Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Perspective
  • Published:

Enhancing computational fluid dynamics with machine learning

Nature Computational Science volume 2, pages 358–366 (2022)Cite this article

Subjects

Abstract

Machine learning is rapidly becoming a core technology for scientific computing, with numerous opportunities to advance the field of computational fluid dynamics. Here we highlight some of the areas of highest potential impact, including to accelerate direct numerical simulations, to improve turbulence closure modeling and to develop enhanced reduced-order models. We also discuss emerging areas of machine learning that are promising for computational fluid dynamics, as well as some potential limitations that should be taken into account.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Summary of some of the most relevant areas where ML can enhance CFD.
Fig. 2: An example of ML-accelerated direct numerical simulation.
Fig. 3: An example of LES modeling where the dissipation coefficient in the Smagorinski model is calculated by means of ML.
Fig. 4: Schematic of NN autoencoders for dimensionality reduction and model identification.

Similar content being viewed by others

References

  1. Godunov, S. & Bohachevsky, I. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–306 (1959).

    MathSciNet  Google Scholar 

  2. Eymard, R., Gallouët, T. & Herbin, R. Finite volume methods. Handb. Numer. Anal. 7, 713–1018 (2000).

    MathSciNet  MATH  Google Scholar 

  3. Zienkiewicz, O. C., Taylor, R. L., Nithiarasu, P. & Zhu, J. Z. The Finite Element Method, 3 (Elsevier, 1977).

  4. Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. Spectral Methods in Fluid Dynamics (Springer Science & Business Media, 2012).

  5. Brunton, S. L. & Kutz, J. N. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems and Control (Cambridge Univ. Press, 2019).

  6. Recht, B. A tour of reinforcement learning: the view from continuous control. Annu. Rev. Control Robot. Auton. Syst. 2, 253–279 (2019).

    Article  Google Scholar 

  7. Vinuesa, R. et al. The role of artificial intelligence in achieving the sustainable development goals. Nat. Commun. 11, 233 (2020).

    Article  Google Scholar 

  8. Noé, F., Tkatchenko, A., Müller, K.-R. & Clementi, C. Machine learning for molecular simulation. Annu. Rev. Phys. Chem. 71, 361–390 (2020).

    Article  Google Scholar 

  9. Niederer, S. A., Sacks, M. S., Girolami, M. & Willcox, K. Scaling digital twins from the artisanal to the industrial. Nat. Comput. Sci. 1, 313–320 (2021).

    Article  Google Scholar 

  10. Samuel, A. L. Some studies in machine learning using the game of checkers. IBM J. Res. Dev. 3, 210–229 (1959).

    Article  MathSciNet  Google Scholar 

  11. Brenner, M., Eldredge, J. & Freund, J. Perspective on machine learning for advancing fluid mechanics. Phys. Rev. Fluids 4, 100501 (2019).

    Article  Google Scholar 

  12. Brunton, S. L., Noack, B. R. & Koumoutsakos, P. Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508 (2020).

    Article  MATH  Google Scholar 

  13. Duraisamy, K., Iaccarino, G. & Xiao, H. Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  14. Ahmed, S. E. et al. On closures for reduced order models—a spectrum of first-principle to machine-learned avenues. Phys. Fluids 33, 091301 (2021).

    Article  Google Scholar 

  15. Wang, B. & Wang, J. Application of artificial intelligence in computational fluid dynamics. Ind. Eng. Chem. Res. 60, 2772–2790 (2021).

    Article  Google Scholar 

  16. Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  17. Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).

    Article  Google Scholar 

  18. Noé, F., Olsson, S., Köhler, J. & Wu, H. Boltzmann generators: sampling equilibrium states of many-body systems with deep learning. Science 365, eaaw1147 (2019).

    Article  Google Scholar 

  19. Vinuesa, R., Hosseini, S. M., Hanifi, A., Henningson, D. S. & Schlatter, P. Pressure-gradient turbulent boundary layers developing around a wing section. Flow. Turbul. Combust. 99, 613–641 (2017).

