Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms come with error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046-dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number $>10^5$ that has a 295,122-dimensional state space.

翻译：Koopman算子是全局线性化非线性动力系统的无限维算子，其谱信息对理解动力学行为具有重要价值。然而，Koopman算子可能具有连续谱和无限维不变子空间，这使得谱信息的计算面临重大挑战。本文提出了具有严格收敛保证的数据驱动算法，用于从轨迹数据计算Koopman算子的谱信息。我们引入残差动态模式分解（ResDMD），该方法首次实现了从快照数据计算一般Koopman算子的谱和伪谱时避免谱污染。通过利用预解算子与ResDMD，我们计算了与一般保测动力系统相关的谱测度的光滑逼近。我们证明了算法的显式收敛定理，即使在混沌系统中计算连续谱和离散谱的密度时也能实现高阶收敛。由于算法具备误差控制能力，ResDMD允许对谱量、Koopman模态分解及学习字典进行后验验证。我们在帐篷映射、圆周旋转、高斯迭代映射、非线性单摆、双摆及Lorenz系统上验证了算法效能。最后，针对高维状态空间的动力系统，我们提供了算法的核化变体。这使得我们能够计算具有20,046维状态空间的蛋白质分子动力学相关的谱测度，并对雷诺数$>10^5$、具有295,122维状态空间的湍流机翼绕流计算带误差界的非线性Koopman模态。