學習不規則域中泊松方程式的格林函數

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陳志煌(Jhih-Huang Chen) 查詢紙本館藏

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數學系

論文名稱

學習不規則域中泊松方程式的格林函數
(Learning the Green’s functions for Poisson equations in irregular domains)

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摘要(中)

本研究的目的是使用物理信息神經網絡和 BI-GreenNet 兩種方法訓練格林函數，以求解泊松方程。泊松方程的解可以透過格林函數的積分式表示，因此我們使用這兩種方法來訓練格林函數。格林函數透過基本解可以分為顯式奇異部分和光滑部分。物理信息神經網絡方法透過定義損失函數，其中包括拉普拉斯方程項的殘差和邊界條件部分，以此來訓練格林函數。在 BI-GreenNet 方法中，先定義單層電勢和雙層電勢，並利用它們自動滿足拉普拉斯方程項的性質，因此損失函數僅包含邊界條件部分。數值結果顯示，無論是物理信息神經網絡還是 BI-GreenNet 方法，當格林函數的點源接邊界時，結果的準確性都會下降。然而，BI-GreenNet 可以通過增加數值積分對邊界分割的數量或是採用更精細的方法處理近奇異積分，來減少誤差，提高結果的精度。

摘要(英)

The purpose of this study is to use two methods, physical information neural network and BI-GreenNet, to train the Green function to solve Poisson equation. The solution of Poisson’s equation can be expressed by the integral form of Green’s function, so we use these two methods to train Green’s function. Green’s function can be divided into an explicit singular part and a smooth part through the basic solution. The physical information neural network method trains the Green’s function by defining a loss function that includes the residual and boundary condition parts of the Laplace equation term. In the BI-GreenNet method, the single-layer potential and the double-layer potential are first defined, and they are used to automatically satisfy
the properties of the Laplace equation terms, so the loss function only contains the boundary condition part. Numerical results show that whether it is the physical information neural network or the BI-GreenNet method, when the point source of the Green’s function approaches the boundary, the accuracy of the results decreases. However, BI-GreenNet can reduce errors and improve the accuracy of results by increasing the number of boundary segmentations by numerical integration or using a more refined method to process near-singular integrals.

關鍵字(中)

★ 格林函數
★ 泊松方程式

關鍵字(英)

★ Green’s functions
★ Poisson equations

論文目次

摘要. . . . . . . . . i
Abstract . . . . . . ii
Table of Contents . . ii
List of Figures . . . iii
Chapter 1 Introduction. . . . 1

Chapter 2 Potential Theory . . . . 3
2.1 2D Green’s function . . . . 3
2.2 Single layer potential and Double layer potential.
. . . 5
Chapter 3 Experimental methods of BI-GreenNet . . . . 7
3.1 Physics Information Neural Networks (PINNs) solve Green’s function . . . . 7
3.2 BI-GreenNet solve Green’s function . . . . 8

Chapter 4 Numerical results . . . . 10
4.1 The Green’s function of the Poisson equation in the unit disc . . . . 10
4.2 The Green’s function of the Poisson equation in the square domain . . . . 14
4.3 The Green’s function of the Poisson equation in the L-shaped domain . . . . 15

Chapter 5 Conclusion . . . . 18

References . . . . 19

參考文獻

[1] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Second Edition. Society for Industrial and Applied Mathematics, 2004.

[2] J. R. Nagel, “Solving the generalized poisson equation using the finite-difference method (fdm),” 2009.

[3] A. Averbuch, M. Israeli, and L. Vozovoi, “A fast poisson solver of arbitrary order accuracy in rectangular regions,” SIAM Journal on Scientific Computing, vol. 19, no. 3, pp.933–952, 1998.

[4] G. Lin, F. Chen, P. Hu, et al., Bi-greennet: Learning green’s functions by boundary integral network, 2022. arXiv: 2204.13247 [cs.LG].

[5] M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations, 2017. arXiv: 1711.
10561 [cs.AI].

[6] M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations, 2017. arXiv: 1711.
10566 [cs.AI].

[7] C. Pozrikidis, A Practical Guide to Boundary Element Methods with the Software Library BEMLIB, 1st. CRC Press, 2002.

[8] G. Lin, P. Hu, F. Chen, et al., Binet: Learning to solve partial differential equations with boundary integral networks, 2021. arXiv: 2110.00352 [math.NA].

[9] 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.

[10] C. Carvalho, S. Khatri, and A. D. Kim, “Asymptotic analysis for close evaluation of layer potentials,” Journal of Computational Physics, vol. 355, pp. 327–341, Feb. 2018.

指導教授

胡偉帆(Wei-Fan Hu)

審核日期

2024-7-18

推文