    Article  Google Scholar 

  20. Choi, H. & Moin, P. Grid-point requirements for large eddy simulation: Chapman’s estimates revisited. Phys. Fluids 24, 011702 (2012).

    Article  Google Scholar 

  21. Bar-Sinai, Y., Hoyer, S., Hickey, J. & Brenner, M. P. Learning data-driven discretizations for partial differential equations. Proc. Natl Acad. Sci. USA 116, 15344–15349 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  22. Stevens, B. & Colonius, T. Enhancement of shock-capturing methods via machine learning. Theor. Comput. Fluid Dyn. 34, 483–496 (2020).

    Article  MathSciNet  Google Scholar 

  23. Jeon, J., Lee, J. & Kim, S. J. Finite volume method network for the acceleration of unsteady computational fluid dynamics: Non-reacting and reacting flows. Int. J. Energy Res. https://doi.org/10.1002/er.7879 (2022).

  24. Stevens, B. & Colonius, T. FiniteNet: a fully convolutional LSTM network architecture for time-dependent partial differential equations. Preprint at https://arxiv.org/abs/2002.03014 (2020).

  25. Kochkov, D. et al. Machine learning-accelerated computational fluid dynamics. Proc. Natl Acad. Sci. USA 118, e2101784118 (2021).

    Article  MathSciNet  Google Scholar 

  26. Chandler, G. J. & Kerswell, R. R. Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  27. Bauer, P., Thorpe, A. & Brunet, G. The quiet revolution of numerical weather prediction. Nature 525, 47–55 (2015).

    Article  Google Scholar 

  28. Schenk, F. et al. Warm summers during the Younger Dryas cold reversal. Nat. Commun. 9, 1634 (2018).

    Article  Google Scholar 

  29. Vinuesa, R. et al. Turbulent boundary layers around wing sections up to Rec = 1,000,000. Int. J. Heat. Fluid Flow. 72, 86–99 (2018).

    Article  MathSciNet  Google Scholar 

  30. Aloy Torás, C., Mimica, P. & Martinez Sober, M. in Artificial Intelligence Research and Development: Current Challenges, New Trends and Applications (eds Falomir, Z. et al.) 59–63 (IOS Press, 2018).

  31. Li, Z. et al. Fourier neural operator for parametric partial differential equations. Preprint at https://arxiv.org/abs/2010.08895 (2020).

  32. Li, Z. et al. Multipole graph neural operator for parametric partial differential equations. In Proc. 34th Int. Conf. on Neural Information Processing Systems 6755–6766 (NIPS, 2020).

  33. Li, Z. et al. Neural operator: graph kernel network for partial differential equations. Preprint at https://arxiv.org/abs/2003.03485 (2020).

  34. Shan, T. et al. Study on a Poisson’s equation solver based on deep learning technique. In Proc. 2017 IEEE Electrical Design of Advanced Packaging and Systems Symposium (EDAPS) 1–3 (IEEE, 2017).

  35. Zhang, Z. et al. Solving Poisson’s equation using deep learning in particle simulation of PN junction. In Proc. 2019 Joint International Symposium on Electromagnetic Compatibility, Sapporo and Asia-Pacific International Symposium on Electromagnetic Compatibility (EMC Sapporo/APEMC) 305–308 (IEEE, 2019).

  36. Bridson, R. Fluid Simulation (A. K. Peters, 2008).

  37. Ajuria, E. et al. Towards a hybrid computational strategy based on deep learning for incompressible flows. In Proc. AIAA AVIATION 2020 Forum 1–17 (AIAA, 2020).

  38. Özbay, A. et al. Poisson CNN: convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh. Data Centric Eng. 2, E6 (2021).

    Article  Google Scholar 

  39. Weymouth, G. D. Data-driven multi-grid solver for accelerated pressure projection. Preprint at https://arxiv.org/abs/2110.11029 (2021).

  40. Fukami, K., Nabae, Y., Kawai, K. & Fukagata, K. Synthetic turbulent inflow generator using machine learning. Phys. Rev. Fluids 4, 064603 (2019).

    Article  Google Scholar 

  41. Morita, Y. et al. Applying Bayesian optimization with Gaussian-process regression to computational fluid dynamics problems. J. Comput. Phys. 449, 110788 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  42. Boussinesq, J. V. Théorie Analytique de la Chaleur: Mise en Harmonie avec la Thermodynamique et avec la Théorie Mécanique de la Lumière T. 2, Refroidissement et Échauffement par Rayonnement Conductibilité des Tiges, Lames et Masses Cristallines Courants de Convection Théorie Mécanique de la Lumière (Gauthier-Villars, 1923).

  43. Slotnick, J. et al. CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences. Technical Report NASA/CR-2014-218178 (NASA, 2014).

  44. Kutz, J. N. Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4 (2017).

    Article  MATH  Google Scholar 

  45. Ling, J., Kurzawski, A. & Templeton, J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  46. Craft, T. J., Launder, B. E. & Suga, K. Development and application of a cubic eddy-viscosity model of turbulence. Int. J. Heat Fluid Flow 17, 108–115 (1996).

    Article  Google Scholar 

  47. Marin, O., Vinuesa, R., Obabko, A. V. & Schlatter, P. Characterization of the secondary flow in hexagonal ducts. Phys. Fluids 28, 125101 (2016).

    Article  Google Scholar 

  48. Spalart, P. R. Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 21, 252–263 (2000).

    Article  Google Scholar 

  49. Vidal, A., Nagib, H. M., Schlatter, P. & Vinuesa, R. Secondary flow in spanwise-periodic in-phase sinusoidal channels. J. Fluid Mech. 851, 288–316 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, J. X., Wu, J. L. & Xiao, H. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2, 034603 (2017).

    Article  Google Scholar 

  51. Wu, J.-L., Xiao, H. & Paterson, E. Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3, 074602 (2018).

    Article  Google Scholar 

  52. Jiang, C. et al. An interpretable framework of data-driven turbulence modeling using deep neural networks. Phys. Fluids 33, 055133 (2021).

    Article  Google Scholar 

  53. Rudin, C. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell. 1, 206–215 (2019).

    Article  Google Scholar 

  54. Vinuesa, R. & Sirmacek, B. Interpretable deep-learning models to help achieve the sustainable development goals. Nat. Mach. Intell. 3, 926 (2021).

    Article  Google Scholar 

  55. Cranmer, M. et al. Discovering symbolic models from deep learning with inductive biases. In Proc. 34th Int. Conf. on Neural Information Processing Systems 17429–17442 (NIPS, 2020)

  56. Weatheritt, J. & Sandberg, R. D. A novel evolutionary algorithm applied to algebraic modifications of the RANS stress-strain relationship. J. Comput. Phys. 325, 22–37 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  57. Koza, J. R. Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, 1992).

  58. Weatheritt, J. & Sandberg, R. D. The development of algebraic stress models using a novel evolutionary algorithm. Int. J. Heat Fluid Flow 68, 298–318 (2017).

    Article  Google Scholar 

  59. Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 3932–3937 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  60. Beetham, S. & Capecelatro, J. Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids 5, 084611 (2020).

    Article  Google Scholar 

  61. Schmelzer, M., Dwight, R. P. & Cinnella, P. Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow Turbul. Combust. 104, 579–603 (2020).

    Article  Google Scholar 

  62. Beetham, S., Fox, R. O. & Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid-particle flows. J. Fluid Mech. 914, A11 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  63. Rezaeiravesh, S., Vinuesa, R. & Schlatter, P. On numerical uncertainties in scale-resolving simulations of canonical wall turbulence. Comput. Fluids 227, 105024 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  64. Emory, M., Larsson, J. & Iaccarino, G. Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures. Phys. Fluids 25, 110822 (2013).

    Article  Google Scholar 

  65. Mishra, A. A. & Iaccarino, G. Uncertainty estimation for Reynolds-averaged Navier-Stokes predictions of high-speed aircraft nozzle jets. AIAA J. 55, 3999–4004 (2017).

