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A practical approach to flow field reconstruction with sparse or incomplete data through physics informed neural network

通过融合物理神经网络重构稀疏或不完整数据流场的实用方法

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Abstract

High-resolution flow field reconstruction is prevalently recognized as a difficult task in the field of experimental fluid mechanics, since the measured data are usually sparse and incomplete in time and space. Specifically, due to the limitations of experimental equipment or measurement techniques, the expected data cannot be measured in some key areas. In this paper, a practical approach is proposed to reconstruct flow field with imperfect data based on the physics informed neural network (PINN), which integrates those known data with the physical principles. The wake flow past a circular cylinder is taken as the test case. Two kinds of the training set are investigated, one is the velocity data with different sparsity, and the other is the velocity data missing in different regions. To accelerate training convergence, the learning rate schedule is discussed, and the cosine annealing algorithm shows excellent performance. Results reveal that the proposed approach not only can reconstruct the true velocity field with high accuracy, but also can predict the pressure field precisely, even when the data sparsity reaches 1% or the core flow area data are truncated away. This study provides encouraging insights that the PINN can serve as a promising data assimilation method for experimental fluid mechanics.

摘要

高分辨率流场重构被普遍认为是实验流体力学领域的一项艰巨任务, 因为测量数据在时间和空间上通常是稀疏或不完整的. 具体而言, 由于实验设备或测量技术的限制, 某些关键区域的数据无法测量. 本文提出了一种基于融合物理神经网络(PINN)的不完美数据重建流场的实用方法, 该网络将已知数据与物理原理相结合. 通过圆柱体的尾流作为测试算例. 研究了两种不完美数据训练集, 一种是不同稀疏度的速度数据, 另一种是不同区域缺失的速度数据. 为了加速训练收敛, 本文采用了余弦退火算法以提高PINN的计算效率.计算结果表明, 该方法不仅可以高精度地重建真实的速度场, 而且即使在数据稀疏度达到1%或核心流动区域数据被截断的情况下, 也可以精确地预测压力场. 这项研究提供了令人鼓舞的结论, 即PINN可以作为实验流体力学的有潜力的数据同化方法.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Jambunathan, S. L. Hartle, S. Ashforth-Frost, and V. N. Fontama, Evaluating convective heat transfer coefficients using neural networks, Int. J. Heat Mass Transfer 39, 2329 (1996).

    Article  MATH  Google Scholar 

  3. G. N. Xie, Q. W. Wang, M. Zeng, and L. Q. Luo, Heat transfer analysis for shell-and-tube heat exchangers with experimental data by artificial neural networks approach, Appl. Thermal Eng. 27, 1096 (2007).

    Article  Google Scholar 

  4. S. Cai, Z. Wang, S. Wang, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks for heat transfer problems, J. Heat Transfer 143, 060801 (2021).

    Article  Google Scholar 

  5. S. Pierret, and R. A. Van den Braembussche, Turbomachinery blade design using a Navier-Stokes solver and artificial neural network, J. Turbomach. 121, 326 (1999).

    Article  Google Scholar 

  6. A. Demeulenaere, A. Ligout, and C. Hirsch, Application of multipoint optimization to the design of turbomachinery blades (2004).

  7. J. Dominique, J. Van den Berghe, C. Schram, and M. A. Mendez, Artificial neural networks modeling of wall pressure spectra beneath turbulent boundary layers, Phys. Fluids 34, 035119 (2022).

    Article  Google Scholar 

  8. J. Svorcan, S. Stupar, S. Trivković, N. Petrašinović, and T. Ivanov, Active boundary layer control in linear cascades using CFD and artificial neural networks, Aerosp. Sci. Tech. 39, 243 (2014).

    Article  Google Scholar 

  9. C. Drygala, B. Winhart, F. di Mare, and H. Gottschalk, Generative modeling of turbulence, Phys. Fluids 34, 035114 (2022).

    Article  Google Scholar 

  10. M. Milano, and P. Koumoutsakos, Neural network modeling for near wall turbulent flow, J. Comput. Phys. 182, 1 (2002).

    Article  MATH  Google Scholar 

  11. D. Schmidt, R. Maulik, and K. Lyras, Machine learning accelerated turbulence modeling of transient flashing jets, Phys. Fluids 33, 127104 (2021).

    Article  Google Scholar 

  12. I. E. Lagaris, A. Likas, and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw. 9, 987 (1998).

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. X. Jin, S. Cai, H. Li, and G. E. Karniadakis, NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations, J. Comput. Phys. 426, 109951 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Laubscher, Simulation of multi-species flow and heat transfer using physics-informed neural networks, Phys. Fluids 33, 087101 (2021).

    Article  Google Scholar 

  16. H. Gao, L. Sun, and J. X. Wang, Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels, Phys. Fluids 33, 073603 (2021).

