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\begin{document}

\title{Data-driven subgrid-scale parameterization of forced 2D turbulence in
the small-data limit}



\author[1]{Yifei Guan}%
\author[1]{Adam Subel}%
\author[1]{Ashesh Chattopadhyay}%
\author[1]{Pedram Hassanzadeh}%
\affil[1]{Rice University}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\begin{abstract}
In this work, we develop a data-driven subgrid-scale (SGS) model using a
fully convolutional neural network (CNN) for large eddy simulation of
forced 2D turbulence. Forced 2D turbulence is a fitting prototype for
many large-scale geophysical and environmental flows (where rotation
and/or stratification dominate) and has been widely used as a testbed
for novel techniques, including machine-learning-based SGS modeling. We
first conduct direct numerical simulation (DNS) and obtain training,
validation, and testing data sets by applying a Gaussian spatial filter
to the DNS solution. With the filtered DNS (FDNS) data in hand, we train
the CNN with the filtered state variables. A priori analysis shows that
the CNN-predicted SGS term accurately captures the inter-scale energy
transfer. A posteriori analysis indicates that the LES-CNN outperforms
the physics-based models in both short-term prediction and long-term
statistics. Although the CNN-based model is promising in predicting the
SGS term, it requires big data to perform satisfactorily. In the
small-data limit, the LES-CNN generates artificial instabilities and
thus leads to unphysical results. We propose three remedies for the CNN
to work in the small-data limit, i.e., data augmentation and group
convolution neural network (GCNN), leveraging the rotational
equivariance of the SGS term and incorporating a physical constraint on
the SGS enstrophy transfer. The SGS term is both translational and
rotational equivariant in a square periodic flow field. While primitive
CNN can capture the translational equivariance, the rotational
equivariance can be accounted for by either including rotated snapshots
in the training data set or by a GCNN that enforces rotational
equivariance as a hard constraint. Additionally, The SGS enstrophy
transfer constraint can be implemented in the loss function of the CNN.
A priori and a posteriori analyses show that the CNN/GCNN with
knowledge/constraints of rotational equivariance and SGS enstrophy
transfer enhances the SGS model and allows the data-driven model to work
stably and accurately in a small-data limit. These findings can
potentially help the ongoing efforts in using machine-learning for SGS
modeling in weather/climate models, where high-quality training data are
scarce and instabilities have been reported in many past studies.%
\end{abstract}%



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