# Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons

arXiv: https://arxiv.org/abs/2202.06988 · HTML: https://mlsanity.com/paper/2202.06988/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 14 Feb 2022.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [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)
- [Towards high-accuracy deep learning inference of compressible turbulent flows over aerofoils](https://mlsanity.com/paper/2109.02183/) (2021)

## Built on by (most cited first, 30 of 39)

- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Scaling physics-informed hard constraints with mixture-of-experts](https://mlsanity.com/paper/2402.13412/) (2024)
- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [Solving Inverse Physics Problems with Score Matching](https://mlsanity.com/paper/2301.10250/) (2023)
- [Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.01174/) (2023)
- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [XLB: A differentiable massively parallel lattice Boltzmann library in Python](https://mlsanity.com/paper/2311.16080/) (2023)
- [NeuralFluid: Neural Fluidic System Design and Control with Differentiable Simulation](https://mlsanity.com/paper/2405.14903/) (2024)
- [A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty](https://mlsanity.com/paper/2307.02432/) (2023)
- [PICT -- A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamics](https://mlsanity.com/paper/2505.16992/) (2025)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [Gradient-free online learning of subgrid-scale dynamics with neural emulators](https://mlsanity.com/paper/2310.19385/) (2023)
- [DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modeling](https://mlsanity.com/paper/2401.00161/) (2023)
- [AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios](https://mlsanity.com/paper/2302.10184/) (2023)
- [Probabilistic Physics-integrated Neural Differentiable Modeling for Isothermal Chemical Vapor Infiltration Process](https://mlsanity.com/paper/2311.07798/) (2023)
- [Stabilizing Backpropagation Through Time to Learn Complex Physics](https://mlsanity.com/paper/2405.02041/) (2024)
- [INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers](https://mlsanity.com/paper/2511.12764/) (2025)
- [Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics](https://mlsanity.com/paper/2604.23874/) (2026)
- [A note on the error analysis of data-driven closure models for large eddy simulations of turbulence](https://mlsanity.com/paper/2405.17612/) (2024)
- [Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systems](https://mlsanity.com/paper/2504.05588/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Differentiable Turbulence: Closure as a partial differential equation constrained optimization](https://mlsanity.com/paper/2307.03683/) (2023)
- [Accurate deep learning sub-grid scale models for large eddy simulations](https://mlsanity.com/paper/2307.10060/) (2023)
- [Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence](https://mlsanity.com/paper/2411.14106/) (2024)
- [Implicit Neural Differential Model for Spatiotemporal Dynamics](https://mlsanity.com/paper/2504.02260/) (2025)
- [Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators](https://mlsanity.com/paper/2506.13120/) (2025)
- [Generalized Neural Operator for Parametric and Boundary-Value Problems](https://mlsanity.com/paper/2607.21932/) (2026)

## Abstract (arXiv)

In this paper, we train turbulence models based on convolutional neural networks. These learned turbulence models improve under-resolved low resolution solutions to the incompressible Navier-Stokes equations at simulation time. Our study involves the development of a differentiable numerical solver that supports the propagation of optimisation gradients through multiple solver steps. The significance of this property is demonstrated by the superior stability and accuracy of those models that unroll more solver steps during training. Furthermore, we introduce loss terms based on turbulence physics that further improve the model accuracy. This approach is applied to three two-dimensional turbulence flow scenarios, a homogeneous decaying turbulence case, a temporally evolving mixing layer, and a spatially evolving mixing layer. Our models achieve significant improvements of long-term a-posteriori statistics when compared to no-model simulations, without requiring these statistics to be directly included in the learning targets. At inference time, our proposed method also gains substantial performance improvements over similarly accurate, purely numerical methods.