    Article  Google Scholar 

  17. H. Wang, Y. Liu, and S. Wang, Dense velocity reconstruction from particle image velocimetry/particle tracking velocimetry using a physics-informed neural network, Phys. Fluids 34, 017116 (2022).

    Article  Google Scholar 

  18. Z. Mao, A. D. Jagtap, and G. E. Karniadakis, Physics-informed neural networks for high-speed flows, Comput. Methods Appl. Mech. Eng. 360, 112789 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, Physics-informed neural networks (PINNs) for fluid mechanics: A review, Acta Mech. Sin. 37, 1727 (2021).

    Article  MathSciNet  Google Scholar 

  20. L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, and S. G. Johnson, Physics-informed neural networks with hard constraints for inverse design, SIAM J. Sci. Comput. 43, B1105 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  21. Y. Chen, L. Lu, G. E. Karniadakis, and L. Dal Negro, Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Opt. Express 28, 11618 (2020).

    Article  Google Scholar 

  22. S. Mishra, and R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs, IMA J. Numer. Anal. 42, 981 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  23. X. Chen, L. Yang, J. Duan, and G. E. Karniadakis, Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks, SIAM J. Sci. Comput. 43, B811 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Kadeethum, T. M. Jørgensen, and H. M. Nick, Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations, PLoS One 15, e0232683 (2020).

    Article  Google Scholar 

  25. L. Yang, X. Meng, and G. E. Karniadakis, B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data, J. Comput. Phys. 425, 109913 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  26. J. P. Molnar, and S. J. Grauer, Flow field tomography with uncertainty quantification using a Bayesian physics-informed neural network, Meas. Sci. Technol. 33, 065305 (2022).

    Article  Google Scholar 

  27. X. Meng, H. Babaee, and G. E. Karniadakis, Multi-fidelity Bayesian neural networks: Algorithms and applications, J. Comput. Phys. 438, 110361 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  28. F. A. C. Viana, and A. K. Subramaniyan, A survey of bayesian calibration and physics-informed neural networks in scientific modeling, Arch. Computat. Methods Eng. 28, 3801 (2021).

    Article  MathSciNet  Google Scholar 

  29. S. Cai, Z. Wang, F. Fuest, Y. J. Jeon, C. Gray, and G. E. Karniadakis, Flow over an espresso cup: Inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks, J. Fluid Mech. 915, A102 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Raissi, A. Yazdani, and G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367, 1026 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Yin, X. Zheng, J. D. Humphrey, and G. E. Karniadakis, Non-invasive inference of thrombus material properties with physics-informed neural networks, Comput. Methods Appl. Mech. Eng. 375, 113603 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Kissas, Y. Yang, E. Hwuang, W. R. Witschey, J. A. Detre, and P. Perdikaris, Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks, Comput. Methods Appl. Mech. Eng. 358, 112623 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  33. Q. Z. He, D. Barajas-Solano, G. Tartakovsky, and A. M. Tartakovsky, Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport, Adv. Water Resour. 141, 103610 (2020).

    Article  Google Scholar 

  34. M. M. Almajid, and M. O. Abu-Al-Saud, Prediction of porous media fluid flow using physics informed neural networks, J. Pet. Sci. Eng. 208, 109205 (2022).

    Article  Google Scholar 

  35. M. A. Nabian, R. J. Gladstone, and H. Meidani, Efficient training of physics-informed neural networks via importance sampling, Comput.-Aided Civil Infrastruct. Eng. 36, 962 (2021).

    Article  Google Scholar 

  36. L. Sun, and J. X. Wang, Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data, Theor. Appl. Mech. Lett. 10, 161 (2020).

    Article  Google Scholar 

  37. L. Sun, H. Gao, S. Pan, and J. X. Wang, Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data, Comput. Methods Appl. Mech. Eng. 361, 112732 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Xu, W. Zhang, and Y. Wang, Explore missing flow dynamics by physics-informed deep learning: The parameterized governing systems, Phys. Fluids 33, 095116 (2021).

    Article  Google Scholar 

  39. T. Wang, Z. Huang, Z. Sun, and G. Xi, Reconstruction of natural convection within an enclosure using deep neural network, Int. J. Heat Mass Transfer 164, 120626 (2021).

    Article  Google Scholar 

  40. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys. 3, 422 (2021).

    Article  Google Scholar 

  41. A. Sciacchitano, R. P. Dwight, and F. Scarano, Navier-Stokes simulations in gappy PIV data, Exp Fluids 53, 1421 (2012).

    Article  Google Scholar 

  42. J. F. G. Schneiders, and F. Scarano, Dense velocity reconstruction from tomographic PTV with material derivatives, Exp. Fluids 57, 139 (2016).

    Article  Google Scholar 

  43. M. K. Bisbo, and B. Hammer, Efficient global structure optimization with a machine-learned surrogate model, Phys. Rev. Lett. 124, 086102 (2020).

