Physics-constrained data-driven subgrid-scale parameterization of 2D turbulence in the small-data regime

In this work, we develop a data-driven subgrid-scale (SGS) model for large eddy simulation (LES) of 2D turbulence using a fully convolutional neural network (CNN). In the small-data regime, the LES-CNN generates artificial instabilities and thus leads to unphysical results. We propose four remedies for the CNN to work in the small-data regime: (1) data augmentation (DA), (2) group equivariant convolution neural network (GCNN), leveraging the rotational equivariance of the SGS term, (3) incorporating a physical constraint on the SGS enstrophy transfer, and (4) variational autoencoder providing stochasticity and uncertainty quantification. The rotational equivariance of SGS terms can be accounted for by either including rotated snapshots in the training data set (DA) 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. Stochasticity can be crucial in modeling backscattering (energy transferred from subgrid scales to resolved scales) of SGS terms. A priori and a posteriori analyses show that the proposed approaches enhance the SGS model and allow the data-driven model to work stably and accurately in a small-data regime.