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Suitable reduced order models (ROMs) are computationally efficient tools incharacterizing key dynamical and statistical features of nature. In this paper,a systematic multiscale stochastic ROM framework is developed for complexsystems with strong chaotic or turbulent behavior. The new ROMs arefundamentally different from the traditional Galerkin ROM (G-ROM) or thosedeterministic ROMs that aim at minimizing the path-wise errors and applyingmainly to laminar systems. Here, the new ROM focuses on recovering thelarge-scale dynamics to the maximum extent while it also exploits cheap buteffective conditional linear functions as the closure terms to capture thestatistical features of the medium-scale variables and its feedback to thelarge scales. In addition, physics constraints are incorporated into the newROM. One unique feature of the resulting ROM is that it facilitates anefficient and accurate scheme for nonlinear data assimilation, the solution ofwhich is provided by closed analytic formulae. Such an analytic solvable dataassimilation solution significantly accelerates the computational efficiencyand allows the new ROM to avoid many potential numerical and sampling issues inrecovering the unobserved states from partial observations. The overall modelcalibration is efficient and systematic via explicit mathematical formulae. Thenew ROM framework is applied to complex nonlinear systems, in which theintrinsic turbulent behavior is either triggered by external random forcing ordeterministic nonlinearity. It is shown that the new ROM significantlyoutperforms the G-ROM in both scenarios in terms of reproducing the dynamicaland statistical features as well as recovering unobserved states via theassociated efficient data assimilation scheme.

An Efficient Data-Driven Multiscale Stochastic Reduced Order Modeling Framework for Complex Systems