Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

We assess the limits of predictability of the large‐scale cloud patterns in the boreal winter Madden‐Julian Oscillation (MJO) as measured through outgoing longwave radiation (OLR) alone, a proxy for convective activity. A recent advanced nonlinear time series technique, nonlinear Laplacian spectral analysis, is applied to the OLR data to define two spatial modes with high intermittency associated with the boreal winter MJO. A recent data‐driven physics‐constrained low‐order stochastic modeling procedure is applied to these time series. The result is a four‐dimensional nonlinear stochastic model for the two observed OLR variables and two hidden variables involving correlated multiplicative noise defined through energy‐conserving nonlinear interaction. Systematic calibration and prediction experiments show the skillful prediction by these models for 40, 25, and 18 days in strong, moderate, and weak MJO winters, respectively. Furthermore, the ensemble spread is an accurate indicator of forecast uncertainty at long lead times.

Solving the Fokker–Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples L, where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of L. The resulting algorithms can efficiently solve the Fokker–Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a 1,000-dimensional stochastic coupled FitzHugh–Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only $L=1$ samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a 240-dimensional two-layer inhomogeneous Lorenz 96 model, using only $L=500$ samples. Significance Solving the Fokker–Planck equation for high-dimensional complex dynamical systems is an important issue. Effective strategies are developed and incorporated into efficient statistically accurate algorithms for solving the Fokker–Planck equations associated with a rich class of high-dimensional nonlinear turbulent dynamical systems with strong non-Gaussian features. These effective strategies exploit a judicious block decomposition of high-dimensional conditional covariance matrices and statistical symmetry to facilitate an extremely efficient parallel computation and a significant reduction of sample numbers. The resulting algorithms can efficiently solve the Fokker–Planck equation in much higher dimensions even with orders in the millions and thus beat the curse of dimension. Skillful behavior of the algorithms is illustrated for highly non-Gaussian systems in excitable media and geophysical turbulence.

Significance: Understanding the role that atmospheric wind bursts play in the initiation and maintenance of the El Niño Southern Oscillation (ENSO) in the tropics is a crucial problem in ocean–atmosphere sciences. We provide insight into the problem by proposing a simple ENSO model, amenable to detailed analysis, where wind burst activity is driven by a simple stochastic jump process that depends on the strength of the western Pacific warm pool. The model captures key features of the observational record, such as the probability density function and power spectrum of eastern Pacific sea surface temperatures. In addition, the varying intensity and strength of El Niño events from the model are also consistent with real observations including super El Niño episodes.