无需假设函数空间封闭，用核方法在线学习非线性系统的动力学算子。
Nonparametric Sparse Online Learning of the Koopman Operator
- 在再生核希尔伯特空间中通过条件均值嵌入建模算子，允许动态过程超出预设空间。
- 提出在线稀疏学习算法，支持轨迹采样并提供渐近与有限时间收敛保证。
- 适用于复杂非线性系统建模，适合控制、机器学习和动力系统研究者。
Koopman算子为一般非线性动力系统提供了强大的表示框架。数据驱动的学习方法通常假设所选函数空间在系统动力下封闭。本文通过再生核希尔伯特空间（RKHS）研究Koopman算子，并探索动态可能超出选定函数空间的误设情形。我们将Koopman算子与条件均值嵌入（CME）算子关联，提出一种算子随机逼近算法，实现对算子的迭代学习，并可控制表示的复杂度。我们提供了基于轨迹采样的在线稀疏学习算法的渐近与有限时间最后迭代保证，分析远比有限维随机逼近复杂。数值实验验证了所提算法的有效性。
原文摘要 · Abstract (English)
The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under system dynamics. In this paper, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and explore the mis-specified scenario where the dynamics may escape the chosen function space. We relate the Koopman operator to the conditional mean embeddings (CME) operator and then present an operator stochastic approximation algorithm to learn the Koopman operator iteratively with control over the complexity of the representation. We provide both asymptotic and finite-time last-iterate guarantees of the online sparse learning algorithm with trajectory-based sampling with an analysis that is substantially more involved than that for finite-dimensional stochastic approximation. Numerical examples confirm the effectiveness of the proposed algorithm.
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