Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics

Area of Science:

• * Physics
• * Applied Mathematics
• * Data Science

Background:

• * Many physical systems are governed by complex nonlinear differential equations that resist full analytical solutions.
• * Traditional methods involve simplifying equations and using statistical approaches to model uninteresting variables.
• * This necessitates robust methods for analyzing complex system dynamics.

Purpose of the Study:

• * To introduce a fully discrete solution method for time-dependent nonlinear differential equations.
• * To simplify data analysis and numerical algorithms for complex physical systems.
• * To connect statistical physics formalisms with practical engineering identification techniques.

Main Methods:

• * Development of a fully discrete solution method for nonlinear differential equations.
• * Application of nonlinear autoregression moving average with exogenous input (NARMAX) for time series identification.
• * Exploration of connections with the Mori-Zwanzig formalism from statistical physics.

Main Results:

• * A novel discrete solution method is presented, simplifying the analysis of complex physical systems.
• * The method yields time series identified by NARMAX, a standard engineering approach.
• * Successful application demonstrated on the Lorenz 96 system, validating the approach.

Conclusions:

• * The proposed discrete method offers an efficient way to approximate solutions for complex nonlinear systems.
• * NARMAX identification provides a practical tool for analyzing the resulting time series.
• * This work bridges theoretical statistical physics with computational methods for physical system modeling.