    Article  Google Scholar 

  66. Poroseva, S., Colmenares, F. J. D. & Murman, S. On the accuracy of RANS simulations with DNS data. Phys. Fluids 28, 115102 (2016).

    Article  Google Scholar 

  67. Wu, J., Xiao, H., Sun, R. & Wang, Q. Reynolds-averaged Navier-Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned. J. Fluid Mech. 869, 553–586 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  68. Obiols-Sales, O., Vishnu, A., Malaya, N. & Chandramowlishwaran, A. CFDNet: a deep learning-based accelerator for fluid simulations. In Proc. 34th ACM Int. Conf. on Supercomputing 1–12 (ACM, 2020).

  69. Spalart. P. & Allmaras, S. A one-equation turbulence model for aerodynamic flows. In 30th Aerospace Sciences Meeting and Exhibit, AIAA Paper 1992-0439 (AIAA, 1992).

  70. Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12, 620–631 (1998).

    Article  Google Scholar 

  71. Gibou, F., Hyde, D. & Fedkiw, R. Sharp interface approaches and deep learning techniques for multiphase flows. J. Comput. Phys. 380, 442–463 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  72. Ma, M., Lu, J. & Tryggvasona, G. Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system. Phys. Fluids 27, 092101 (2015).

    Article  Google Scholar 

  73. Mi, Y., Ishii, M. & Tsoukalas, L. H. Flow regime identification methodology with neural networks and two-phase flow models. Nucl. Eng. Des. 204, 87–100 (2001).

    Article  Google Scholar 

  74. Smagorinsky, J. General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weather Rev. 91, 99–164 (1963).

    Article  Google Scholar 

  75. Beck, A. D., Flad, D. G. & Munz, C.-D. Deep neural networks for data-driven LES closure models. J. Comput. Phys. 398, 108910 (2019).

    Article  MathSciNet  Google Scholar 

  76. Lapeyre, C. J., Misdariis, A., Cazard, N., Veynante, D. & Poinsot, T. Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates. Combust. Flame 203, 255–264 (2019).

    Article  Google Scholar 

  77. Maulik, R., San, O., Rasheed, A. & Vedula, P. Subgrid modelling for two-dimensional turbulence using neural networks. J. Fluid Mech. 858, 122–144 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  78. Kraichnan, R. H. Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423 (1967).

    Article  Google Scholar 

  79. Vollant, A., Balarac, G. & Corre, C. Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. J. Turbul. 18, 854–878 (2017).

    Article  MathSciNet  Google Scholar 

  80. Gamahara, M. & Hattori, Y. Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2, 054604 (2017).

    Article  Google Scholar 

  81. Maulik, R. & San, O. A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831, 151–181 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  82. Reissmann, M., Hasslbergerb, J., Sandberg, R. D. & Klein, M. Application of gene expression programming to a-posteriori LES modeling of a Taylor Green vortex. J. Comput. Phys. 424, 109859 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  83. Novati, G., de Laroussilhe, H. L. & Koumoutsakos, P. Automating turbulence modelling by multi-agent reinforcement learning. Nat. Mach. Intell. 3, 87–96 (2021).

    Article  Google Scholar 

  84. Hutchins, N., Chauhan, K., Marusic, I., Monty, J. & Klewicki, J. Towards reconciling the large-scale structure of turbulent boundary layers in the atmosphere and laboratory. Bound. Layer Meteorol. 145, 273–306 (2012).

    Article  Google Scholar 

  85. Britter, R. E. & Hanna, S. R. Flow and dispersion in urban areas. Annu. Rev. Fluid Mech. 35, 469–496 (2003).

    Article  MATH  Google Scholar 

  86. Giometto, M. G. et al. Spatial characteristics of roughness sublayer mean flow and turbulence over a realistic urban surface. Bound. Layer Meteorol. 160, 425–452 (2016).

    Article  Google Scholar 

  87. Bou-Zeid, E., Meneveau, C. & Parlange, M. A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17, 025105 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  88. Moeng, C. A large-eddy-simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci. 13, 2052–2062 (1984).

    Article  Google Scholar 

  89. Mizuno, Y. & Jiménez, J. Wall turbulence without walls. J. Fluid Mech. 723, 429–455 (2013).

    Article  MATH  Google Scholar 

  90. Encinar, M. P., García-Mayoral, R. & Jiménez, J. Scaling of velocity fluctuations in off-wall boundary conditions for turbulent flows. J. Phys. Conf. Ser. 506, 012002 (2014).

    Article  Google Scholar 

  91. Sasaki, K., Vinuesa, R., Cavalieri, A. V. G., Schlatter, P. & Henningson, D. S. Transfer functions for flow predictions in wall-bounded turbulence. J. Fluid Mech. 864, 708–745 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  92. Arivazhagan, G. B. et al. Predicting the near-wall region of turbulence through convolutional neural networks. Preprint at https://arxiv.org/abs/2107.07340 (2021).

  93. Milano, M. & Koumoutsakos, P. Neural network modeling for near wall turbulent flow. J. Comput. Phys. 182, 1–26 (2002).

    Article  MATH  Google Scholar 

  94. Moriya, N. et al. Inserting machine-learned virtual wall velocity for large-eddy simulation of turbulent channel flows. Preprint at https://arxiv.org/abs/2106.09271 (2021).

  95. Bae, H. J. & Koumoutsakos, P. Scientific multi-agent reinforcement learning for wall-models of turbulent flows. Nat. Commun. 13, 1443 (2022).

    Article  Google Scholar 

  96. Taira, K. et al. Modal analysis of fluid flows: an overview. AIAA J. 55, 4013–4041 (2017).

    Article  Google Scholar 

  97. Rowley, C. W. & Dawson, S. T. Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387–417 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  98. Taira, K. et al. Modal analysis of fluid flows: applications and outlook. AIAA J. 58, 998–1022 (2020).

    Article  Google Scholar 

  99. Lumley, J. L. in Atmospheric Turbulence and Wave Propagation (eds Yaglom, A. M. & Tatarski, V. I.) 166–178 (1967).

  100. Schmid, P. J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  101. Baldi, P. & Hornik, K. Neural networks and principal component analysis: learning from examples without local minima. Neural Netw. 2, 53–58 (1989).

    Article  Google Scholar 

  102. Murata, T., Fukami, K. & Fukagata, K. Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J. Fluid Mech. 882, A13 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  103. Eivazi, H., Le Clainche, S., Hoyas, S. & Vinuesa, R. Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows. Expert Syst. Appl. 202, 117038 (2022).

    Article  Google Scholar 

  104. Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  105. Lee, K. & Carlberg, K. T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404, 108973 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  106. Benner, P., Gugercin, S. & Willcox, K. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57, 483–531 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  107. Carlberg, K., Barone, M. & Antil, H. Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction. J. Comput. Phys. 330, 693–734 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  108. Cenedese, M., Axås, J., Bäuerlein, B., Avila, K. & Haller, G. Data-driven modeling and prediction of nonlinearizable dynamics via spectral submanifolds. Nat. Commun. 13, 872 (2022).

    Article  Google Scholar 

  109. Lopez-Martin, M., Le Clainche, S. & Carro, B. Model-free short-term fluid dynamics estimator with a deep 3D-convolutional neural network. Expert Syst. Appl. 177, 114924 (2021).

    Article  Google Scholar 

  110. Vlachas, P. R., Byeon, W., Wan, Z. Y., Sapsis, T. P. & Koumoutsakos, P. Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc. R. Soc. A 474, 20170844 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  111. Srinivasan, P. A., Guastoni, L., Azizpour, H., Schlatter, P. & Vinuesa, R. Predictions of turbulent shear flows using deep neural networks. Phys. Rev. Fluids 4, 054603 (2019).

    Article  Google Scholar 

  112. Abadía-Heredia, R. et al. A predictive hybrid reduced order model based on proper orthogonal decomposition combined with deep learning architectures. Expert Syst. Appl. 187, 115910 (2022).

