Complex (dynamical) structures: Their higher order correlations, their traces in the Fourier domain and their prediction

Abstract

A defining property of complex (dynamical) structures is the presences of higher order correlations (HOC) in them. We start with presenting some methods for measuring the information contained in the HOCs focusing on Minkowski-functionals and -tensors. We outline how the HOCs are represented in the Fourier domain. We introduce the method of surrogates and demonstrate how the type and strength of HOCs can be narrowed down by specially devised statistical tests involving surrogate data sets. We further present reservoir computing (RC) as an efficient and interpretable AI-method and its application to the classification of complex structures and the prediction of their dynamics. It turns out that prediction tasks are best performed if the only remaining tunable non-linearity of the reservoir matches with the degree of non-linearity in the data. Possible further applications of the presented methodologies for hyperuniform data sets and material sciences are briefly outlined.