Uncertainty quantification and extreme event analysis for turbulent flows using energy-preserving data-driven closure schemes *

We present a reduced-order modelling scheme for computing the statistics of incompressible turbulent flows. The scheme involves evolution equations for the mean and covariance of the system, in tandem with a neural-network modelling of the higher-order closure terms. The closure terms are approximated using spatio-temporally nonlocal neural-networks with appropriate constraints enforced during training. These constraints allow the reduced-order model to adhere to the physical properties of the reference system. In more detail, energy preservation constraints ensure that the energy transfer between scales, as modelled by the nonlinear terms of the full system, are properly included in our model. The constraints are essential for our approximate model to reach the appropriate statistical equilibrium that corresponds to the full reference system. We showcase that our computationally tractable model is able to robustly approximate effectively complicated and expensive turbulent flows in the ocean and atmosphere. We test the method in its ability to approximate important statistics of these systems such as its mean and energy spectrum. We also approximate the probability density functions of quantities of interest, showing that the model can estimate the probability of intermittent extreme events occuring. Turbulent flows with Gaussian and strongly non-Gaussian statistics are examined.