论文标题
用储层计算机近似Koopman操作员的两种方法
Two methods to approximate the Koopman operator with a reservoir computer
论文作者
论文摘要
Koopman操作员为动态系统的数据驱动分析提供了强大的框架。在过去的几年中，已经提出了提供操作员有限维近似值的大量数值方法（例如，扩展动态模式分解（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很好地选择了Koopman操作员的有效近似值。但是，这种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储层计算机的新方法来训练字典。这些方法仅依赖于线性凸优化。在数据重建，预测和计算Koopman操作员频谱的情况下，我们用几个数值示例说明了该方法的效率。这些结果为在Koopman操作员框架中使用储层计算机的使用铺平了道路。
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 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 to the use of the reservoir computer in the Koopman operator framework.
