Geometric Dictionary Learning for Operator Representations of Dynamical Systems

Operator-theoretic representations provide a powerful framework for learning dynamical systems from data and interpreting their behavior: through spectral decomposition, they characterize the dynamics in terms of modes, frequencies, and time scales. Yet existing approaches typically estimate an operator independently for each system, even when data from a set of related dynamics are available, thereby failing to leverage population-level information to obtain more compact representations and more data-efficient estimators. In this work, we posit that the operator representations of related dynamical systems concentrate near a low-dimensional manifold. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that approximates this population manifold through a dictionary of characteristic operators. The resulting dictionary yields compact and interpretable representations of individual systems while preserving their dynamical properties. Beyond representation learning, DOODL enables operator estimation from short trajectories by constraining new operators to be combinations of the learned operators. Experiments on metastable Langevin dynamics and turbulent plasma simulations show that DOODL captures meaningful variations across populations of complex multiscale systems while enabling accurate operator estimation in low-data regimes, with errors one to two orders of magnitude lower than independent operator estimation methods.