Data-Driven Low-Order Stochastic Models

Abstract

Data-driven low-order stochastic models have broad applications in reality. They can be utilized to effectively model the time evolution of each spectral mode of a complex spatially extended system, where the complicated nonlinearity is replaced by suitable stochastic noise that facilitates efficient data assimilation and forecast. They can also be used to model and predict large-scale features of many physical phenomena described by low-dimensional indices. In addition, simple low-order models provide cheap and powerful ways to validate various hypotheses in developing more complicated models. In this chapter, several practically useful data-driven low-order models are presented, aiming at resolving these problems. The models introduced here include complex Ornstein–Uhlenbeck (OU) processes, linear models with multiplicative noise, simple stochastic parameterized models, and physics-constrained nonlinear regression models. The development of these models, their physical and statistical properties, systematic model calibrations, and applications to real-world problems are all highlighted. The procedure of utilizing a suite of these simple stochastic models to reproduce the statistical results of complex PDEs is also illustrated. Finally, a discussion of the linear versus Gaussian approximations for nonlinear systems is included, indicating the fundamental differences in applying these approximations in solving practical problems.