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

arXiv: https://arxiv.org/abs/2307.02432 · HTML: https://mlsanity.com/paper/2307.02432/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 102 references that are in the map.
- First posted on arXiv 5 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils](https://mlsanity.com/paper/1608.03990/) (2016)
- [Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks](https://mlsanity.com/paper/1807.02901/) (2018)
- [Emerging trends in machine learning for computational fluid dynamics](https://mlsanity.com/paper/2211.15145/) (2022)
- [Physics-constrained, data-driven discovery of coarse-grained dynamics](https://mlsanity.com/paper/1802.03824/) (2018)
- [A fully-differentiable compressible high-order computational fluid dynamics solver](https://mlsanity.com/paper/2112.04979/) (2021)

## Built on by (most cited first, 7 of 7)

- [Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2312.05320/) (2023)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Multi-fidelity Constrained Optimization for Stochastic Black Box Simulators](https://mlsanity.com/paper/2311.15137/) (2023)
- [From concrete mixture to structural design -- a holistic optimization procedure in the presence of uncertainties](https://mlsanity.com/paper/2312.03607/) (2023)
- [Simulating, Fast and Slow: Learning Policies for Black-Box Optimization](https://mlsanity.com/paper/2406.04261/) (2024)
- [The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems](https://mlsanity.com/paper/2502.07415/) (2025)
- [Deep Learning-based Algebraic Reynolds Stress Closures for RANS Simulations of Turbulent Flows](https://mlsanity.com/paper/2605.26358/) (2026)

## Abstract (arXiv)

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.