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In this paper, we propose and assess several stochastic parametrizations for data‐driven modeling of the two‐dimensional Euler equations using coarse‐grid SPDEs. The framework of Stochastic Advection by Lie Transport (SALT) (Cotter et al., 2019, https://doi.org/10.1137/18m1167929) is employed to define a stochastic forcing that is decomposed in terms of a deterministic basis (empirical orthogonal functions, EOFs) multiplied by temporal traces, here regarded as stochastic processes. The EOFs are obtained from a fine‐grid data set and are defined in conjunction with corresponding deterministic time series. We construct stochastic processes that mimic properties of the measured time series. In particular, the processes are defined such that the underlying probability density functions (pdfs) or the estimated correlation time of the time series are retained. These stochastic models are compared to stochastic forcing based on Gaussian noise, which does not use any information of the time series. We perform uncertainty quantification tests and compare stochastic ensembles in terms of mean and spread. Reduced uncertainty is observed for the developed models. On short timescales, such as those used for data assimilation (Cotter et al., 2020a, https://doi.org/10.1007/s10955-020-02524-0), the stochastic models show a reduced ensemble mean error and a reduced spread. Particularly, using estimated pdfs yields stochastic ensembles which rarely fail to capture the reference solution on small time scales, whereas introducing correlation into the stochastic models improves the quality of the coarse‐grid predictions with respect to Gaussian noise. Plain Language Summary: Turbulent flows often contain small‐scale fluctuations that behave in a seemingly random way. Predicting the behavior of such a flow is challenging, since simulating the flow in full detail is computationally expensive. To reduce the computational costs, one can initially ignore the small‐scale fluctuations and subsequently try to include the effects of these scales by including an additional term into the equations that describe the flow. We propose and assess various models that represent the influence of the small‐scales through a stochastic (random) forcing term. We compare three types of stochastic processes that use information from high‐resolution data. It is found that using more information from the data leads to a reduced spread and ensemble mean error. Key Points: High‐resolution numerical simulation data are used to extract small‐scale features of the 2D Euler equationsAn empirical orthogonal function (EOF)‐based stochastic forcing is proposed, where the EOF time series serve to define data‐driven stochastic processes for each EOFThe data‐driven processes are found to produce ensembles with reduced mean error and spread, compared to using Gaussian noise [ABSTRACT FROM AUTHOR]

Small-scale features of shallow water flow obtained from direct numerical simulation (DNS) with two different computational codes for the shallow water equations are gathered offline and subsequently employed with the aim of constructing a reduced-order correction. This is used to facilitate high-fidelity online flow predictions at much reduced costs on coarse meshes. The resolved small-scale features at high resolution represent subgrid properties for the coarse representation. Measurements of the subgrid dynamics are obtained as the difference between the evolution of a coarse grid solution and the corresponding DNS result. The measurements are sensitive to the particular numerical methods used for the simulation on coarse computational grids and can be used to approximately correct the associated discretization errors. The subgrid features are decomposed into empirical orthogonal functions (EOFs), after which a corresponding correction term is constructed. By increasing the number of EOFs in the approximation of the measured values the correction term can in principle be made arbitrarily accurate. Both computational methods investigated here show a significant decrease in the simulation error already when applying the correction based on the dominant EOFs only. The error reduction accounts for the particular discretization errors that are incurred and are hence specific to the particular simulation method that is adopted. This improvement is also observed for very coarse grids, which may be used for computational model reduction in geophysical and turbulent flow problems. [ABSTRACT FROM AUTHOR]