Uncertainty quantification of turbulent systems via physically consistent and data-informed reduced-order models

This work presents a data-driven, energy-conserving closure method for the coarse-scale evolution of the mean and covariance of turbulent systems. Spatiotemporally non-local neural networks are employed for calculating the impact of non-Gaussian effects to the low-order statistics of dynamical systems with an energy-preserving quadratic nonlinearity. This property, which characterizes the advection term of turbulent flows, is encoded via an appropriate physical constraint in the training process of the data-informed closure. This condition is essential for the stability and accuracy of the simulations as it appropriately captures the energy transfers between unstable and stable modes of the system. The numerical scheme is implemented for a variety of turbulent systems, with prominent forward and inverse energy cascades. These problems include prototypical models such as an unstable triad system and the Lorentz-96 system, as well as more complex models: The two-layer quasi-geostrophic flows and incompressible, anisotropic jets where passive inertial tracers are being advected on. Training data are obtained through high-fidelity direct numerical simulations. In all cases, the hybrid scheme displays its ability to accurately capture the energy spectrum and high-order statistics of the systems under discussion. The generalizability properties of the trained closure models in all the test cases are explored, using out-of-sample realizations of the systems. The presented method is compared with existing first-order closure schemes, where only the mean equation is evolved. This comparison showcases that correctly evolving the covariance of the system outperforms first-order schemes in accuracy, at the expense of increased computational cost. (C) 2022 Author(s).