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 (Rigged DMD can be used with any rigging). 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：本文提出装备动态模态分解算法，该算法可计算Koopman算子的广义本征函数分解。通过考虑观测量的演化，Koopman算子将复杂非线性动力学转化为适合谱分析的线性框架。尽管传统动态模态分解技术功能强大，但常难以处理连续谱问题。装备动态模态分解通过数据驱动方法应对这些挑战，利用系统演化过程的快照数据逼近Koopman算子的预解式及其广义本征函数。该算法的核心在于：通过将保测延拓动态模态分解与高阶平滑核函数相结合，构建广义Koopman本征函数和模态的波包近似。这提供了包含离散与连续谱元素的鲁棒分解方法。我们推导了广义本征函数与谱测度的高阶收敛定理。此外，提出基于时滞嵌入构建装备希尔伯特空间的新框架，显著扩展了算法适用性（装备动态模态分解可与任意装备结构配合使用）。通过包括Lebesgue谱系统、可积哈密顿系统、Lorenz系统以及二维方腔高雷诺数顶盖驱动流在内的算例，验证了装备动态模态分解的收敛性、高效性与普适性。本研究为连续谱分解的未来研究与应用奠定了基础。