025-07-28

AI Summary

This study addresses the challenge of quantitatively characterizing complex spatiotemporal dynamics in hierarchical soft-matter systems—exemplified by liquid-crystal skyrmion arrays. We propose Ψ-functions, a novel topological descriptor integrating geometric and topological information, combined with image-driven vector field analysis, persistent homology (TDA), multiscale geometric modeling, and numerical topological feature extraction. This framework enables cross-scale dynamic analysis—from individual soliton morphology to collective spatial organization. Our method establishes a quantitative mapping between experimental images and underlying physical mechanisms, enabling, for the first time, systematic characterization of the nonlinear response of skyrmion arrays. It further achieves classification and evolution prediction of multiscale dynamical behaviors. The approach provides a generalizable computational framework for dynamical modeling of topological soft matter, bridging experimental observation, geometric intuition, and topological quantification.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 non-living 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.

Problem

• Analyzing multiscale dynamics in soft matter systems
• Developing topological descriptors for spatiotemporal pattern changes
• Linking image data to physical and biological processes

Innovation

• Multiscale geometric and topological learning
• Ψ-function for topological descriptor
• Vector field analysis of skyrmion images