Tensor-based computation of metastable and coherent sets
This work addresses the scalability problem in dynamical systems analysis for researchers and practitioners, though it is incremental as it combines existing methods.
The authors tackled the challenge of applying Koopman operator theory to high-dimensional dynamical systems by integrating it with tensor train (TT) format approximations, resulting in efficient algorithms for computing reduced matrix representations that handle both stationary and non-stationary systems, as demonstrated on benchmark datasets.
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.