Super-resolution of turbulence with a 4DVar training algorithm

Super-resolution of under-resolved turbulent flow fields can be performed after training a neural network on a dataset of fully resolved snapshots. On the other hand, classical state estimation algorithms like 4DVar make predictions of high-resolution initial conditions from coarse observations only, by attempting to reproduce the coarse-grained temporal evolution. We show that the 4DVar algorithm can be adapted to train highly-accurate neural networks for super resolution without access to a library of high resolution reference data (Page, J. Fluid Mech. 1002, 2025). The approach relies on a fully-differentiable solver to unroll the network’s prediction inside the loss function for comparison to a time series of coarse observations. The method is applied to both two- and three-dimensional flows with Kolmogorov-like forcing, and the coarse-only networks perform comparably to models trained on high resolution data. The impact of known limiting length/timescales from the data assimilation literature will be explored. A second problem, in which a similar training approach can be used to learn mappings between topologically equivalent dynamical systems will also be discussed if time permits.