Tensor-based EDMD for the Koopman analysis of high-dimensional systems

Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory -- with extended dynamic mode decomposition (EDMD) being a cornerstone of the field. 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 EDMD and the TT format, enabling the application of EDMD to high-dimensional problems in conjunction with a large set of features. We present the construction of different TT representations of tensor-structured data arrays. Furthermore, we also derive efficient algorithms to solve the EDMD eigenvalue problem based on those representations and to project the data into a low-dimensional representation defined by the eigenvectors. We prove that there is a physical interpretation of the procedure and demonstrate its capabilities by applying the method to benchmark data sets of molecular dynamics simulation.