Finite-Data Error Bounds for Koopman-Based Prediction and Control
Finite-Data Error Bounds for Koopman-Based Prediction and Control
复制标题
基于库普曼的预测和控制的有限数据误差界
DOI：
10.1007/s00332-022-09862-1
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发表时间：
2021
影响因子：
3
通讯作者：
K. Worthmann
中科院分区：
文献类型：
--
作者：
F. Nüske;Sebastian Peitz;F. Philipp;M. Schaller;K. Worthmann
Koopman算子已经成为动态(控制)系统的数据驱动近似的基本工具，例如通过扩展的动态模式分解。尽管它很受欢迎，但收敛结果，特别是误差界仍然很少。在这篇文章中，我们得到了当使用遍历轨迹或I.I.D.时，常微分方程组和随机微分方程解的逼近误差和预测误差与训练数据点的数目有关的概率界。样本。我们通过一个Ornstein-Uhlenbeck过程的例子来说明这些界限。此外，我们将我们的分析推广到(随机)非线性控制-仿射系统。我们证明了以前提出的一种方法的误差估计，该方法利用Koopman生成器的线性来获得双线性代理控制系统，从而绕过了维度诅咒，因为系统不是通过控制输入来增加状态来自治的。据我们所知，这是第一次在随机和/或控制环境中进行有限数据误差分析。最后，我们通过与最新技术的比较，证明了双线性方法的有效性，表明了当状态和控制耦合时双线性方法的优越性。
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.
影响因子：
10.2
作者：
Brunton, Steven L.;Budisic, Marko;Kutz, J. Nathan
通讯作者：
Kutz, J. Nathan
影响因子：
7.8
作者：
Mamakoukas, Giorgos;Castano, Maria L.;Murphey, Todd D.
通讯作者：
Murphey, Todd D.