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There is growing interest in discovering interpretable, closed-form equationsfor subgrid-scale (SGS) closures/parameterizations of complex processes inEarth system. Here, we apply a common equation-discovery technique withexpansive libraries to learn closures from filtered direct numericalsimulations of 2D forced turbulence and Rayleigh-B\'enard convection (RBC).Across common filters, we robustly discover closures of the same form formomentum and heat fluxes. These closures depend on nonlinear combinations ofgradients of filtered variables (velocity, temperature), with constants thatare independent of the fluid/flow properties and only depend on filtertype/size. We show that these closures are the nonlinear gradient model (NGM),which is derivable analytically using Taylor-series expansions. In fact, wesuggest that with common (physics-free) equation-discovery algorithms,regardless of the system/physics, discovered closures are always consistentwith the Taylor-series. Like previous studies, we find that large-eddysimulations with NGM closures are unstable, despite significant similaritiesbetween the true and NGM-predicted fluxes (pattern correlations $> 0.95$). Weidentify two shortcomings as reasons for these instabilities: in 2D, NGMproduces zero kinetic energy transfer between resolved and subgrid scales,lacking both diffusion and backscattering. In RBC, backscattering of potentialenergy is poorly predicted. Moreover, we show that SGS fluxes diagnosed fromdata, presumed the "truth" for discovery, depend on filtering procedures andare not unique. Accordingly, to learn accurate, stable closures fromhigh-fidelity data in future work, we propose several ideas around usingphysics-informed libraries, loss functions, and metrics. These findings arerelevant beyond turbulence to closure modeling of any multi-scale system.

Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges