An equivariant autoencoder for turbulent flows

The Navier-Stokes equation is equivariant under Euclidean symmetry which is broken by both boundary conditions and external body forces. Forcing does not break any symmetry at all if it is promoted to an independent field that transforms simultaneously with the flow. I use this property to construct an autoencoder for turbulent 2D flows on a square domain. This architecture is equivariant with respect to continuous translations and the discrete symmetries of a square and exactly accounts for all possible symmetries of body forcing. This group equivariance ensures zero-shot generalization to all symmetry-related data and removes any need to augment data during training. I use an example of 2D Kolmogorov flow to show that shallow equivariant networks can successfully map the vorticity field to a latent space of very low dimension.