2024年5月1日 · The KDMD allows implicit observable functions; only the kernel function’s form needs to be given. Although the introduction of nonlinear measurements helps to capture the more complex nonlinear transients, the EDMD and KDMD still have shortcomings.

其中 x(n) 为 n 时刻的状态， x \in \mathbb{\mathcal{M}} \subseteq \mathbb{R}^{N} 状态空间, n \in \mathbb{Z} 离散时间， \boldsymbol{F} : \mathcal{M} \rightarrow \mathcal{M} 为演变算子(动力学系统)， F 是作用于状态 \boldsymbol{x} \in \mathcal{M} ，跟 F 不同, Koopman算子 \mathcal{K} 作用 …

difference between kDMD and the present work is that our goal is to implicitly model the (non-square) nonlinear dynamics in (1) in terms of the original state x, enabling the robust extraction of the linear component L, as opposed to analysing the …

Comparing DMD and KDMD for Slow manifold dynamics. 1. Applying DMD; 2. Applying KDMD; Validate Koopman eigenfunction with an unseen trajectory; Validate the learned Koopman eigenfunction; Extended DMD with control for chaotic duffing oscillator; Extended DMD with control for Van der Pol oscillator; Hankel Alternative View of Koopman Operator ...

DMD 是直接基于 SVD 分解来求得 Koopman values 和 Koopman modes 的，因此其逼近能力是有限的。 KDMD 是通过手动设置 dictionary functions (lifted/basic functions)，常用是用 RBF 核函数。 先将数据经过核函数处理，再基于 SVD 分解。 我也没有深入研究 DMD， KDMD，不在讨论范围内。 EDMD 在特征提取这一块和 KDMD 是类似的，都是先通过手动设置字典函数。 后面不是采用 SVD 分解，而是在有限高维下用一个 K 矩阵来近似无限维的 Koopman operator \mathcal …

classical nonlinear Koopman analysis method (e.g., EDMD, KDMD) suffers from having hundreds to thousands of approximated Koopman triplets. How to choose an accurate and informative Koopman invariant subspace in Extended/Kernel DMD?

The Koopman operator is a linear, in nite-dimensional operator that governs the dynamics of system observables; Extended Dynamic Mode Decomposition (EDMD) is a data-driven method for approximating the Koopman operator using functions (features) of the system state snapshots.

We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlin-ear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.

今天讲的这篇论文是发表在 Automatica 上的使用 EDMD 近似Koopman operator的文章 [1]，这篇文章的方法不再只用于非受力系统，受力系统也可以近似。 koopman 是使用线性系统用来近似非线性系统的一个符号算子，使用线性系统近似非线性系统后，就可以使用线性系统的控制理论来控制非线性系统了。 值得注意的是： Koopman 算子是一个无穷维的线性变换，而我们要做的往往是用一个有限维的 K 矩阵去近似 Koopman 算子， 我们要求的也就是这个矩阵，现在常用的方 …

An essential difference between kDMD and the present work is that our goal is to implicitly model the (non-square) nonlinear dynamics in in terms of the original state x, enabling the robust extraction of the linear component L, as opposed to analysing the Koopman operator over measurement functions. Further differences between LANDO and other ...