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.

近年来，基于 Koopman 算子理论和相关方法的动态系统数据驱动分析取得了快速进展。另一方面，低秩张量积近似——特别是张量训练（TT）格式——已成为解决许多领域大规模问题的宝贵工具。在这项工作中，我们将基于 Koopman 的模型和 TT 格式结合起来，将它们与一组丰富的基函数或特征结合起来应用于高维问题。我们推导出有效的算法，从数据的适当低秩表示开始，获得系统进化算子的简化矩阵表示。这些算法可以应用于静止和非静止系统。我们建立了这些矩阵表示的无限数据限制，