    Article  Google Scholar 

  44. C. Ma, B. Zhu, X. Q. Xu, and W. Wang, Machine learning surrogate models for Landau fluid closure, Phys. Plasmas 27, 042502 (2020).

    Article  Google Scholar 

  45. X. Yan, J. Zhu, M. Kuang, and X. Wang, Aerodynamic shape optimization using a novel optimizer based on machine learning techniques, Aerosp. Sci. Tech. 86, 826 (2019).

    Article  Google Scholar 

  46. J. Li, M. Zhang, J. R. R. A. Martins, and C. Shu, Efficient aerodynamic shape optimization with deep-learning-based geometric filtering, AIAA J. 58, 4243 (2020).

    Article  Google Scholar 

  47. N. Umetani, and B. Bickel, Learning three-dimensional flow for interactive aerodynamic design, ACM Trans. Graph. 37, 1 (2018).

    Article  Google Scholar 

  48. B. Wang, W. Zhang, and W. Cai, Multi-scale deep neural network (MscaleDNN) methods for oscillatory Stokes flows in complex domains, Commun. Comput. Phys. 28, 2139 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Mattheakis, D. Sondak, and P. Protopapas, Physical symmetries embedded in neural networks, Bull. Am. Phys. Soc. 64, (2019).

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

    Article  MathSciNet  MATH  Google Scholar 

  51. A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Automatic differentiation in machine learning: A survey, J. Mach. Learn. Res. 18, 1 (2018).

    MathSciNet  MATH  Google Scholar 

  52. S. Wang, Y. Teng, and P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-informed neural networks, SIAM J. Sci. Comput. 43, A3055 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  53. S. Ioffe, and C. Szegedy, Batch normalization: Accelerating deep network training by reducing internal covariate shift, arXiv: 1502.03167.

  54. K. Li, J. Kou, and W. Zhang, Deep learning for multi-fidelity aerodynamic distribution modeling from experimental and simulation data, arXiv: 2109.12966.

  55. M. D. Zeiler, Adadelta: An adaptive learning rate method, arXiv: 1212.5701.

  56. I. Loshchilov, and F. Hutter, Sgdr: Stochastic gradient descent with warm restarts, arXiv: 1608.03983.

  57. D. J. Tritton, Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech. 6, 547 (1959).

    Article  MATH  Google Scholar 

  58. A. Roshko, Experiments on the flow past a circular cylinder at very high Reynolds number, J. Fluid Mech. 10, 345 (1961).

    Article  MATH  Google Scholar 

  59. S. Behara, and S. Mittal, Flow past a circular cylinder at low Reynolds number: Oblique vortex shedding, Phys. Fluids 22, 054102 (2010).

    Article  MATH  Google Scholar 

  60. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics informed deep learning (part II): Data-driven discovery of nonlinear partial differential equations, arXiv: 1711.10566.

  61. S. Siegel, K. Cohen, and T. McLaughlin, Feedback control of a circular cylinder wake in experiment and simulation, AIAA Paper No. 2003–3569, 2003.

  62. D. P. Kingma, and J. Ba, Adam: A method for stochastic optimization, arXiv: 1412.6980.

  63. R. J. Adrian, Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23, 261 (1991).

    Article  Google Scholar 

  64. T. Keller, J. Henrichs, K. Hochkirch, and A. C. Hochbaum, Numerical simulations of a surface piercing a-class catamaran hydrofoil and comparison against model tests, J. Sailing Technol. 2, 1 (2017).

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52006232) and the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No. 2019020). The two-dimensional wake flow data adopted in this paper are open sourced by Raissi on Github at https://github.com/maziarraissi/PINNs. In addition, all codes and results used in this manuscript are available on Github at https://github.com/Shengfeng233/PINN-for-NS-equation.

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Authors and Affiliations

  1. Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Shengfeng Xu  (许盛峰), Zhenxu Sun  (孙振旭), Renfang Huang  (黄仁芳), Dilong Guo  (郭迪龙), Guowei Yang  (杨国伟) & Shengjun Ju  (鞠胜军)

  2. School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shengfeng Xu  (许盛峰)

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  1. Shengfeng Xu  (许盛峰)
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  2. Zhenxu Sun  (孙振旭)
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  3. Renfang Huang  (黄仁芳)
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  4. Dilong Guo  (郭迪龙)
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Correspondence to Zhenxu Sun  (孙振旭) or Renfang Huang  (黄仁芳).

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Shengfeng Xu, Zhenxu Sun and Renfang Huang designed the research. Shengfeng Xu wrote the first draft of the manuscript. Zhenxu Sun, Dilong Guo and Guowei Yang helped organize the manuscript and provided supervision. Shengjun Ju revised and edited the final version.

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Xu, S., Sun, Z., Huang, R. et al. A practical approach to flow field reconstruction with sparse or incomplete data through physics informed neural network. Acta Mech. Sin. 39, 322302 (2023). https://doi.org/10.1007/s10409-022-22302-x

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