Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.
翻译:扩展动态模式分解( EDMD) 是一种流行的数据驱动方法,用来估计 Koopman 操作员在功能字典所覆盖的线性功能空间上的行动。 EDMD 模型的准确性关键取决于特定字典的跨度的质量,具体取决于该词典在 Koopman 操作员之下离无动性有多近。 观察到EDMD 的剩余差错通常用于字典学习,并不编码功能空间的质量,而且对依据的选择十分敏感,因此我们引入了新的一致性指数概念。 我们显示,根据使用 EDMD 前向和后向,这一计量具有一些可取的品质,使之适合于动态系统的数据驱动模型:它测量功能空间的质量,在选择的基础上是无动性,可以从数据中以封闭的形式进行计算,并且对词典整个范围内所有函数预测的相对根值平均方差有一个紧的上限。