Colloquium

Nonlinear dynamic phenomena often require a large number of dynamical variables for their description, only a small fraction of which are scientifically relevant. Reduced models that use only these relevant variables can be very useful both for computational efficiency and dynamical insights. In recent years, data-driven model reduction methods have grown in popularity, in part because they promise general applicability with minimal dynamical assumptions. In this talk, I will review a discrete-time version of the Mori-Zwanzig formalism of nonequilibrium statistical mechanics, which provides a general framework for model reduction. I will explain how the problem of data-driven model reduction, for both chaotic and stochastic dynamical systems, can be given a clear and precise formulation within this framework. Among other things, the framework enables one to derive the NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous inputs) representation of stochastic processes, widely used in time series analysis and data-driven modeling, from an underlying dynamical model. I will illustrate these ideas on a prototypical model of spatiotemporal chaos and a stochastically-forced PDE.