Learning theory for dynamical systems

The task of modelling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks - extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. We present a mathematical framework in which the dynamical information is represented in the form of an embedding. The framework combines the two subtasks using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms - delay-coordinates and reservoir computing. We use this framework as a platform for two other investigations of the reconstructed system - its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying system - the behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap between universally observed behavior of dynamics modelling; and the spectral, differential and ergodic properties intrinsic to the dynamics.