Fourier neural operator with boundary conditions for efficient prediction of steady airfoil flows

doi:10.1007/s10483-023-3050-9

Abstract

Abstract: An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator (FNO), which is a new framework of neural networks. Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO, which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions. The proposed approach runs several orders of magnitude faster than the traditional numerical methods. The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach. Furthermore, the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids, thereby improving the accuracy in the near-wall region. There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.

Key words: deep learning (DL), Fourier neural operator (FNO), steady airfoil flow

2010 MSC Number:

76G25

Yuanjun DAI, Yiran AN, Zhi LI, Jihua ZHANG, Chao YU. Fourier neural operator with boundary conditions for efficient prediction of steady airfoil flows. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 2019-2038.

TrendMD

References

[1] KRIZHEVSKY, A., SUTSKEVER, I., and HINTON, G. E. ImageNet classification with deep convolutional neural networks. Communications of the ACM, 60(6), 84–90(2017)
[2] DEVLIN, J., CHANG, M., LEE, K., and TOUTANOVA, K. BERT: pre-training of deep bidirectional transformers for language understanding. Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Association for Computational Linguistics, Minneapolis, 4171–4186(2019)
[3] DAHL, G. E., YU, D., DENG, L., and ACERO, A. Context-dependent pre-trained deep neural networks for large-vocabulary speech recognition. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 20(1), 30–42(2012)
[4] WU, H., LIU, X., AN, W., CHEN, S., and LYU, H. A deep learning approach for efficiently and accurately evaluating the flow field of supercritical airfoils. Computers & Fluids, 198, 104393(2020)
[5] HUI, X., BAI, J., WANG, H., and ZHANG, Y. Fast pressure distribution prediction of airfoils using deep learning. Aerospace Science and Technology, 105, 105949(2020)
[6] LI, K., KOU, J., and ZHANG, W. Unsteady aerodynamic reduced-order modeling based on machine learning across multiple airfoils. Aerospace Science and Technology, 119, 107173(2021)
[7] WAN, Z. Y., VLACHAS, P., KOUMOUTSAKOS, P., and SAPSIS, T. Data-assisted reduced-order modeling of extreme events in complex dynamical systems. PLoS One, 13(5), e0197704(2018)
[8] VLACHAS, P. R., BYEON, W., WAN, Z. Y., SAPSIS, T., and KOUMOUTSAKOS, P. Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 447(2213), 20170844(2018)
[9] GUO, X., LI, W., and IORIO, F. Convolutional neural networks for steady flow approximation. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, San Francisco, 481–490(2016)
[10] BHATNAGAR, S., AFSHAR, Y., PAN, S., DURAISAMY, K., and KAUSHIK, S. Prediction of aerodynamic flow fields using convolutional neural networks. Computational Mechanics, 64, 525–545(2019)
[11] THUEREY, N., WEIβENOW, K., PRANTL, L., and HU, X. Deep learning methods for Reynolds-averaged Navier-Stokes simulations of airfoil flows. AIAA Journal, 58, 25–36(2020)
[12] GOODFELLOW, I., POUGET-ABADIE, J., MIRZA, M., XU, B., WARDE-FARLEY, D., OZAIR, S., COURVILLE, A., and BENGIO, Y. Generative adversarial nets. Proceedings of the 27th International Conference on Neural Information Processing Systems, MIT Press, Cambridge, 2672–2680(2014)
[13] HOCHREITER, S. and SCHMIDHUBER, J. Long short-term memory. Neural Computation, 9(8), 1735–1780(1997)
[14] RONNEBERGER, O., FISCHER, P., and BROX, T. U-net: convolutional networks for biomedical image segmentation. Medical Image Computing and Computer-Assisted Intervention MICCAI-2015, Springer, Berlin, 234–241(2015)
[15] PFAFF, T., FORTUNATO, M., SANCHEZ-GONZALEZ, A., and BATTAGLIA, P. Learning mesh-based simulation with graph networks. arXiv Preprint, arXiv: 2010.03409(2020) https://doi.org/10.48550/arXiv.2010.03409
[16] BELBUTE-PERES, F., ECONOMON, T. D., and KOLTER, J. Z. Combining differentiable PDE solvers and graph neural networks for fluid flow prediction. arXiv Preprint, arXiv: 2007.04439v1(2020) https://doi.org/10.48550/arXiv.2007.04439
[17] SANCHEZ-LENGELING, B., REIF, E., PEARCE, A., and WILTSCHKO, A. A gentle introduction to graph neural networks. Distill (2021) https://distill.pub/2021/gnn-intro/
[18] LI, Q., HAN, Z., and WU, X. Deeper insights into graph convolutional networks for semisupervised learning. Proceedings of the 32nd AAAI Conference on Artificial Intelligence, AIAA Press, New Orleans, 3538–3545(2018)
