TY - JOUR A1 - Klus, Stefan A1 - Nüske, Feliks A1 - Koltai, Peter A1 - Wu, Hao A1 - Kevrekidis, Ioannis A1 - Schütte, Christof A1 - Noé, Frank T1 - Data-driven model reduction and transfer operator approximation JF - Journal of Nonlinear Science Y1 - 2018 UR - https://link.springer.com/article/10.1007/s00332-017-9437-7 U6 - https://doi.org/10.1007/s00332-017-9437-7 VL - 28 IS - 3 SP - 985 EP - 1010 ER - TY - JOUR A1 - Lücke, Marvin A1 - Nüske, Feliks T1 - tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format JF - Journal of Nonlinear Science N2 - Extracting information about dynamical systems from models learned off simulation data has become an increasingly important research topic in the natural and engineering sciences. Modeling the Koopman operator semigroup has played a central role in this context. As the approximation quality of any such model critically depends on the basis set, recent work has focused on deriving data-efficient representations of the Koopman operator in low-rank tensor formats, enabling the use of powerful model classes while avoiding over-fitting. On the other hand, detailed information about the system at hand can be extracted from models for the infinitesimal generator, also called Kolmogorov backward operator for stochastic differential equations. In this work, we present a data-driven method to efficiently approximate the generator using the tensor train (TT) format. The centerpiece of the method is a TT representation of the tensor of generator evaluations at all data sites. We analyze consistency and complexity of the method, present extensions to practically relevant settings, and demonstrate its applicability to benchmark numerical examples. Y1 - 2022 U6 - https://doi.org/10.1007/s00332-022-09801-0 VL - 32 ER - TY - JOUR A1 - Klus, Stefan A1 - Nüske, Feliks A1 - Peitz, Sebastian A1 - Niemann, Jan-Hendrik A1 - Clementi, Cecilia A1 - Schütte, Christof T1 - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control JF - Physica D: Nonlinear Phenomena Y1 - 2020 U6 - https://doi.org/10.1016/j.physd.2020.132416 VL - 406 ER -