Learning Koopman Eigenfunctions and Invariant Subspaces From Data: Symmetric Subspace Decomposition
Learning Koopman Eigenfunctions and Invariant Subspaces From Data: Symmetric Subspace Decomposition
复制标题
从数据中学习库普曼特征函数和不变子空间：对称子空间分解
DOI：
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发表时间：
2019
影响因子：
6.8
通讯作者：
J. Cort'es
中科院分区：
文献类型：
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作者：
Masih Haseli;J. Cort'es
本文发展了数据驱动的方法来识别与动力系统相关的Koopman算子的特征函数以及在该算子下不变的子空间。我们建立在扩展动态模式分解(EDMD)的基础上，这是一种数据驱动的方法，它在预定义的函数字典的范围内找到Koopman算子的有限维近似。给出了基于时间向前和向后应用的EDMD识别Koopman特征函数的充要条件。此外，我们还提出了对称子空间分解(SSD)算法，这是一种迭代方法，可以证明在字典范围内识别最大Koopman不变子空间和Koopman特征函数。我们还介绍了流SSD算法，这是SSD的在线扩展，只需要很小的固定内存，并在接收到新数据时合并。最后，我们提出了一种扩展的SSD，当字典不包含足够的信息特征函数时，它逼近Koopman特征函数和不变子空间。
This article develops data-driven methods to identify eigenfunctions of the Koopman operator associated with a dynamical system and subspaces that are invariant under the operator. We build on Extended Dynamic Mode Decomposition (EDMD), a data-driven method that finds a finite-dimensional approximation of the Koopman operator on the span of a predefined dictionary of functions. We propose a necessary and sufficient condition to identify Koopman eigenfunctions based on the application of EDMD forward and backward in time. Moreover, we propose the Symmetric Subspace Decomposition (SSD) algorithm, an iterative method that provably identifies the maximal Koopman-invariant subspace and the Koopman eigenfunctions in the span of the dictionary. We also introduce the Streaming SSD algorithm, an online extension of SSD that only requires a small fixed memory and incorporates new data as is received. Finally, we propose an extension of SSD that approximates Koopman eigenfunctions and invariant subspaces when the dictionary does not contain sufficient informative eigenfunctions.