逼近Koopman算子的有限数据误差界：采样测度、超多项式收敛与正则化
Finite-Data Error Bounds for Approximating the Koopman Operator: Sampling Measures, Super-Polynomial Convergence and Regularization
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中文总结 AI 辅助
本研究为有限数据下EDMD逼近Koopman算子建立误差界，证明蒙特卡洛速率，并在特定结构下实现超多项式收敛，同时为欠采样情形提供基于压缩感知的恢复保证。
中文摘要 AI 辅助
Koopman算子是一个成熟的框架，用于将非线性动力系统提升到无限维空间，在该空间中动力学是线性的。扩展动态模态分解（EDMD）是数据驱动动力学中广泛使用的方法，因为它提供了Koopman算子在给定可观测函数字典的有限维张成空间上的Galerkin逼近。特别地，EDMD仅需要来自动力系统的样本，无需了解底层动力学即可描述Koopman算子。尽管许多研究分析了EDMD在数据方面的渐近收敛结果，但在一般设置下，更实用的有限数据收敛结果问题仍不完整。在本工作中，我们证明了有限数据EDMD对Koopman算子的Galerkin逼近的误差界，其速率与样本数的根成反比（蒙特卡洛速率）。这些结果适用于多类问题，即离散或连续动力学，包括随机系统。在某些更具结构性的设置中，我们证明超多项式速率在理论上和计算上都是可实现的。最后，我们推导了受压缩感知技术启发的欠采样机制下EDMD变体的恢复保证。
英文摘要
The Koopman operator is a well-established framework for lifting nonlinear dynamical systems to an infinite-dimensional space where dynamics are linear. Extended Dynamic Mode Decomposition (EDMD) is a widely used method in data-driven dynamics as it provides a Galerkin approximation of the Koopman operator on the finite-dimensional span of a prescribed dictionary of observable functions. In particular, EDMD only requires samples from the dynamical system, describing the Koopman operator without knowledge of the underlying dynamics. While many studies analyze asymptotic convergence results for EDMD in terms of data, the question of more practical finite-data convergence results remains incomplete in general settings. In this work, we prove bounds for the finite-data EDMD Galerkin approximation of the Koopman operator by a rate proportional to the inverse of the root of the number of samples (the Monte Carlo rate). These results apply to multiple classes of problems, i.e., for discrete or continuous dynamics, including stochastic systems. In certain more structured setting, we demonstrate super-polynomial rates are achievable both theoretically and computationally. Finally, we derive recovery guarantees for an EDMD variant in the undersampled regime inspired by compressed sensing techniques.
发表机构
- Concordia University（康考迪亚大学）
机构由 AI 辅助整理，请以论文原文为准。