    Article  Google Scholar 

  113. Pathak, J., Hunt, B., Girvan, M., Lu, Z. & Ott, E. Model-free prediction of large spatiotemporally chaotic systems from data: a reservoir computing approach. Phys. Rev. Lett. 120, 024102 (2018).

    Article  Google Scholar 

  114. Kaiser, E. et al. Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365–414 (2014).

    Article  MATH  Google Scholar 

  115. Peherstorfer, B. & Willcox, K. Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Meth. Appl. Mech. Eng. 306, 196–215 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  116. Benner, P., Goyal, P., Kramer, B., Peherstorfer, B. & Willcox, K. Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Comput. Meth. Appl. Mech. Eng. 372, 113433 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  117. Qian, E., Kramer, B., Peherstorfer, B. & Willcox, K. Lift & Learn: physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D 406, 132401 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  118. Loiseau, J.-C. & Brunton, S. L. Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  119. Loiseau, J.-C. Data-driven modeling of the chaotic thermal convection in an annular thermosyphon. Theor. Comput. Fluid Dyn. 34, 339–365 (2020).

    Article  MathSciNet  Google Scholar 

  120. Guan, Y., Brunton, S. L. & Novosselov, I. Sparse nonlinear models of chaotic electroconvection. R. Soc. Open Sci. 8, 202367 (2021).

    Article  Google Scholar 

  121. Deng, N., Noack, B. R., Morzynski, M. & Pastur, L. R. Low-order model for successive bifurcations of the fluidic pinball. J. Fluid Mech. 884, A37 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  122. Deng, N., Noack, B. R., Morzynski, M. & Pastur, L. R. Galerkin force model for transient and post-transient dynamics of the fluidic pinball. J. Fluid Mech. 918, A4 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  123. Callaham, J. L., Rigas, G., Loiseau, J.-C. & Brunton, S. L. An empirical mean-field model of symmetry-breaking in a turbulent wake. Sci. Adv. 8, eabm4786 (2022).

    Article  Google Scholar 

  124. Callaham, J. L., Brunton, S. L. & Loiseau, J.-C. On the role of nonlinear correlations in reduced-order modelling. J. Fluid Mech. 938, A1 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  125. Champion, K., Lusch, B., Kutz, J. N. & Brunton, S. L. Data-driven discovery of coordinates and governing equations. Proc. Natl Acad. Sci. USA 116, 22445–22451 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  126. Yeung, E., Kundu, S. & Hodas, N. Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. Preprint at https://arxiv.org/abs/1708.06850 (2017).

  127. Takeishi, N., Kawahara, Y. & Yairi, T. Learning Koopman invariant subspaces for dynamic mode decomposition. In Advances in Neural Information Processing Systems 1130–1140 (ACM, 2017).

  128. Lusch, B., Kutz, J. N. & Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9, 4950 (2018).

    Article  Google Scholar 

  129. Mardt, A., Pasquali, L., Wu, H. & No‚, F. VAMPnets: deep learning of molecular kinetics. Nat. Commun. 9, 5 (2018).

    Article  Google Scholar 

  130. Otto, S. E. & Rowley, C. W. Linearly-recurrent autoencoder networks for learning dynamics. SIAM J. Appl. Dyn. Syst. 18, 558–593 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  131. Wang, R., Walters, R. & Yu, R. Incorporating symmetry into deep dynamics models for improved generalization. Preprint at https://arxiv.org/abs/2002.03061 (2020).

  132. Wang, R., Kashinath, K., Mustafa, M., Albert, A. & Yu, R. Towards physics-informed deep learning for turbulent flow prediction. In Proc. 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining 1457–1466 (ACM, 2020).

  133. Frezat, H., Balarac, G., Le Sommer, J., Fablet, R. & Lguensat, R. Physical invariance in neural networks for subgrid-scale scalar flux modeling. Phys. Rev. Fluids 6, 024607 (2021).

    Article  Google Scholar 

  134. Erichson, N. B., Muehlebach, M. & Mahoney, M. W. Physics-informed autoencoders for Lyapunov-stable fluid flow prediction. Preprint at https://arxiv.org/abs/1905.10866 (2019).

