Turbulence closure modeling with data-driven techniques: Investigation of generalizable deep neural networks
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Generalizability of machine-learning (ML) based turbulence closures to accurately predict unseen practical flows remains an important challenge. It is well recognized that the neural network (NN) architecture and training protocol profoundly influence the generalizability characteristics. At the Reynolds-averaged NavierStokes level, NNbased turbulence closure modeling is rendered difficult due to two important reasons: inherent complexity of the constitutive relation arising from flow-dependent non-linearity and bifurcations; and, inordinate difficulty in obtaining high-fidelity data covering the entire parameter space of interest. Thus, a predictive turbulence model must be robust enough to perform reasonably outside the domain of training. In this context, the objective of the work is to investigate the approximation capabilities of standard moderate-sized fully connected NNs. We seek to systematically investigate the effects of (i) intrinsic complexity of the solution manifold; (ii) sampling procedure (interpolation vs extrapolation); and (iii) optimization procedure. To overcome the data acquisition challenges, three proxy-physics turbulence surrogates of different degrees of complexity (yet significantly simpler than turbulence physics) are employed to generate the parameter-to-solution maps. Lacking a strong theoretical basis for finding the globally optimal NN architecture and hyperparameters in the presence of non-linearity and bifurcations, a brute-force parameter-space sweep is performed to determine a locally optimal solution. Even for this simple proxy-physics system, it is demonstrated that feed-forward NNs require more degrees of freedom than the original proxy-physics model to accurately approximate the true model even when trained with data over the entire parameter space (interpolation). Additionally, if deep fully connected NNs are trained with data only from part of the parameter space (extrapolation), their approximation capability reduces considerably and it is not straightforward to find an optimal architecture. Overall, the findings provide a realistic perspective on the utility of ML turbulence closures for practical applications and identify areas for improvement.
