Representation of a dynamical system in terms of simplifying modes is a central premise of reduced order modelling and a primary concern of the increasingly popular DMD (dynamic mode decomposition) empirical interpretation of Koopman operator analysis of complex systems. In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants are to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. That Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. Corresponding to freedom to arbitrarily define functions on a data surface, for each eigenvalue, there are infinitely many eigenfunctions emanating along characteristics. We focus on contrasting cardinality and equivalence. In particular, we introduce an equivalence class, “primary eigenfunctions,’ consisting of those eigenfunctions with identical sets of level sets, that helps contrast algebraic multiplicity from other geometric aspects. Popularly, Koopman methods and notably dynamic mode decomposition (DMD) and variants, allow data-driven study of how measurable functions evolve along orbits. As far as we know, there has not been an in depth study regarding the underlying geometry as related to an efficient representation. We present a construction that leads to functions on the data surface whose corresponding eigenfunctions are efficient in a least squares sense. We call this construction optimal Koopman eigenfunction DMD, (oKEEDMD), and we highlight with examples.

以简化模式表示动态系统是降阶建模的主要前提，也是对复杂系统的考夫曼算子分析越来越流行的DMD（动态模式分解）经验解释的主要关注点。本着最佳逼近和降阶建模的精神，DMD方法和变体的目标是尽可能将动态演化描述为在适当转换的较低秩空间中的线性演化。Koopman本征函数遵循线性PDE，该线性PDE通过特征方法可求解，从而在几何和代数性质之间产生了一些有趣的关系。对应于针对每个特征值在数据表面上任意定义函数的自由，沿着特征散发出无穷的本征函数。我们专注于对比基数和等效性。特别是，我们引入了一个等价类“主要特征函数”，该类由具有相同水平集集合的那些特征函数组成，可帮助从其他几何方面对比代数多重性。普遍地，考夫曼方法，尤其是动态模式分解（DMD）和变体，允许以数据驱动的方式研究可测量功能如何沿轨道发展。据我们所知，尚未对与有效表示相关的基础几何进行深入研究。我们提出了一种导致数据表面上函数的结构，其对应的本征函数在最小二乘意义上是有效的。我们称这种构造为最佳Koopman特征函数DMD（oKEEDMD），