物理信息神經網絡求解二維納維-斯托克斯流

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姓名

陳宗興(Tsung-Hsing Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

物理信息神經網絡求解二維納維-斯托克斯流
(Physics-Informed Neural Network Approach for Solving 2D Navier-Stokes Flows)

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摘要(中)

在本論文中，我們提出了一種修改的投影方法，利用物理信息神經網絡來求解不可壓縮的納維-斯托克斯方程。我們首先應用有限差分法結合投影方法來解決泰勒-格林渦旋，並將結果與解析解進行比較。我們的結果表明，該方法能夠以二階收斂速率準確預測泰勒-格林渦旋的流動和壓力場。隨後，我們使用結合投影方法的物理信息神經網絡來解決泰勒-格林渦旋。然而，我們的實驗結果表明，直接使用投影法會導致速度場的預測結果較差。為了解決這個問題，我們提出了一種修改的投影方法，同時求解流體函數和勢函數，並通過流體函數來更新速度場。我們的數值結果表明，這種方法能夠在方形、橢圓形和L形區域中準確預測泰勒-格林渦旋的流動和壓力場。

摘要(英)

In this thesis, we propose a modified projection method for solving the incompressible Navier-Stokes equations using physics-informed neural networks (PINNs). We begin by applying the finite difference method combined with the projection method to solve the Taylor- Green vortex and compare the results with the analytical solution. Our results demonstrate that this approach accurately predicts the flow and pressure fields of the Taylor-Green vortex with a second-order convergence rate. We then use PINNs with the projection method to solve the Taylor-Green vortex. However, our experimental results indicate that, direct usage of the projection method leads to poor prediction results of the velocity field. To address this, we propose a modified projection method that simultaneously solves the stream function and potential function. Our numerical results show that this approach accurately predicts the flow and pressure fields of the Taylor-Green vortex in square, ellptical and L-shaped domains.

關鍵字(中)

★ 物理信息神經網絡
★ 納維-斯托克斯流
★ 泰勒-格林渦旋

關鍵字(英)

★ Physics-Informed Neural Network
★ Navier-Stokes Flows
★ Taylor-Green vortex

論文目次

中文摘要 - i
英文摘要 - ii
Table of Contents - iii
List of Figures - iv
List of Tables - v
Chapter 1 Introduction - 1
Chapter 2 Research Context and Methods - 3
2.1 Navier-Stokes Equations - 3
2.2 Projection Method - 4
2.3 Modified Projection Method - 5
Chapter 3 Physics-Informed Neural Networks - 7
Chapter 4 Numerical Simulations - 11
4.1 The Results of the Projection Method and the Modified Projection Method - 12
4.2 Modified Projection Method Results - 14
4.2.1 Numerical Results: Square Domain - 14
4.2.2 Numerical Results: Elliptical Domain - 16
4.2.3 Numerical Results: L-shaped Domain - 18
Chapter 5 Conclusions - 20
Bibliography - 21

參考文獻

[1] A. J. Chorin, “Numerical solution of the navier-stokes equations,” Mathematics of computation, vol. 22, no. 104, pp. 745–762, 1968.
[2] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations: theory and algorithms. Springer Science & Business Media, 2012, vol. 5.
[3] J. Marshall, A. Adcroft, C. Hill, L. Perelman, and C. Heisey, “A finite-volume, incompressible navier stokes model for studies of the ocean on parallel computers,” Journal of Geophysical Research: Oceans, vol. 102, no. C3, pp. 5753–5766, 1997.
[4] D. L. Brown, R. Cortez, and M. L. Minion, “Accurate projection methods for the incompressible navier-stokes equations,” Journal of Computational Physics, vol. 168, no. 2, pp. 464–499, 2001.
[5] X. Jin, S. Cai, H. Li, and G. E. Karniadakis, “Nsfnets (navier-stokes flow nets): Physicsinformed neural networks for the incompressible navier-stokes equations,” Journal of Computational Physics, vol. 426, p. 109 951, Feb. 2021.
[6] S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, “Physics-informed neural networks (pinns) for fluid mechanics: A review,” Acta Mechanica Sinica, vol. 37, no. 12, pp. 1727–1738, 2021.
[7] S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, “Scientific machine learning through physics–informed neural networks: Where we are and what＇s next,” Journal of Scientific Computing, vol. 92, no. 3, p. 88, 2022.
[8] M. Raissi, P. Perdikaris, and G. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019.
[9] Y.-H. Tseng, “The simulation of incompressible flow around a solid body by volume-offluid approach,” M.S. thesis, National Chiao Tung University, 2005.
[10] S. Wang, S. Sankaran, H. Wang, and P. Perdikaris, An expert’s guide to training physicsinformed neural networks, 2023. arXiv: 2308.08468 [cs.LG].
[11] D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” Journal of the Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431–441, 1963.

指導教授

胡偉帆(Wei-Fan Hu)

審核日期

2024-7-10

推文