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We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes(RANS) simulations that incorporates aleatoric, model uncertainty. The proposedclosure consists of two parts. A parametric one, which utilizes previouslyproposed, neural-network-based tensor basis functions dependent on the rate ofstrain and rotation tensor invariants. This is complemented by latent, randomvariables which account for aleatoric model errors. A fully Bayesianformulation is proposed, combined with a sparsity-inducing prior in order toidentify regions in the problem domain where the parametric closure isinsufficient and where stochastic corrections to the Reynolds stress tensor areneeded. Training is performed using sparse, indirect data, such as meanvelocities and pressures, in contrast to the majority of alternatives thatrequire direct Reynolds stress data. For inference and learning, a StochasticVariational Inference scheme is employed, which is based on Monte Carloestimates of the pertinent objective in conjunction with the reparametrizationtrick. This necessitates derivatives of the output of the RANS solver, forwhich we developed an adjoint-based formulation. In this manner, the parametricsensitivities from the differentiable solver can be combined with the built-in,automatic differentiation capability of the neural network library in order toenable an end-to-end differentiable framework. We demonstrate the capability ofthe proposed model to produce accurate, probabilistic, predictive estimates forall flow quantities, even in regions where model errors are present, on aseparated flow in the backward-facing step benchmark problem.

A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty