Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Differentiable Turbulence: Closure as a partial differential equation constrained optimization (2307.03683v2)

Published 7 Jul 2023 in physics.flu-dyn and cs.LG

Abstract: Deep learning is increasingly becoming a promising pathway to improving the accuracy of sub-grid scale (SGS) turbulence closure models for large eddy simulations (LES). We leverage the concept of differentiable turbulence, whereby an end-to-end differentiable solver is used in combination with physics-inspired choices of deep learning architectures to learn highly effective and versatile SGS models for two-dimensional turbulent flow. We perform an in-depth analysis of the inductive biases in the chosen architectures, finding that the inclusion of small-scale non-local features is most critical to effective SGS modeling, while large-scale features can improve pointwise accuracy of the \textit{a-posteriori} solution field. The velocity gradient tensor on the LES grid can be mapped directly to the SGS stress via decomposition of the inputs and outputs into isotropic, deviatoric, and anti-symmetric components. We see that the model can generalize to a variety of flow configurations, including higher and lower Reynolds numbers and different forcing conditions. We show that the differentiable physics paradigm is more successful than offline, \textit{a-priori} learning, and that hybrid solver-in-the-loop approaches to deep learning offer an ideal balance between computational efficiency, accuracy, and generalization. Our experiments provide physics-based recommendations for deep-learning based SGS modeling for generalizable closure modeling of turbulence.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (62)
  1. S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).
  2. H. Tennekes and J. L. Lumley, A First Course in Turbulence, The MIT Press (MIT Press, London, England, 2018).
  3. P. Moin and K. Mahesh, “DIRECT NUMERICAL SIMULATION: A tool in turbulence research,” Annual Review of Fluid Mechanics 30, 539–578 (1998).
  4. C. Argyropoulos and N. Markatos, “Recent advances on the numerical modelling of turbulent flows,” Applied Mathematical Modelling 39, 693–732 (2015).
  5. T. Gatski and C. Rumsey, “Linear and nonlinear eddy viscosity models,” in Closure Strategies for Turbulent and Transitional Flows (Cambridge University Press, 2001) pp. 9–46.
  6. P. Sagaut, Large Eddy Simulation for Incompressible Flows, 3rd ed., Scientific Computation (Springer, New York, NY, 2006).
  7. S. B. Pope, “Ten questions concerning the large-eddy simulation of turbulent flows,” New Journal of Physics 6, 35–35 (2004).
  8. J. Schalkwijk, H. J. J. Jonker, A. P. Siebesma,  and E. V. Meijgaard, “Weather forecasting using GPU-based large-eddy simulations,” Bulletin of the American Meteorological Society 96, 715–723 (2015).
  9. Z. Shen, A. Sridhar, Z. Tan, A. Jaruga,  and T. Schneider, “A library of large-eddy simulations forced by global climate models,” Journal of Advances in Modeling Earth Systems 14 (2022), 10.1029/2021ms002631.
  10. P. R. Spalart, “Philosophies and fallacies in turbulence modeling,” Progress in Aerospace Sciences 74, 1–15 (2015).
  11. B. J. Geurts and J. Fröhlich, “A framework for predicting accuracy limitations in large-eddy simulation,” Physics of Fluids 14, L41–L44 (2002).
  12. W. C. Reynolds, “The potential and limitations of direct and large eddy simulations,” in Whither Turbulence? Turbulence at the Crossroads (Springer Berlin Heidelberg) pp. 313–343.
  13. K. Duraisamy, G. Iaccarino,  and H. Xiao, “Turbulence modeling in the age of data,” Annual Review of Fluid Mechanics 51, 357–377 (2019).
  14. A. Beck and M. Kurz, “A perspective on machine learning methods in turbulence modeling,” GAMM-Mitteilungen 44 (2021), 10.1002/gamm.202100002.
  15. J. N. Kutz, “Deep learning in fluid dynamics,” Journal of Fluid Mechanics 814, 1–4 (2017).
  16. J. Ling, A. Kurzawski,  and J. Templeton, “Reynolds averaged turbulence modelling using deep neural networks with embedded invariance,” Journal of Fluid Mechanics 807, 155–166 (2016).
  17. W. Liu, J. Fang, S. Rolfo, C. Moulinec,  and D. R. Emerson, “An iterative machine-learning framework for RANS turbulence modeling,” International Journal of Heat and Fluid Flow 90, 108822 (2021).
