A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics
It provides a practical, data-driven approach for reducing complex nonlinear systems, relevant to researchers in dynamical systems and engineering.
The paper develops a discrete-time framework for stochastic parametrization and dimensional reduction in nonlinear dynamics, using NARMAX representations to model unresolved variables. Applied to the Lorenz 96 system, the method demonstrates effective reduction while capturing stochastic effects.
Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate solvable approximate equations for the variables of greater interest, and use data and statistical methods to account for the impact of the other variables. In the present paper the problem is considered in a fully discrete-time setting, which simplifies both the analysis of the data and the numerical algorithms. The resulting time series are identified by a NARMAX (nonlinear autoregression moving average with exogenous input) representation familiar from engineering practice. The connections with the Mori-Zwanzig formalism of statistical physics are discussed, as well as an application to the Lorenz 96 system.