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：我们提出了一种用于雷诺平均纳维-斯托克斯（RANS）模拟的数据驱动封闭模型，该模型融入了偶然模型不确定性。所提出的封闭模型由两部分组成：一部分为参数化模型，利用先前提出的基于神经网络、依赖应变率和旋转张量不变量的张量基函数；另一部分由潜在随机变量补充，用于解释偶然模型误差。我们提出了全贝叶斯公式，并结合稀疏诱导先验，以识别问题域中参数化封闭模型不足且需要对雷诺应力张量进行随机修正的区域。训练采用稀疏间接数据（如平均速度和压力），与多数需要直接雷诺应力数据的替代方法形成对比。在推理和学习中，采用基于相关目标的蒙特卡洛估计与重参数化技巧的随机变分推理方案。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，来自可微求解器的参数灵敏度可与神经网络库内置的自动微分能力相结合，从而构建端到端可微框架。我们在后向台阶基准问题的分离流上展示了所提模型的能力：即便在存在模型误差的区域，它也能对所有流动量产生准确、概率性的预测估计。