Pattern recognition in complex systems via vector-field representations of spatio-temporal data

A complex system comprises multiple interacting entities whose interdependencies form a unified whole, exhibiting emergent behaviours not present in individual components. Examples include the human brain, living cells, soft matter, Earth's climate, ecosystems, and the economy. These systems exhibit high-dimensional, non-linear dynamics, making their modelling, classification, and prediction particularly challenging. Advances in information technology have enabled data-driven approaches to studying such systems. However, the sheer volume and complexity of spatio-temporal data often hinder traditional methods like dimensionality reduction, phase-space reconstruction, and attractor characterisation. This paper introduces a geometric framework for analysing spatio-temporal data from complex systems, grounded in the theory of vector fields over discrete measure spaces. We propose a two-parameter family of metrics suitable for data analysis and machine learning applications. The framework supports time-dependent images, image gradients, and real- or vector-valued functions defined on graphs and simplicial complexes. We validate our approach using data from numerical simulations of biological and physical systems on flat and curved domains. Our results show that the proposed metrics, combined with multidimensional scaling, effectively address key analytical challenges. They enable dimensionality reduction, mode decomposition, phase-space reconstruction, and attractor characterisation. Our findings offer a robust pathway for understanding complex dynamical systems, especially in contexts where traditional modelling is impractical but abundant experimental data are available.