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Discretize first, filter next: learning divergence-consistent closure models for large-eddy simulation (2403.18088v2)

Published 26 Mar 2024 in math.NA, cs.NA, and physics.flu-dyn

Abstract: We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm "discretize first, filter and close next". This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the closure models can learn to account for the discretization. Furthermore, we employ a divergence-consistent discrete filter defined through face-averaging and provide novel theoretical and numerical filter analysis. This filter preserves the discrete divergence-free constraint by construction, unlike general discrete filters such as volume-averaging filters. We show that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures.

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References (72)
  1. S. D. Agdestein and B. Sanderse. Discretize first, filter next – a new closure model approach. In 8th European Congress on Computational Methods in Applied Sciences and Engineering. CIMNE, 2022.
  2. Numerical and modeling error assessment of large-eddy simulation using direct-numerical-simulation-aided large-eddy simulation. arXiv:2208.02354, 2022.
  3. A.R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory, 39(3):930–945, May 1993.
  4. Deep neural networks for data-driven les closure models. Journal of Computational Physics, 398:108910, 2019.
  5. A perspective on machine learning methods in turbulence modeling. GAMM-Mitteilungen, 44(1):e202100002, 2021.
  6. Toward discretization-consistent closure schemes for large eddy simulation using reinforcement learning. Physics of Fluids, 35(12):125122, 12 2023.
  7. Neural network–based closure models for large–eddy simulations with explicit filtering. Available at SSRN: https://ssrn.com/abstract=4462714, 2023.
  8. Mathematics of Large Eddy Simulation of Turbulent Flows. Springer-Verlag, 2006.
  9. Rapid software prototyping for heterogeneous and distributed platforms. Advances in Engineering Software, 132:29–46, 2019.
  10. Effective extensible programming: Unleashing Julia on GPUs. IEEE Transactions on Parallel and Distributed Systems, 2018.
  11. Julia: A fresh approach to numerical computing. SIAM Review, 59(1):65–98, Sep 2017.
  12. On the properties of discrete spatial filters for CFD. Journal of Computational Physics, 326:474–498, 2016.
  13. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Transactions on Neural Networks, 6(4):911–917, 1995.
  14. cudnn: Efficient primitives for deep learning. arXiv:1410.0759, 2014.
  15. Alexandre Joel Chorin. Numerical solution of the Navier-Stokes equations. Mathematics of computation, 22(104):745–762, 1968.
  16. JuliaGPU/KernelAbstractions.jl: v0.9.10. Zenodo, Oct 2023.
  17. Makie.jl: Flexible high-performance data visualization for Julia. Journal of Open Source Software, 6(65):3349, 2021.
  18. Karthik Duraisamy. Perspectives on machine learning-augmented reynolds-averaged and large eddy simulation models of turbulence. Physical Review Fluids, 6(5), May 2021.
  19. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 51(1):357–377, January 2019.
  20. Residual and solution filtering for explicitly-filtered large-eddy simulations. In AIAA Scitech 2019 Forum. American Institute of Aeronautics and Astronautics, Jan 2019.
  21. Affordable explicitly filtered large-eddy simulation for reacting flows. AIAA Journal, 57(2):809–823, Feb 2019.
  22. Simulation and modeling of turbulent flows. Oxford University Press, 1996.
  23. M. Germano. Differential filters for the large eddy numerical simulation of turbulent flows. The Physics of Fluids, 29(6):1755–1757, 06 1986.
  24. Commutator errors in large-eddy simulation. Journal of Physics A: Mathematical and General, 39(9):2213–2229, February 2006.
  25. Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena, 443, 2023.
  26. Solving ordinary differential equations II: Stiff and Differential-Algebraic Problems. Springer, 1991.
  27. The numerical solution of differential-algebraic systems by Runge-Kutta methods, volume 1409. Springer, 2006.
  28. Numerical calculation of time‐dependent viscous incompressible flow of fluid with free surface. The Physics of Fluids, 8(12):2182–2189, 1965.
  29. Phiflow: A differentiable PDE solving framework for deep learning via physical simulations. In NeurIPS workshop, volume 2, 2020.
  30. Michael Innes. Don’t unroll adjoint: Differentiating SSA-form programs. CoRR, abs/1810.07951, 2018.
  31. On the identification of a vortex. Journal of Fluid Mechanics, 285:69–94, 1995.
  32. Autonomic closure for turbulence simulations. Physical Review E, 93(3):031301, 2016.
  33. Adam: A method for stochastic optimization. arXiv:1412.6980v9, 2017.
  34. Turbulent flow simulation using autoregressive conditional diffusion models. arXiv: 2309.01745, Sep 2023.
  35. M. Kurz and A. Beck. Investigating model-data inconsistency in data-informed turbulence closure terms. In 14th WCCM-ECCOMAS Congress. CIMNE, 2021.
