New Strategies for Reduced-order Models for Predicting the Statistical Responses

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among 4 many complex systems in science and engineering including climate, material, and neural science. The existence of a strange attractor in the turbulent systems 5 containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties from imperfect modeling equations or perturbations 6 in initial values, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in turbulent 7 dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in 8 their nonlinear responses to changes in external forcing parameters or uncertain initial data. In the development of a proper UQ scheme for systems of high 9 or infinite dimensionality with instabilities, significant model errors compared with the true natural signal are always unavoidable due to both the imperfect 10 understanding of the underlying physical processes and the limited computational resources available through direct Monte-Carlo integration. One central issue 11 in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium 12 (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity). 13 Here we discuss a general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical 14 variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems. There are three 15 stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase using only data in the more complex perfect 16 model statistics; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal 17 class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several 18 mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory 19 and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total 20 statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. 21 The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic 22 turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical 23 response for these complex models are accurately captured by the reduced-order models with only 2× 102 modes in a highly turbulent system with 1× 105 24 degrees of freedom. Less than 0.15% of the total spectral modes are needed in the reduced-order models. 25