Learning complex systems dynamics from vector fields over discrete measure spaces

Abstract

A complex system is formed by entities that, through their interactions and dependencies, give rise to a unified whole with properties and behaviour distinct from those of its constituent parts. Examples of complex systems include the human brain, living cells, organisms, soft matter materials, the Earth’s global climate, ecosystems and the economy. A large number of entities and dependencies produce high dimensional non-linear dynamics, which make the modelling, classification and prediction of complex systems dynamics a major challenge. Advances in modern information technology have made possible a successful use of data-driven approaches to the study of dynamical systems. However, in the particular case of complex systems there are still important open questions to address such as large and complex data sets analysis, dimensionality reduction, phase space reconstruction and global attractor classification. In this paper we introduce the theory of vector fields over discrete measure spaces to analyse data structurally complex, such as images, image gradients, and real and vector valued functions over simplicial complexes. We apply our framework to the analysis of data obtained from numerical solutions of equations commonly used in biology and physics to model a variety of complex systems. We show that our geometric framework, together with multidimensional scaling, an unsupervised learning method, can be successfully used in the analysis of large data sets for dimensionality reduction, mode decomposition and global attractor characterisation of complex systems dynamics. These results pave the way towards the characterisation and understanding of a number of complex dynamical systems.