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We parameterize sub-grid scale (SGS) fluxes in sinusoidally forcedtwo-dimensional turbulence on the $\beta$-plane at high Reynolds numbers(Re$\sim$25000) using simple 2-layer Convolutional Neural Networks (CNN) havingonly O(1000)parameters, two orders of magnitude smaller than recent studiesemploying deeper CNNs with 8-10 layers; we obtain stable, accurate, andlong-term online or a posteriori solutions at 16X downscaling factors. Ourmethodology significantly improves training efficiency and speed of onlineLarge Eddy Simulations (LES) runs, while offering insights into the physics ofclosure in such turbulent flows. Our approach benefits from extensivehyperparameter searching in learning rate and weight decay coefficient space,as well as the use of cyclical learning rate annealing, which leads to morerobust and accurate online solutions compared to fixed learning rates. Our CNNsuse either the coarse velocity or the vorticity and strain fields as inputs,and output the two components of the deviatoric stress tensor. We minimize aloss between the SGS vorticity flux divergence (computed from thehigh-resolution solver) and that obtained from the CNN-modeled deviatoricstress tensor, without requiring energy or enstrophy preserving constraints.The success of shallow CNNs in accurately parameterizing this class ofturbulent flows implies that the SGS stresses have a weak non-local dependenceon coarse fields; it also aligns with our physical conception that small-scalesare locally controlled by larger scales such as vortices and their strainedfilaments. Furthermore, 2-layer CNN-parameterizations are more likely to beinterpretable and generalizable because of their intrinsic low dimensionality.

Turbulence closure with small, local neural networks: Forced two-dimensional and $\beta$-plane flows