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.
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