  18. A. P. Singh, K. Duraisamy,  and Z. J. Zhang, “Augmentation of turbulence models using field inversion and machine learning,” in 55th AIAA Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, 2017).
  19. Y. Bin, L. Chen, G. Huang,  and X. I. A. Yang, “Progressive, extrapolative machine learning for near-wall turbulence modeling,” Physical Review Fluids 7 (2022), 10.1103/physrevfluids.7.084610.
  20. R. Maulik, O. San, A. Rasheed,  and P. Vedula, “Subgrid modelling for two-dimensional turbulence using neural networks,” Journal of Fluid Mechanics 858, 122–144 (2018).
  21. Z. Wang, K. Luo, D. Li, J. Tan,  and J. Fan, “Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation,” Physics of Fluids 30, 125101 (2018).
  22. H. Frezat, G. Balarac, J. L. Sommer, R. Fablet,  and R. Lguensat, “Physical invariance in neural networks for subgrid-scale scalar flux modeling,” Physical Review Fluids 6 (2021), 10.1103/physrevfluids.6.024607.
  23. Y. Guan, A. Chattopadhyay, A. Subel,  and P. Hassanzadeh, “Stable a posteriori LES of 2d turbulence using convolutional neural networks: Backscattering analysis and generalization to higher re via transfer learning,” Journal of Computational Physics 458, 111090 (2022).
  24. Y. Guan, A. Subel, A. Chattopadhyay,  and P. Hassanzadeh, “Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES,” Physica D: Nonlinear Phenomena 443, 133568 (2023).
  25. A. Subel, Y. Guan, A. Chattopadhyay,  and P. Hassanzadeh, “Explaining the physics of transfer learning in data-driven turbulence modeling,” PNAS Nexus 2 (2023), 10.1093/pnasnexus/pgad015.
  26. G. Novati, H. L. de Laroussilhe,  and P. Koumoutsakos, “Automating turbulence modelling by multi-agent reinforcement learning,” Nature Machine Intelligence 3, 87–96 (2021).
  27. Y. Zhao, H. D. Akolekar, J. Weatheritt, V. Michelassi,  and R. D. Sandberg, “RANS turbulence model development using CFD-driven machine learning,” Journal of Computational Physics 411, 109413 (2020).
  28. O. Obiols-Sales, A. Vishnu, N. Malaya,  and A. Chandramowliswharan, “CFDNet,” in Proceedings of the 34th ACM International Conference on Supercomputing (ACM, 2020).
  29. S. Pandey, J. Schumacher,  and K. R. Sreenivasan, “A perspective on machine learning in turbulent flows,” Journal of Turbulence 21, 567–584 (2020).
  30. R. Vinuesa and S. L. Brunton, “Enhancing computational fluid dynamics with machine learning,” Nature Computational Science 2, 358–366 (2022).
  31. K. Stachenfeld, D. B. Fielding, D. Kochkov, M. Cranmer, T. Pfaff, J. Godwin, C. Cui, S. Ho, P. Battaglia,  and A. Sanchez-Gonzalez, “Learned coarse models for efficient turbulence simulation,”  (2022), arXiv:2112.15275 [physics.flu-dyn] .
  32. A. T. Mohan and D. V. Gaitonde, “A deep learning based approach to reduced order modeling for turbulent flow control using lstm neural networks,”  (2018), arXiv:1804.09269 [physics.comp-ph] .
  33. R. Wang, K. Kashinath, M. Mustafa, A. Albert,  and R. Yu, “Towards physics-informed deep learning for turbulent flow prediction,” in Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (ACM, 2020).
  34. D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner,  and S. Hoyer, “Machine learning–accelerated computational fluid dynamics,” Proceedings of the National Academy of Sciences 118 (2021), 10.1073/pnas.2101784118.
  35. P. Holl, N. Thuerey,  and V. Koltun, “Learning to control pdes with differentiable physics,” in International Conference on Learning Representations (2020).
  36. D. A. Bezgin, A. B. Buhendwa,  and N. A. Adams, “Jax-fluids: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows,” Computer Physics Communications , 108527 (2022).
  37. A. Jameson, “Optimum aerodynamic design using CFD and control theory,” in 12th Computational Fluid Dynamics Conference (American Institute of Aeronautics and Astronautics, 1995).
  38. J.-D. Müller and P. Cusdin, “On the performance of discrete adjoint CFD codes using automatic differentiation,” International Journal for Numerical Methods in Fluids 47, 939–945 (2005).
