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Reduced-order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. In fluid dynamics, ROMs have been successful in providing efficient and relatively accurate solutions for the numerical simulation of laminar flows. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models in order to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, instead of reviewing all the ROM closures and stabilizations, we took a more modest step and focused on one particular type of ROM closure and stabilization that is inspired by large eddy simulation (LES), a classical strategy in computational fluid dynamics (CFD). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Indeed, LES-ROMs are modular and generally require minimal modifications to standard (“legacy”) ROM formulations. Furthermore, the computational overhead of these modifications is minimal. Finally, carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in science and engineering applications. LES-ROMs are constructed by leveraging spatial filtering, which is the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also “bridges” two distinct research fields (LES and ROMs) that have been disconnected until now. This paper is a review of LES-ROMs, with a particular focus on the LES concepts and models that enable the construction of LES-inspired ROMs and the bridging of LES and reduced-order modeling. This paper starts with a description of a versatile LES strategy called evolve–filter–relax (EFR) that has been successfully used as a full-order method for both incompressible and compressible convection-dominated flows. We present evidence of this success. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs (e.g., EFR-ROM). Several applications of LES-ROMs to the numerical simulation of incompressible and compressible convection-dominated flows are presented. Finally, we draw conclusions and outline several research directions and open questions in LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which we believe has a lot of potential in the practical numerical simulation of convection-dominated flows in science, engineering, and medicine. 

Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends the MFEnKF into to make use of a broader class of surrogate model such as those based on machine learning methods such as autoencoders non-linear couplings in between the model hierarchies. We identify the right-invertibility property for autoencoders as being a key predictor of success in the forecasting power of autoencoder-based reduced order models. We propose a methodology that allows us to construct reduced order surrogate models that are more accurate than the ones obtained via conventional linear methods. Numerical experiments with the canonical Lorenz'96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity ensemble Kalman filtering. We additionality show a large-scale proof-of-concept result with the quasi-geostrophic equations, showing the competitiveness of the method with a traditional reduced order model-based MFEnKF.