Equivariant Machine Learning of Sub-Grid Scale Closure Models for Large Eddy Simulation

Several data-driven techniques have been recently proposed for modelling the sub-grid scale stress (SGS) tensor in large eddy simulation (LES). In the broader turbulence modelling context, the Reynolds-averaged Navier-Stokes (RANS) community has adopted equivariance as a requirement when predicting the Reynolds stress tensor. Equivariance means that a model's output stress tensor transforms along with its input tensors under symmetry transformations of the Navier-Stokes equations. While some machine learning-based SGS models for LES enforce equivariance via learning invariant scalars or canonicalizing (e.g., training in the eigenframe of the strain-rate tensor), many studies do not consider equivariance in the model design or training procedure. In this study, we explore the question of whether equivariance is a necessary property for an SGS closure model to be accurate and generalizable. Further, we compare the accuracy, generalizability, and training difficulty of various techniques for achieving equivariant SGS closure mappings, such as including equivariance as an architectural inductive bias, learning bias, or using data augmentation in the training loop. Our results will inform the design of future machine learning-based LES closure models and their training procedures.