  36. A machine learning framework for LES closure terms. arxiv:2010.03030v1, Oct 2020. Electronic transactions on numerical analysis ETNA 56 (2022) 117-137.
  37. D. K. Lilly. On the numerical simulation of buoyant convection. Tellus, 14(2):148–172, 1962.
  38. D. K. Lilly. The representation of small-scale turbulence in numerical simulation experiments. Proc. IBM Sci. Comput. Symp. on Environmental Science, pages 195–210, 1967.
  39. Douglas K Lilly. On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems. Monthly Weather Review, 93(1):11–25, 1965.
  40. How temporal unrolling supports neural physics simulators. arXiv:2402.12971, 2024.
  41. Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons. Journal of Fluid Mechanics, 949:A25, 2022.
  42. David G. Luenberger. The conjugate residual method for constrained minimization problems. SIAM Journal on Numerical Analysis, 7(3):390–398, Sep 1970.
  43. T.S. Lund. The use of explicit filters in large eddy simulation. Computers & Mathematics with Applications, 46(4):603–616, 2003. Turbulence Modelling and Simulation.
  44. Embedded training of neural-network subgrid-scale turbulence models. Phys. Rev. Fluids, 6:050502, May 2021.
  45. A neural network approach for the blind deconvolution of turbulent flows. Journal of Fluid Mechanics, 831:151–181, 2017.
  46. A stable and scale-aware dynamic modeling framework for subgrid-scale parameterizations of two-dimensional turbulence. Computers & Fluids, 158:11–38, 2017. Large-eddy and direct numerical simulations of oceanic and atmospheric flows.
  47. Sub-grid modelling for two-dimensional turbulence using neural networks. arXiv:1808.02983v1, Aug 2018.
  48. Comparison of neural closure models for discretised PDEs. Computers & Mathematics with Applications, 143:94–107, 2023.
  49. Paolo Orlandi. Fluid flow phenomena: a numerical toolkit, volume 55. Springer Science & Business Media, 2000.
  50. Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett., 28:76–79, Jan 1972.
  51. Avik Pal. Lux: Explicit Parameterization of Deep Neural Networks in Julia, April 2023.
  52. Toward neural-network-based large eddy simulation: application to turbulent channel flow. Journal of Fluid Mechanics, 914, March 2021.
  53. J Blair Perot. Discrete conservation properties of unstructured mesh schemes. Annual review of fluid mechanics, 43:299–318, 2011.
  54. Stephen B Pope. Turbulent Flows. Cambridge University Press, Cambridge, England, Aug 2000.
  55. R. S. Rogallo. Numerical experiments in homogeneous turbulence. NASA-TM-81315, 1981.
  56. R S Rogallo and P Moin. Numerical simulation of turbulent flows. Annual Review of Fluid Mechanics, 16(1):99–137, 1984.
  57. Pierre Sagaut. Large eddy simulation for incompressible flows: an introduction. Springer Science & Business Media, 2005.
  58. High-order methods for decaying two-dimensional homogeneous isotropic turbulence. Computers & Fluids, 63:105–127, Jun 2012.
  59. B. Sanderse. Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations. Journal of Computational Physics, 233:100–131, 2013.
  60. Scientific machine learning for closure models in multiscale problems: A review, March 2024.
  61. Differentiable physics-enabled closure modeling for Burgers’ turbulence. arXiv:2209.11614v1, (arXiv:2209.11614), September 2022.
  62. Wei Shyy. A study of finite difference approximations to steady-state, convection-dominated flow problems. Journal of Computational Physics, 57(3):415–438, February 1985.
  63. PDE-constrained models with neural network terms: Optimization and global convergence. arXiv:2105.08633v6, May 2021.
  64. Dynamic deep learning LES closures: Online optimization with embedded DNS. arXiv:2303.02338v1, Mar 2023.
  65. DPM: A deep learning PDE augmentation method with application to large-eddy simulation. Journal of Computational Physics, 423:109811, 2020.
  66. J. Smagorinsky. General circulation experiments with the primitive equations: I. the basic experiment. Monthly Weather Review, 91(3):99 – 164, 1963.
  67. Development of a large-eddy simulation subgrid model based on artificial neural networks: a case study of turbulent channel flow. Geoscientific Model Development, 14(6):3769–3788, June 2021.
  68. On the construction of discrete filters for symmetry-preserving regularization models. Computers & Fluids, 40(1):139–148, 2011.
  69. Solver-in-the-loop: Learning from differentiable physics to interact with iterative PDE-solvers. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 6111–6122. Curran Associates, Inc., 2020.
  70. Energy-Conserving Neural Network for Turbulence Closure Modeling, February 2023.
  71. R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-preserving discretization of turbulent flow. J. Comput. Phys., 187(1):343–368, May 2003.
  72. Direct numerical simulation of turbulence at lower costs. Journal of Engineering Mathematics, 32(2/3):143–159, 1997.
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