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In this paper, we train turbulence models based on convolutional neuralnetworks. These learned turbulence models improve under-resolved low resolutionsolutions to the incompressible Navier-Stokes equations at simulation time. Ourstudy involves the development of a differentiable numerical solver thatsupports the propagation of optimisation gradients through multiple solversteps. The significance of this property is demonstrated by the superiorstability and accuracy of those models that unroll more solver steps duringtraining. Furthermore, we introduce loss terms based on turbulence physics thatfurther improve the model accuracy. This approach is applied to threetwo-dimensional turbulence flow scenarios, a homogeneous decaying turbulencecase, a temporally evolving mixing layer, and a spatially evolving mixinglayer. Our models achieve significant improvements of long-term a-posterioristatistics when compared to no-model simulations, without requiring thesestatistics to be directly included in the learning targets. At inference time,our proposed method also gains substantial performance improvements oversimilarly accurate, purely numerical methods.

Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons