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| 研究生： |
黃振維 Huang, Chen-Wei |
|---|---|
| 論文名稱： |
利用神經網路解微分方程 Neural Network Methods for Solving Differential Equation |
| 指導教授： | 符聖珍 |
| 口試委員: |
符聖珍
曾睿彬 李宣緯 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2019 |
| 畢業學年度： | 107 |
| 語文別： | 中文 |
| 論文頁數： | 44 |
| 中文關鍵詞： | 微分方程 、神經網路 |
| DOI URL： | http://doi.org/10.6814/NCCU201900919 |
| 相關次數： | 點閱：66 下載：26 |
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本文是在敘述利用前饋人工神經網路的數值方法去近似微分方程的解，其中分別利用邊界條件或是初始條件去造出試驗函數去讓神經網路去近似，或是試驗函數不隱含初始條件或邊界條件，直接把初始條件與邊界條件當作神經網路的目標函數的優化條件，利用SGD和ADAM優化器去更新神經網路參數，再分別做比較。
其中在常微分方程分別去試驗了邊界值問題、特徵值問題、初始值問題、生態系統、及三種經典的偏微分方程，依照不同的方法去滿足不同的條件，進一步的去降低數值解的誤差。
This paper descirbes how to use the feed forward artificial neural network method to find the approximate solution of differential equations. Two types of the trial funcitons are used, and the objective function is minimized by SGD and ADAM methods respectively.
We test the boundary value problem, eigenvalue problem, initial value problem, two types of the ecological systems, and three classical types of the partial differential equations. We illustrate some examples and give some comparison results in Chapter 4.
Contents
致謝 i
中文摘要 ii
Abstract iii
Contents iv
List of Tables vi
List of Figures vii
1 Introduction 1
2 Feed Forward Artificial Neural Network 3
2.1 Architecture 3
2.2 Mathematical Model of Artificial Neural Network 4
2.3 Activation Function 4
2.4 Optimizers 5
2.4.1 Stochastic Gradient Decent(SGD) 5
2.4.2 Adaptive Moment Estimation(ADAM) 5
3 Objective Functions and Algorithm 7
3.1 Trial function method of type1 7
3.2 Trial function method of type2 8
3.3 Algorithm 8
4 Using Neural Network Method to Solve Differential Equations
4.1 Boundary Value Problem 9
4.1.1 Construction of the trail function and objective function 10
4.1.2 Select the number of hidden layers 13
4.1.3 Select the optimizers of the neural network 14
4.2 The Eigenvalue Problem 14
4.3 Initial Value Problem 21
4.3.1 Lane-Emden Equation 21
4.4 Systems of Differential Equations 26
4.4.1 Rabbit versus Sheep Problem 26
4.4.2 Lokta-Volterra Model 29
4.5 Partial Differential Equations 33
4.5.1 Laplace’s Equation 33
4.5.2 Heat Equation 36
4.5.3 Wave Equation 39
Bibliography 43
Bibliography
[1] Ravi P Agarwal and Donal O’Regan. An introduction to ordinary differential equations. Springer Science & Business Media, 2008.
[2] Jerrold Bebernes and David Eberly. Mathematical problems from combustion theory,volume 83. Springer Science & Business Media, 2013.
[3] Richard L Burden and J Douglas Faires. Numerical analysis(7th). Brooks/Cole, 2001.
[4] Matt Curnan, Siddharth Deshpande, Hari Thirumalai, Zhaofeng Chen, John Michael, et al. Solving odes with a neural network and autograd. https://kitchingroup.cheme.cmu.edu/blog/2017/11/28/Solving-ODEs-with-a-neural-network-and-autograd/.
[5] Vivek Dua. An artificial neural network approximation based decomposition approach for parameter estimation of system of ordinary differential equations. Computers & chemical
[6] Ji-Huan He. Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation, 114(2-3):115–123, 2000.
[7] Hamid A Jalab, Rabha W Ibrahim, Shayma A Murad, Amera I Melhum, and Samir B Hadid. Numerical solution of lane-emden equation using neural network. In AIP Conference Proceedings, volume 1482, pages 414–418. AIP, 2012.
[8] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
[9] Hans Petter Langtangen and Hans Petter Langtangen. A primer on scientific programming with Python, volume 6. Springer, 2011.
[10] Sashank J Reddi, Satyen Kale, and Sanjiv Kumar. On the convergence of adam and beyond. arXiv preprint arXiv:1904.09237, 2019.
[11] Shagi-Di Shih et al. The period of a lotka-volterra system1. Taiwanese Journal of Mathematics, 1(4):451–470, 1997.
[12] Steven H Strogatz. Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, 2018.
[13] Luma NM Tawfiq and Othman M Salih. Design feed forward neural network to solve eigenvalue problems with dirishlit boundary conditions. Int. J. Modern Math. Sci, 11(2):58–68, 2014.
[14] Neha Yadav, Anupam Yadav, Manoj Kumar, et al. An introduction to neural network methods for differential equations. Springer, 2015.
