Data-driven stochastic parameterization and dimension reduction in nonlinear dynamics

Prediction for high-dimensional nonlinear dynamic systems often encounters difficulties: such systems are expensive to solve in high resolution, and initial data are often missing because only the large scale slow variables, which are of direct interest, are observed. The development of reduced models for these slow variables is thus needed. The challenges come from the nonlinear interactions between different scales, and the difficulties in quantifying uncertainties from discrete data.

We address these challenges by developing discrete nonlinear stochastic reduced systems, in which one formulates discrete solvable approximate equations for the large scale variables and uses data and statistical methods to account for the impact of the unresolved variables. A key ingredient in the construction of the stochastic reduced systems is a discrete-time stochastic parametrization based on the NARMAX (nonlinear autoregression moving average with exogenous input) model. Our examples include the Lorenz 96 system and the Kuramoto-Sivashinsky equation.