  We demonstrate how incorporating physics constraints into convolutional
neural networks (CNNs) enables learning subgrid-scale (SGS) closures for stable
and accurate large-eddy simulations (LES) in the small-data regime (i.e., when
the availability of high-quality training data is limited). Using several
setups of forced 2D turbulence as the testbeds, we examine the {\it a priori}
and {\it a posteriori} performance of three methods for incorporating physics:
1) data augmentation (DA), 2) CNN with group convolutions (GCNN), and 3) loss
functions that enforce a global enstrophy-transfer conservation (EnsCon). While
the data-driven closures from physics-agnostic CNNs trained in the big-data
regime are accurate and stable, and outperform dynamic Smagorinsky (DSMAG)
closures, their performance substantially deteriorate when these CNNs are
trained with 40x fewer samples (the small-data regime). We show that CNN with
DA and GCNN address this issue and each produce accurate and stable data-driven
closures in the small-data regime. Despite its simplicity, DA, which adds
appropriately rotated samples to the training set, performs as well or in some
cases even better than GCNN, which uses a sophisticated equivariance-preserving
architecture. EnsCon, which combines structural modeling with aspect of
functional modeling, also produces accurate and stable closures in the
small-data regime. Overall, GCNN+EnCon, which combines these two physics
constraints, shows the best {\it a posteriori} performance in this regime.
These results illustrate the power of physics-constrained learning in the
small-data regime for accurate and stable LES.
