Head(s): Prof. Dr. Jens Eisert (FU Berlin), Prof. Dr. Christof Schütte (FU Berlin, stand-in for Dr. Stefan Klus now at the University of Surrey, UK)
Project member(s): Dr. Patrick Gelß
Participating institution(s): FU Berlin

Project Summary

The main focus of this new project is the data-driven and tensor-based analysis of complex dynamical systems exhibiting multiple time scales. Our goal is to obtain relevant information about global properties of the underlying system such as almost invariant sets or a decomposition into fast and slow processes. Global information can be obtained by analysing the eigenvalues and eigenfunctions of the Perron–Frobenius or Koopman operator associated with the system. In what follows, we will refer to both operators as transfer operators.
Several different data-driven methods for the approximation of transfer operators have been developed over the last years. The main advantage is that these methods can be applied to measurement or simulation data generated by systems whose governing equations might be unknown. Based purely on available data, these methods extract the governing equations or transfer operators associated with the system. Examples of such methods are Dynamic Mode Decomposition (DMD), Extended Dynamic Mode Decomposition (EDMD), and Sparse Identification of Nonlinear Dynamics (SINDy). These and related methods have been successfully applied and generalised within project B03 and project A04. Due to the curse of dimensionality, however, it is often not possible to analyse complex dynamical systems with sufficient accuracy. Different approaches have been proposed to mitigate this problem, including tensor-based methods, kernel-based methods, and model reduction strategies relying on coarse approximations of the eigenfunctions of transfer operators. Additionally, compressed sensing techniques can be used to penalise nonsparse solutions. This is typically accomplished by adding 𝓁₁ regularisation terms.

These different methods and their extensions have, however, not been analysed systematically and the convergence properties are not well understood. In this project, we will extend and generalise tensor decomposition approaches and also kernel-based methods for computing eigenfunctions of transfer operators associated with high-dimensional systems. We will combine these approaches with compressed sensing techniques to find parsimonious representations of the system or its eigenfunctions. Furthermore, we will study the convergence properties of these methods. The overall goal is to identify methodologies for analysing high-dimensional systems and to apply these novel methods to problems relevant for the CRC. The applicability of the aforementioned approaches will depend on the characteristic properties of the system. It will thus be necessary to identify suitable strategies for specific applications first. Collaborations are planned with the projects A01 (atmospheric flows), A04 (molecular dynamics), A06 (reproducing kernel Hilbert spaces), B03 (reaction coordinates), and B07 (turbulent flows).

Publications B06