  135. Kaptanoglu, A. A., Callaham, J. L., Hansen, C. J., Aravkin, A. & Brunton, S. L. Promoting global stability in data-driven models of quadratic nonlinear dynamics. Phys. Rev. Fluids 6, 094401 (2021).

    Article  Google Scholar 

  136. Vinuesa, R., Lehmkuhl, O., Lozano-Durán, A. & Rabault, J. Flow control in wings and discovery of novel approaches via deep reinforcement learning. Fluids 865, 281–302 (2019).

    Google Scholar 

  137. Guastoni, L. et al. Convolutional-network models to predict wall-bounded turbulence from wall quantities. J. Fluid Mech. 928, A27 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  138. Kim, H., Kim, J., Won, S. & Lee, C. Unsupervised deep learning for super-resolution reconstruction of turbulence. J. Fluid Mech. 910, A29 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  139. Fukami, K., Fukagata, K. & Taira, K. Super-resolution reconstruction of turbulent flows with machine learning. J. Fluid Mech. 870, 106–120 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  140. Güemes, A. et al. From coarse wall measurements to turbulent velocity fields through deep learning. Phys. Fluids 33, 075121 (2021).

    Article  Google Scholar 

  141. Fukami, K., Nakamura, T. & Fukagata, K. Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data. Phys. Fluids 32, 095110 (2020).

    Article  Google Scholar 

  142. Raissi, M., Yazdani, A. & Karniadakis, G. E. Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367, 1026–1030 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  143. Eivazi, H., Tahani, M., Schlatter, P. & Vinuesa, R. Physics-informed neural networks for solving Reynolds-averaged Navier-Stokes equations. Preprint at https://arxiv.org/abs/2107.10711 (2021).

  144. Kim, Y., Choi, Y., Widemann, D. & Zohdi, T. A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder. J. Comput. Phys. 451, 110841 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  145. Eivazi, H. & Vinuesa, R. Physics-informed deep-learning applications to experimental fluid mechanics. Preprint at https://arxiv.org/abs/2203.15402 (2022).

  146. Markidis, S. The old and the new: can physics-informed deep-learning replace traditional linear solvers? Front. Big Data https://doi.org/10.3389/fdata.2021.669097 (2021).

  147. Kim, J., Moin, P. & Moser, R. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1987).

    Article  MATH  Google Scholar 

  148. Fukagata, K. Towards quantum computing of turbulence. Nat. Comput. Sci. 2, 68–69 (2022).

    Article  Google Scholar 

  149. Barba, L. A. The hard road to reproducibility. Science 354, 142–142 (2016).

    Article  Google Scholar 

  150. Mesnard, O. & Barba, L. A. Reproducible and replicable computational fluid dynamics: it’s harder than you think. Comput. Sci. Eng. 19, 44–55 (2017).

    Article  Google Scholar 

Download references

Acknowledgements

R.V. acknowledges financial support from the Swedish Research Council (VR) and from ERC grant no. ‘2021-CoG-101043998, DEEPCONTROL’. S.L.B. acknowledges funding support from the Army Research Office (ARO W911NF-19-1-0045; programme manager M. Munson).

Author information

Authors and Affiliations

  1. FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden

    Ricardo Vinuesa

  2. Swedish e-Science Research Centre (SeRC), Stockholm, Sweden

    Ricardo Vinuesa

  3. Department of Mechanical Engineering, University of Washington, Seattle, WA, USA

    Steven L. Brunton

Authors
  1. Ricardo Vinuesa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven L. Brunton
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed equally to the ideation of the study and the writing process.

Corresponding author

Correspondence to Ricardo Vinuesa.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Computational Science thanks Michael Brenner and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Principal Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vinuesa, R., Brunton, S.L. Enhancing computational fluid dynamics with machine learning. Nat Comput Sci 2, 358–366 (2022). https://doi.org/10.1038/s43588-022-00264-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s43588-022-00264-7

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing: AI and Robotics