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\begin{document}

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



\author[1]{Karan Jakhar}%
\author[1]{Yifei Guan}%
\author[1]{Rambod Mojgani}%
\author[1]{Ashesh Chattopadhyay}%
\author[1]{Pedram Hassanzadeh}%
\author[2]{Laure Zanna}%
\affil[1]{Rice University}%
\affil[2]{New York University}%


\vspace{-1em}



  
  \date{June 16, 2024}


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\begin{abstract}
There is growing interest in discovering interpretable, closed-form
equations for subgrid-scale (SGS) closures/parameterizations of complex
processes in Earth system. Here, we apply a common equation-discovery
technique with expansive libraries to learn closures from filtered
direct numerical simulations of 2D forced turbulence and Rayleigh-Benard
convection (RBC). Across common filters, we robustly discover closures
of the same form for momentum and heat fluxes. These closures depend on
nonlinear combinations of gradients of filtered variables (velocity,
temperature), with constants that are independent of the fluid/flow
properties and only depend on filter type/size. We show that these
closures are the nonlinear gradient model (NGM), which is derivable
analytically using Taylor-series expansions. In fact, we suggest that
with common (physics-free) equation-discovery algorithms, regardless of
the system/physics, discovered closures are always consistent with the
Taylor-series. Like previous studies, we find that large-eddy
simulations with NGM closures are unstable, despite significant
similarities between the true and NGM-predicted fluxes (pattern
correlations \textgreater{} 0.95). We identify two shortcomings as
reasons for these instabilities: in 2D, NGM produces zero kinetic energy
transfer between resolved and subgrid scales, lacking both diffusion and
backscattering. In RBC, backscattering of potential energy is poorly
predicted. Moreover, we show that SGS fluxes diagnosed from data,
presumed the `truth' for discovery, depend on filtering procedures and
are not unique. Accordingly, to learn accurate, stable closures from
high-fidelity data in future work, we propose several ideas around using
physics-informed libraries, loss functions, and metrics. These findings
are relevant beyond turbulence to closure modeling of any multi-scale
system.%
\end{abstract}%



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