End-to-end differentiable learning of turbulence models from indirect observations

Ensemble Kalman method for learning turbulence models from indirect observation data

“…Such an algorithmic modification is crucial for accelerating convergence and improving robustness of the learning, which can make an otherwise intractable learning problem with the adjoint method (Michelén Ströfer & Xiao 2021) computationally feasible with the ensemble method. We show that, by incorporating Hessian information with adaptive stepping, the ensemble Kalman method exceeds the performance of the adjoint-based learning (Michelén Ströfer & Xiao 2021) in both accuracy and robustness. Specifically, the present method successfully learned a generalizable nonlinear eddy viscosity model for the separated flows over periodic hills (Section 4), which the adjoint method was not able to achieve due to the lack of robustness.…”

“…This is in stark contrast to the traditional method of training neural networks that learns from direct data (output of the neural network, i.e., Reynolds stresses in this case), where the gradients can be directly obtained from back-propagation. In model-consistent training, one must typically resort to adjoint solvers to obtain the RANS solver-contributed gradient (sensitivity of velocity with respect to Reynolds stresses), as the full model consists of both the neural network and the RANS solver (Holland et al 2019;Michelén Ströfer & Xiao 2021). The adjoint sensitivity is then multiplied to the neural network gradient according to the chain rule to yield the full gradient.…”

“…In view of the difficulty in developing adjoint solvers (particularly for legacy codes and multi-physics coupled solvers), Michelén Ströfer et al (2021b) explored ensemble-based gradient approximation as an alternative to the adjoint solver used in Michelén Ströfer & Xiao (2021) to learn turbulence model from indirect data. Such a gradient is combined with that from the neural network via chain rule and then used in an explicit gradientdescent training.…”

“…Specifically, the present method successfully learned a generalizable nonlinear eddy viscosity model for the separated flows over periodic hills (Section 4), which the adjoint method was not able to achieve due to the lack of robustness. We emphasize that all these improvements are achieved at a much lower computational cost (measured in wall-time) and with a significantly lower implementation effort compared to the adjoint method (Michelén Ströfer & Xiao 2021). Both methods used the same representation of Reynolds stresses based on the tensor basis neural network (Ling et al 2016).…”

A PDE-free, neural network-based eddy viscosity model coupled with RANS equations

“…Studies have shown that the RANS equation can be very sensitive to the Reynolds stresses [27], which poses an additional challenge to the closure modelling. An alternative strategy for improving the robustness is to incorporate the RANS solver in the training process, but that would introduce major challenges in the training process by requiring adjoint solver [28] or ensemble simulations [29].…”