We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.

翻译：我们提出了一种数据驱动的闭合模型，用于雷诺平均Navier-Stokes（RANS）模拟，该模型融入了偶然性模型不确定性。所提出的闭合由两部分组成：参数化部分利用先前提出的基于神经网络的张量基函数，该函数依赖于应变率和旋转张量不变量；而潜在随机变量则补充刻画了偶然性模型误差。我们采用全贝叶斯框架，并结合稀疏性诱导先验，以识别问题域中参数化闭合不足的区域，以及需要对雷诺应力张量进行随机修正的区域。与多数需要直接雷诺应力数据的替代方法不同，本训练基于稀疏间接数据（如平均速度和压力）。在推断与学习过程中，我们采用随机变分推断方案，该方案基于相关目标函数的蒙特卡洛估计并结合重参数化技巧。这需要RANS求解器输出的导数，为此我们开发了伴随公式。通过这种方式，可微求解器的参数敏感性可与神经网络库内置的自动微分能力相结合，从而实现端到端可微框架。我们在后向台阶基准问题的分离流中验证了该模型的能力：即使在存在模型误差的区域，它也能对所有流动量产生准确且具有概率意义的预测估计。