On the approximations of the Koopman operator and applications to spectral identification of networks

Abstract

This dissertation investigates the Koopman operator as a powerful tool for the analysis of nonlinear dynamical systems and its applications to spectral network identification.

The Koopman operator is an infinite-dimensional linear operator that captures the evolution of observables along the trajectories of a dynamical system.
The first part of the dissertation reviews the historical background and introduces the theory of the Koopman operator and its finite-dimensional approximations using data-driven methods.
Then, two novel methods based on reservoir computing are proposed to construct approximations of the Koopman operator from time series data.

The second part of the dissertation focuses on spectral network identification, which is the problem of inferring structural properties of a network from measurements of its dynamics by exploiting the properties of the associated Koopman operator and its eigenvalues.
Some concepts of spectral graph theory are given to introduce the basics of spectral network identification.
Afterward, an existing framework for spectral network identification with a scalar-valued diffusive coupling is extended to include vector-valued diffusive and additive couplings.
Finally, a new framework for spectral network identification with an impulsive coupling is developed and illustrated.

Date of Award	3 Nov 2023
Original language	English
Awarding Institution	University of Namur
Sponsors	University of Namur
Supervisor	Alexandre Mauroy (Supervisor), Teo Carletti (Co-Supervisor), Joseph Winkin (President), Julien Hendrickx (Jury) & Felix Dietrich (Jury)