In turbulence modeling, and more particularly in the Large-Eddy Simulation (LES) framework, we are concerned with finding closure models that represent the effect of the unresolved “subgrid” scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we extend the system of equations describing the filtered solution with a set of equations that describe the evolution of (a compressed version of) the energy of the subgrid scales. Having an estimate of this energy, we can use the concept of energy conservation and derive stability. The compressed variables are determined via a data-driven technique in such a way that the energy of the subgrid-scales is matched. For the extended system, the closure model should be energy-conserving, and a new skew-symmetric convolutional neural network architecture is proposed that has this property. Stability is thus guaranteed, independent of the actual weights and biases of the network. Importantly, our framework allows energy exchange between resolved scales and compressed subgrid scales and thus enables backscatter. For more information, check our article.

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