mlsanityData to 14 Oct 2026

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

arXiv 2202.06988 ↗First posted 14 Feb 2022physics.flu-dyn cs.LG physics.comp-phcited by 39

Not on the map. Why?

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.

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

17 of 58 references.

References · 17

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2020Learning to Simulate Complex Physics with Graph Networkscited by 880
  4. 2017Understanding the Effective Receptive Field in Deep Convolutional Neural Networkscited by 553
  5. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
  6. 2021Machine learning accelerated computational fluid dynamicscited by 471
  7. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  8. 2018Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flowscited by 150
  9. 2019Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networkscited by 145
  10. 2020Learning to Control PDEs with Differentiable Physicscited by 132
  11. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 91
  12. 2021Physics-based Deep Learningcited by 68
5 more
  1. 2020Convolutional-network models to predict wall-bounded turbulence from wall quantitiescited by 47
  2. 2019DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)cited by 41
  3. 2018Deep Neural Networks for Data-Driven Turbulence Modelscited by 35
  4. 2021Embedded training of neural-network sub-grid-scale turbulence modelscited by 13
  5. 2021Towards high-accuracy deep learning inference of compressible turbulent flows over aerofoilscited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 13; 2024: 7; 2025: 8; 2026: 10.

39 papers.

Sort:

By area:

  1. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 91
  2. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
  3. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  4. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
  5. 2025Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulationcited by 19
  6. 2023Solving Inverse Physics Problems with Score Matchingcited by 16
  7. 2023Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equationscited by 13
  8. 2023A Multifidelity deep operator network approach to closure for multiscale systemscited by 11
  9. 2023XLB: A differentiable massively parallel lattice Boltzmann library in Pythoncited by 11
  10. 2024NeuralFluid: Neural Fluidic System Design and Control with Differentiable Simulationcited by 10
  11. 2023A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertaintycited by 7
  12. 2025PICT -- A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamicscited by 6
27 more
  1. 2025EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scalecited by 6
  2. 2023Gradient-free online learning of subgrid-scale dynamics with neural emulatorscited by 5
  3. 2023DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modelingcited by 5
  4. 2023AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarioscited by 4
  5. 2023Probabilistic Physics-integrated Neural Differentiable Modeling for Isothermal Chemical Vapor Infiltration Processcited by 3
  6. 2024Stabilizing Backpropagation Through Time to Learn Complex Physicscited by 3
  7. 2025INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solverscited by 3
  8. 2026Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamicscited by 3
  9. 2024A note on the error analysis of data-driven closure models for large eddy simulations of turbulencecited by 2
  10. 2025Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systemscited by 2
  11. 2025Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modelingcited by 2
  12. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1
  13. 2023Differentiable Turbulence: Closure as a partial differential equation constrained optimizationcited by 1
  14. 2023Accurate deep learning sub-grid scale models for large eddy simulationscited by 1
  15. 2024Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulencecited by 1
  16. 2025Implicit Neural Differential Model for Spatiotemporal Dynamicscited by 1
  17. 2025Accelerating PDE-Constrained Optimization by the Derivative of Neural Operatorscited by 1
  18. 2026Generalized Neural Operator for Parametric and Boundary-Value Problemscited by 1
  19. 2024A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equationscited by 0
  20. 2026Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulationscited by 0
  21. 2026Bias-Constrained Diffusion Schedules for PDE Emulations: Reconstruction Error Minimization and Efficient Unrolled Trainingcited by 0
  22. 2026A neural operator framework for data-driven discovery of stability and receptivity in physical systemscited by 0
  23. 2026Deep Learning-based Algebraic Reynolds Stress Closures for RANS Simulations of Turbulent Flowscited by 0
  24. 2026A foundation model of numerical intelligence with cross-disciplinary generalizationcited by 0
  25. 2026Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulationcited by 0
  26. 2026Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEscited by 0
  27. 2026Improving Reduced-Order Rotating Detonation Engine Models with Data Assimilation and Machine Learningcited by 0