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

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. Dynamical systems are widely used across scientific and engineering disciplines to model state variables' evolution over time (Lasota & Mackey, 2013). Nonlinear ordinary or partial differential equations typically govern these systems and may incorporate stochastic components (Meyn & Tweedie, 2012). However, in many practical situations, analytical models are unavailable or intractable, motivating the use of data-driven approaches to infer the underlying dynamics from sampled trajectories. In this context, Koopman and transfer operator regressions have emerged as a powerful framework for learning and interpreting dynamical systems from data (Brunton et al., 2022). Rather than directly modeling the evolution of state variables, these operators advance observables (scalar functions defined on the state space) by mapping each to its expected future value conditioned on the current state. Crucially, these operators are linear even when the underlying systems are not linear.