  39. G. K. Kenway, C. A. Mader, P. He,  and J. R. Martins, “Effective adjoint approaches for computational fluid dynamics,” Progress in Aerospace Sciences 110, 100542 (2019).
  40. O. Tonomura, M. Kano,  and S. Hasebe, “Shape optimization of microchannels using CFD and adjoint method,” in Computer Aided Chemical Engineering (Elsevier, 2010) pp. 37–42.
  41. W. Liu, R. Duan, C. Chen, C.-H. Lin,  and Q. Chen, “Inverse design of the thermal environment in an airliner cabin by use of the CFD-based adjoint method,” Energy and Buildings 104, 147–155 (2015).
  42. M. P. Rumpfkeil and D. W. Zingg, “The optimal control of unsteady flows with a discrete adjoint method,” Optimization and Engineering 11, 5–22 (2008).
  43. M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbulence and Combustion 65, 393–415 (2000).
  44. K. Um, R. Brand, Yun, Fei, P. Holl,  and N. Thuerey, “Solver-in-the-loop: Learning from differentiable physics to interact with iterative pde-solvers,”  (2021), arXiv:2007.00016 [physics.comp-ph] .
  45. B. List, L.-W. Chen,  and N. Thuerey, “Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons,” Journal of Fluid Mechanics 949 (2022), 10.1017/jfm.2022.738.
  46. J. Sirignano, J. F. MacArt,  and J. B. Freund, “DPM: A deep learning PDE augmentation method with application to large-eddy simulation,” Journal of Computational Physics 423, 109811 (2020).
  47. H. Frezat, J. L. Sommer, R. Fablet, G. Balarac,  and R. Lguensat, “A posteriori learning for quasi-geostrophic turbulence parametrization,” Journal of Advances in Modeling Earth Systems 14 (2022), 10.1029/2022ms003124.
  48. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne,  and Q. Zhang, “JAX: composable transformations of Python+NumPy programs,”  (2018).
  49. C. A. Mader, J. R. R. A. Martins, J. J. Alonso,  and E. van der Weide, “ADjoint: An approach for the rapid development of discrete adjoint solvers,” AIAA Journal 46, 863–873 (2008).
  50. Z. Zhou, G. He, S. Wang,  and G. Jin, “Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network,” Computers & Fluids 195, 104319 (2019).
  51. J. SMAGORINSKY, “GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS,” Monthly Weather Review 91, 99–164 (1963).
  52. Y. Zang, R. L. Street,  and J. R. Koseff, “A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows,” Physics of Fluids A: Fluid Dynamics 5, 3186–3196 (1993).
  53. D. K. Lilly, “A proposed modification of the germano subgrid-scale closure method,” Physics of Fluids A: Fluid Dynamics 4, 633–635 (1992).
  54. N. Adams and S. Stolz, “A subgrid-scale deconvolution approach for shock capturing,” Journal of Computational Physics 178, 391–426 (2002).
  55. P. M. Milani, J. Ling,  and J. K. Eaton, “On the generality of tensor basis neural networks for turbulent scalar flux modeling,” International Communications in Heat and Mass Transfer 128, 105626 (2021).
  56. S. Berrone and D. Oberto, “An invariances-preserving vector basis neural network for the closure of reynolds-averaged navier–stokes equations by the divergence of the reynolds stress tensor,” Physics of Fluids 34, 095136 (2022).
  57. S. B. Pope, “A more general effective-viscosity hypothesis,” Journal of Fluid Mechanics 72, 331 (1975).
  58. R. Stoffer, C. M. van Leeuwen, D. Podareanu, V. Codreanu, M. A. Veerman, M. Janssens, O. K. Hartogensis,  and C. C. van Heerwaarden, “Development of a large-eddy simulation subgrid model based on artificial neural networks: a case study of turbulent channel flow,” Geoscientific Model Development 14, 3769–3788 (2021).
  59. P. C. D. Leoni, T. A. Zaki, G. Karniadakis,  and C. Meneveau, “Two-point stress–strain-rate correlation structure and non-local eddy viscosity in turbulent flows,” Journal of Fluid Mechanics 914 (2021), 10.1017/jfm.2020.977.
  60. Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart,  and A. Anandkumar, “Fourier neural operator for parametric partial differential equations,”  (2021), arXiv:2010.08895 [cs.LG] .
  61. D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”  (2017), arXiv:1412.6980 [cs.LG] .
  62. E. D. Fylladitakis, “Kolmogorov flow: Seven decades of history,” Journal of Applied Mathematics and Physics 06, 2227–2263 (2018).
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now