  In this paper, we train turbulence models based on convolutional neural
networks. These learned turbulence models improve under-resolved low resolution
solutions to the incompressible Navier-Stokes equations at simulation time. Our
study involves the development of a differentiable numerical solver that
supports the propagation of optimisation gradients through multiple solver
steps. The significance of this property is demonstrated by the superior
stability and accuracy of those models that unroll more solver steps during
training. Furthermore, we introduce loss terms based on turbulence physics that
further improve the model accuracy. This approach is applied to three
two-dimensional turbulence flow scenarios, a homogeneous decaying turbulence
case, a temporally evolving mixing layer, and a spatially evolving mixing
layer. Our models achieve significant improvements of long-term a-posteriori
statistics when compared to no-model simulations, without requiring these
statistics to be directly included in the learning targets. At inference time,
our proposed method also gains substantial performance improvements over
similarly accurate, purely numerical methods.

Fluid Dynamics;Machine Learning;Computational Physics