Gulina Marvyn, Mauroy Alexandre
Department of Mathematics and Namur Institute for Complex Systems (naXys), University of Namur, 5000 Namur, Belgium.
Chaos. 2021 Feb;31(2):023116. doi: 10.1063/5.0026380.
The Koopman operator provides a powerful framework for data-driven analysis of dynamical systems. In the last few years, a wealth of numerical methods providing finite-dimensional approximations of the operator have been proposed [e.g., extended dynamic mode decomposition (EDMD) and its variants]. While convergence results for EDMD require an infinite number of dictionary elements, recent studies have shown that only a few dictionary elements can yield an efficient approximation of the Koopman operator, provided that they are well-chosen through a proper training process. However, this training process typically relies on nonlinear optimization techniques. In this paper, we propose two novel methods based on a reservoir computer to train the dictionary. These methods rely solely on linear convex optimization. We illustrate the efficiency of the method with several numerical examples in the context of data reconstruction, prediction, and computation of the Koopman operator spectrum. These results pave the way for the use of the reservoir computer in the Koopman operator framework.
库普曼算子为动力系统的数据驱动分析提供了一个强大的框架。在过去几年中，已经提出了大量提供该算子有限维近似的数值方法[例如，扩展动态模态分解（EDMD）及其变体]。虽然EDMD的收敛结果需要无限数量的字典元素，但最近的研究表明，只要通过适当的训练过程精心选择，只需几个字典元素就能对库普曼算子进行有效的近似。然而，这个训练过程通常依赖于非线性优化技术。在本文中，我们提出了两种基于回声状态网络来训练字典的新方法。这些方法仅依赖于线性凸优化。我们在数据重建、预测和库普曼算子谱计算的背景下，用几个数值例子说明了该方法的有效性。这些结果为在库普曼算子框架中使用回声状态网络铺平了道路。
