Super-resolution of turbulence with dynamics in the loss

Super-resolution of turbulence is a term used to describe the prediction of high-resolution snapshots of a flow from coarse-grained observations. This is typically accomplished with a deep neural network and training usually requires a dataset of high-resolution images. An approach is presented here in which robust super-resolution can be performed without access to high-resolution reference data, as might be expected in an experiment. The training procedure is similar to data assimilation, wherein the model learns to predict an initial condition that leads to accurate coarse-grained predictions at later times, while only being shown coarse-grained observations. Implementation of the approach requires the use of a fully differentiable flow solver in the training loop to allow for time-marching of predictions. A range of models are trained on data generated from forced, two-dimensional turbulence. The networks have reconstruction errors which are similar to those obtained with ‘standard’ super-resolution approaches using high-resolution data. Furthermore, the methods are comparable to the performance of standard data assimilation for state estimation on individual trajectories, outperforming these variational approaches at initial time and remaining robust when unrolled in time where performance of the standard data-assimilation algorithm improves.