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

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key mechanisms. In the absence of special features such as scale separation or symmetries, time evolution these variables typically exhibits memory effects. Recent work has found a variety data-driven model be effective for representing non-Markovian dynamics, but their scope underpinning remain incompletely understood. Here, we study from systems perspective. For both chaotic randomly-forced systems, show problem can naturally formulated within framework Koopman operators Mori-Zwanzig projection operator formalism. We give heuristic derivation NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input) an underlying model. The is based on simple construction call Wiener projection, which links theory classical filtering. apply ideas Kuramoto-Sivashinsky spatiotemporal chaos viscous Burgers equation stochastic forcing.