Embedded training of neural-network subgrid-scale turbulence models

The weights of a deep neural-network model are optimized over the governing flow equations to provide a model for the subgrid-scale stresses in a turbulent plane jet. The training, which is done in $a{{0}{0ex}}p{{0}{0ex}}o{{0}{0ex}}s{{0}{0ex}}t{{0}{0ex}}e{{0}{0ex}}r{{0}{0ex}}i{{0}{0ex}}o{{0}{0ex}}r{{0}{0ex}}i$ large-eddy simulations (LES), solves the adjoint Navier-Stokes equations to provide end-to-end sensitivities. Out-of-sample testing on multiple dual-jet configurations confirms that the required grid resolution is half that needed by dynamic models for comparable accuracy. The coupled formulation is generalized to train based only on the mean flow and Reynolds stresses, which are more readily available from experiments.