Transfer operators such as Perron–Frobenius and Koopman operator play a key role in modeling and analysis of complex dynamical systems, which allow linear representations of nonlinear dynamics by transforming the original state variables to feature spaces. However, it remains challenging to identify the optimal low-dimensional feature mappings from data. The variational approach for Markov processes (VAMP) provides a comprehensive framework for the evaluation and optimization of feature mappings based on the variational estimation of modeling errors, but it still suffers from a flawed assumption on the transfer operator and therefore sometimes fails to capture the essential structure of system dynamics. In this paper, we develop a powerful alternative to VAMP, called kernel embedding based variational approach for dynamical systems (KVAD). By using the distance measure of functions in the kernel embedding space, KVAD effectively overcomes theoretical and practical limitations of VAMP. In addition, we develop a data-driven KVAD algorithm for seeking the ideal feature mapping within a subspace spanned by given basis functions, and numerical experiments show that the proposed algorithm can significantly improve the modeling accuracy compared to VAMP.

中文翻译：

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基于核嵌入的动态系统低维逼近的变分方法

Perron-Frobenius 和 Koopman 算子等传递算子在复杂动力系统的建模和分析中起着关键作用，它们通过将原始状态变量转换为特征空间来实现非线性动力学的线性表示。然而，从数据中识别最佳低维特征映射仍然具有挑战性。马尔可夫过程的变分方法 (VAMP) 为基于建模误差变分估计的特征映射评估和优化提供了一个综合框架，但它仍然存在对转移算子的有缺陷的假设，因此有时无法捕捉到本质系统动力学的结构。在本文中，我们开发了一种强大的 VAMP 替代方案，称为基于内核嵌入的动态系统变分方法 (KVAD)。通过使用内核嵌入空间中函数的距离度量，KVAD 有效地克服了 VAMP 的理论和实践局限性。此外，我们开发了一种数据驱动的 KVAD 算法，用于在给定基函数跨越的子空间内寻找理想的特征映射，数值实验表明，与 VAMP 相比，所提出的算法可以显着提高建模精度。