The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. Here, we present a unifying framework for blending mechanistic and machine-learning approaches for identifying dynamical systems from data. This framework is agnostic to the chosen machine learning model parameterization, and casts the problem in both continuous- and discrete-time.

We will focus on recent developments that fuse data assimilation with auto-differentiable ODE solvers which, when combined, allow us to learn from noisy, partial observations.

We will also present comments on reservoir computers and their connections to random feature (and hence, kernel) methods. Joint work with Andrew Stuart.