Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method
Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method
复制标题
使用库普曼半群的无穷小生成元辨识非线性系统——Mauroy-Goncalves 方法的数值实现
DOI：
10.3390/math9172075
复制
发表时间：
2021
期刊：
影响因子：
2.4
通讯作者：
Ryan Mohr
中科院分区：
文献类型：
--
作者：
Z. Drmač;I. Mezić;Ryan Mohr
在数据驱动环境中推断复杂非线性动力系统的潜在结构是一个具有挑战性的数学问题，在科学和工程中的应用范围不断增加。基于Koopman算子的线性化提供了一个强大的框架，适用于各种情况下的非线性系统的识别。Mauroy和Goncalves最近提出的方法是基于提升数据快照到一个合适的有限维函数空间和识别的Koopman半群的无穷小生成元。这种优雅和数学吸引力的方法具有良好的分析（收敛）性能，但数值实验表明，该方法的软件实现有一定的局限性。更确切地说，随着维数的增加，保证理论上更好的近似和最终收敛，数值实现可能变得不稳定，甚至可能崩溃。数值困难的主要来源是压缩Koopman算子的矩阵表示及其对数的计算。本文讨论了微妙的数值细节，并提出了一个新的实现算法，解决这些问题。
Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.