Tensor-based computation of metastable and coherent sets

TL;DR

This paper introduces a novel approach combining Koopman operator theory with tensor train (TT) formats to efficiently analyze high-dimensional dynamical systems, enabling scalable computation of metastable and coherent sets.

Contribution

It develops algorithms that integrate Koopman models with TT formats for large-scale, high-dimensional data analysis, extending applicability to non-stationary systems.

Findings

Algorithms efficiently compute reduced operators from low-rank data representations.

Methods work for both stationary and non-stationary systems.

Demonstrated effectiveness on benchmark datasets.

Abstract

Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine Koopman-based models and the TT format, enabling their application to high-dimensional problems in conjunction with a rich set of basis functions or features. We derive efficient algorithms to obtain a reduced matrix representation of the system's evolution operator starting from an appropriate low-rank representation of the data. These algorithms can be applied to both stationary and non-stationary systems. We establish the infinite-data limit of these matrix representations, and demonstrate our methods' capabilities using several benchmark data sets.