We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional Dynamic Mode Decomposition (DMD) techniques often struggle with continuous spectra. Rigged DMD addresses these challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions using snapshot data from the system's evolution. At its core, Rigged DMD builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements. We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. Additionally, we propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity, demonstrating Rigged DMD's convergence, efficiency, and versatility. This work paves the way for future research and applications of decompositions with continuous spectra.

翻译：本文提出赋权动态模态分解（Rigged DMD）算法，用于计算库普曼算子的广义本征函数分解。通过考察可观测量的演化过程，库普曼算子将复杂的非线性动力学转化为适用于谱分析的线性框架。传统动态模态分解（DMD）技术虽功能强大，但在处理连续谱时往往面临挑战。赋权动态模态分解采用数据驱动方法，利用系统演化过程中的快照数据近似库普曼算子的预解式及其广义本征函数，从而有效应对上述挑战。该算法的核心在于将保测扩展动态模态分解与高阶平滑核相结合，构建广义库普曼本征函数和模态的波包近似，进而实现包含离散谱与连续谱元素的稳健分解。我们推导了广义本征函数与谱测度的显式高阶收敛定理，并提出了利用时间延迟嵌入构造赋权希尔伯特空间的新框架，显著拓展了算法的适用范围。通过涵盖勒贝格谱系统、可积哈密顿系统、洛伦兹系统以及二维方腔高雷诺数顶盖驱动流动在内的多个算例，验证了赋权动态模态分解的收敛性、高效性与通用性。本工作为后续基于连续谱分解的研究与应用开辟了新路径。