[19] RAISSI, M., PERDIKARIS, P., and KARNIADAKIS, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal Computational Physics, 378, 686–707(2019)
[20] RAISSI, M., WANG, Z., TRIANTAFYLLOU, M. S., and KARNIADAKIS, G. E. Deep learning of vortex-induced vibrations. Journal of Fluid Mechanics, 861, 119–137(2019)
[21] HE, Y. C., WANG, Z. C., XIANG, H., JIANG, X. M., and TANG, D. W. An artificial viscosity augmented physics-informed neural network for incompressible flow. Applied Mathematics and Mechanics (English Edition), 44(7), 1101–1110(2023) https://doi.org/10.1007/s10483-023-2993-9
[22] MAO, Z., JAGTAP, A. D., and KARNIADAKIS, G. E. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360, 112789(2020)
[23] YIN, M., ZHENG, X., HUMPHREY, J. D., and KARNIADAKIS, G. E. Non-invasive inference of thrombus material properties with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 375, 113603(2021)
[24] LUCOR, D., AGRAWAL, A., and SERGENT, A. Physics-aware deep neural networks for surrogate modeling of turbulent natural convection. arXiv Preprint, arXiv: 2103.03565(2021) https://doi.org/10.48550/arXiv.2103.03565
[25] EIVAZI, H., TAHANI, M., SCHLATTER, P., and VINUESA, R. Physics-informed neural networks for solving Reynolds-averaged Navier-Stokes equations. Physics of Fluids, 34, 075117(2022)
[26] WANG, S., TENG, Y., and PERDIKARIS, P. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43, 3055–3081(2021)
[27] WANG, S., YU, X., and PERDIKARIS, P. When and why PINNs fail to train: a neural tangent kernel perspective. Journal Computational Physics, 449, 110768(2022)
[28] SUN, L., GAO, H., PAN, S., and WANG, J. X. Surrogate modeling for fluid flows based on physicsconstrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 361, 112732(2020)
[29] LU, L., JIN, P., and KARNIADAKIS, G. E. DeepONet: learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv Preprint, arXiv: 1910.03193v3(2019) https://doi.org/10.48550/arXiv.1910.03193
[30] BHATTACHARYA, K., HOSSEINI, B., KOVACHKI, N. B., and STUART, A. M. Model reduction and neural networks for parametric PDEs. SMAI Journal of Computational Mathematics, 7, 121–157(2021)
[31] LI, Z., KOVACHKI, N., AZIZZADENESHELI, K., LIU, B., BHATTACHARYA, K., STUART, A., and ANANDKUMAR, A. Neural operator: graph kernel network for partial differential equations. arXiv Preprint, arXiv: 2003.03485(2020) https://doi.org/10.48550/arXiv.2003.03485
[32] LI, Z., KOVACHKI, N., AZIZZADENESHELI, K., LIU, B., BHATTACHARYA, K., STUART, A., and ANANDKUMAR, A. Fourier neural operator for parametric partial differential equations. arXiv Preprint, arXiv: 2010.08895(2020) https://doi.org/10.48550/arXiv.2010.08895
[33] PATEL, R. G., TRASK, N. A., WOOD, M. A., and CYR, E. C. A physics-informed operator regression framework for extracting data-driven continuum models. Computer Methods in Applied Mechanics and Engineering, 373, 113500(2021)
[34] KOVACHKI, N., LANTHALER, S., and MISHRA, S. On universal approximation and error bounds for Fourier neural operators. Journal of Machine Learning Research, 22, 1–76(2021)
[35] HE, K., ZHANG, X., REN, S., and SUN, J. Deep residual learning for image recognition. Proceedings of 2016 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, Las Vegas, 770–778(2016)
[36] LI, Z., ZHENG, H., and KOVACHKI, N. Physics-informed neural operator for learning partial differential equations. arXiv Preprint, arXiv: 2111.03794(2021) https://doi.org/10.48550/arXiv.2111.03794
[37] LI, Z., PENG, W., YUAN, Z., and WANG, J. Fourier neural operator approach to large eddy simulation of three-dimensional turbulence. Theoretical and Applied Mechanics Letters, 12(6), 100389(2022)
[38] WEN, G., LI, Z., AZIZZADENESHELI, K., ANANDKUMAR, A., and BENSON, M. B. U-FNOan enhanced Fourier neural operator based deep-learning model for multiphase flow. Advances in Water Resources, 163, 104180(2022)
[39] GUAN, S., HSU, K. T., and CHITNIS, P. Fourier neural operator network for fast photoacoustic wave simulations. Algorithms, 16, 124(2023)
[40] ZHONG, M., YAN, Z., and TIAN, S. Data-driven parametric soliton-rogon state transitions for nonlinear wave equations using deep learning with Fourier neural operator. Communications in Theoretical Physics, 75, 025001(2023)
[41] ZHANG, Q., MILLETARI, M., ORUGANTI, Y., and WITTE, P. Surrogate modeling for methane dispersion simulations using fourier neural operator. NeurIPS 2022 Workshop on Tackling Climate Change with Machine Learning.
[42] JOHNNY, W., BRIGIDO, H., LADEIRA, M., and SOUZA, J. Fourier neural operator for image classification. 17th Iberian Conference on Information Systems and Technologies (CISTI), IEEE, Madrid, 1–6(2022)