标题： 一种用于动态系统算子表示的谱-格拉斯曼 Wasserstein 度量 显示英文标题

标题： A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems

摘要： 从轨迹数据中估计动态系统的几何结构是机器学习应用中的一个主要挑战。 Koopman 算子和转移算子通过其谱分解提供了非线性动力学的线性表示，为比较提供了一个自然的框架。 我们提出了一种新方法，将每个系统表示为其联合算子本征值和谱投影器的分布，并通过最优传输定义系统之间的度量。 所提出的度量对轨迹的采样频率具有不变性。 它在计算上也是高效的，由有限样本收敛保证支持，并能够计算 Fréchet 均值，从而在动态系统之间进行插值。 在模拟和真实世界数据集上的实验表明，我们的方法在机器学习应用中，包括降维和分类，始终优于标准的基于算子的距离，并在动态系统之间提供了有意义的插值。 显示英文摘要

摘要： The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors and defining a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fr\'echet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.