# Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies

arXiv: https://arxiv.org/abs/2208.03498 · HTML: https://mlsanity.com/paper/2208.03498/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 6 Aug 2022.

## Builds on (most cited first, 7 of 7)

- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Deep Neural Networks for Data-Driven Turbulence Models](https://mlsanity.com/paper/1806.04482/) (2018)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence](https://mlsanity.com/paper/2105.08633/) (2021)
- [Comparative analysis of machine learning methods for active flow control](https://mlsanity.com/paper/2202.11664/) (2022)

## Built on by (most cited first, 11 of 11)

- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Walrus: A Cross-Domain Foundation Model for Continuum Dynamics](https://mlsanity.com/paper/2511.15684/) (2025)
- [PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations](https://mlsanity.com/paper/2505.24717/) (2025)
- [Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics](https://mlsanity.com/paper/2604.23874/) (2026)
- [oRANS: Online optimisation of RANS machine learning models with embedded DNS data generation](https://mlsanity.com/paper/2510.02982/) (2025)
- [Neural network-based Godunov corrections for approximate Riemann solvers using bi-fidelity learning](https://mlsanity.com/paper/2503.13248/) (2025)
- [OGF: An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows](https://mlsanity.com/paper/2507.05149/) (2025)
- [Physics-Based Machine Learning Closures and Wall Models for Hypersonic Transition-Continuum Boundary Layer Predictions](https://mlsanity.com/paper/2507.08986/) (2025)
- [Active Control of Turbulent Airfoil Flows Using Adjoint-based Deep Learning](https://mlsanity.com/paper/2510.07106/) (2025)
- [Addressing A Posteriori Performance Degradation in Neural Network Subgrid Stress Models](https://mlsanity.com/paper/2511.17475/) (2025)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)

## Abstract (arXiv)

A deep learning (DL) closure model for large-eddy simulation (LES) is developed and evaluated for incompressible flows around a rectangular cylinder at moderate Reynolds numbers. Near-wall flow simulation remains a central challenge in aerodynamic modeling: RANS predictions of separated flows are often inaccurate, while LES can require prohibitively small near-wall mesh sizes. The DL-LES model is trained using adjoint PDE optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. It is then evaluated out-of-sample (i.e., for new aspect ratios and Reynolds numbers not included in the training data) and compared against a standard LES model (the dynamic Smagorinsky model). The DL-LES model outperforms dynamic Smagorinsky and is able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS grid by a factor of four in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, mean flow, and Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, the time-averaged mean velocity baru(x) = displaystyle limt rightarrow ∞ frac1t int0t u(s,x) ds. Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain; it is a non-trivial question whether an unsteady partial differential equation model whose functional form is defined by a deep neural network can remain stable and accurate on t ∈ [0, ∞). Our results demonstrate that the DL-LES model is accurate and stable over large physical time spans, enabling the estimation of the steady-state statistics for the velocity, fluctuations, and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamic applications.