Data-driven model reduction, Wiener projections, and the Mori-Zwanzig formalism

First-principles models of complex dynamic phenomena often have many degrees of freedom, only a small fraction of which may be scientifically relevant or observable. Reduced models distill such phenomena to their essence by modeling only relevant variables, thus decreasing computational cost and clarifying dynamical mechanisms. Here, we consider data-driven model reduction for nonlinear dynamical systems without sharp scale separation. Motivated by a discrete-time version of the Mori-Zwanzig projection operator formalism and the Wiener filter, we propose a simple and flexible mathematical formulation based on Wiener projection, which decomposes a nonlinear dynamical system into a component predictable by past values of relevant variables and its orthogonal complement. Wiener projection is equally applicable to deterministic chaotic dynamics and randomly-forced systems, and provides a natural starting point for systematic approximations. In particular, we use it to derive NARMAX models from an underlying dynamical system, thereby clarifying the scope of these widely-used tools in time series analysis. We illustrate its versatility on the Kuramoto-Sivashinsky model of spatiotemporal chaos and a stochastic Burgers equation.