A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics

TL;DR

This paper introduces a discrete-time framework for stochastic parametrization and dimensional reduction in nonlinear systems, utilizing NARMAX models and connecting to the Mori-Zwanzig formalism, demonstrated on the Lorenz 96 system.

Contribution

It presents a novel discrete-time approach to stochastic parametrization, linking NARMAX models with the Mori-Zwanzig formalism for nonlinear dynamics.

Findings

Effective reduction of complex nonlinear systems

Application to Lorenz 96 demonstrates practical utility

Connections established between NARMAX and Mori-Zwanzig

Abstract

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.