A coupled framework for symbolic turbulence models from deep-learning

Improvements in turbulence modelling in the recent years has seen an increasing prominence of various machine-learning algorithms. In this work, two different algorithms: tensor basis neural networks (TBNNs) and gene-expression programming (GEP), are combined to extract interpretable Reynolds stress closures. Representations of the high-fidelity Reynolds stress, obtained from deep-learning, are used to learn symbolic expressions for the stress closures. This is in contrast to the previously developed approaches of using either neural networks or symbolic regression algorithms to close Reynolds stresses from high-fidelity data. The involvement of TBNNs as an intermediate step in developing symbolic expressions stems from their ability to maintain the complex relationships between input features and output fields resulting from multiple datasets of different types of flows, which symbolic regression algorithms currently struggle with. The a priori and a posteriori results show that the closures developed by informing GEP with TBNN predictions produce results with similar accuracy to closures developed with GEP purely from high-fidelity data. Additionally, if the high-fidelity database contains multiple flows with uniquely different features, GEP closures developed from TBNN predictions yield better prediction accuracy than if the high-fidelity database (comprising of the multiple flows) was used to train GEP closures. On the point of interpretability, the closures were also examined for the purposes of complexity reduction as well as understanding which terms were responsible for the improvements seen. These results indicate that there is promise in building complex neural networks comprising a catalogue of wide-ranging flow types, which can then be used to extract symbolic closures on a case-by-case basis, thereby reducing the dependency on high-fidelity data for closure modelling.

Automated summary

Recent improvements in turbulence modeling have led to the increasing use of machine-learning algorithms. In this study, tensor basis neural networks (TBNNs) and gene-expression programming (GEP) algorithms are combined to develop interpretable Reynolds stress closures. The high-fidelity Reynolds stress representations obtained from deep-learning are used to learn symbolic expressions for the closures. The results show that using TBNN predictions to inform GEP produces closures with similar accuracy to those developed solely from high-fidelity data, and better prediction accuracy can be achieved when using TBNN predictions for GEP closures compared to using a high-fidelity database comprising multiple flows.