簡易檢索 / 詳目顯示

研究生: 徐捷耀
Syu, Jie-Yao
論文名稱: 透過計算和深度學習探索非局部奇異擾動微分方程的解
Exploring the Solution of Singular Differential Equation with Non-local Boundary Conditions by Calculating and Deep Learning
指導教授: 李俊璋
Lee, Chiun-Chang
陳人豪
Chen, Jen-Hao
口試委員: 林得勝
Lin, Te-Sheng
吳昌鴻
Wu, Chang-Hong
學位類別: 碩士
Master
系所名稱: 理學院 - 計算與建模科學研究所
Institute of Computational and Modeling Science
論文出版年: 2023
畢業學年度: 111
語文別: 英文
論文頁數: 51
中文關鍵詞: 非局部微分方程奇點深度學習
外文關鍵詞: non-local differential equation, singularity, deep learning
相關次數: 點閱:70下載:0
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報
  • N/A


    In this thesis, we address the issue of boundary influence by interior points in the context of the non-local singularly perturbed equation. On the other hand, we provide an example with exact solutions and employ neural networks to analyze its solutions. Moreover, we identify several potential challenges that may arise. In the first section, we explore computational approaches to estimate the behavior of the solution and its derivatives. In the second section, we investigate the problems that may occur when using neural networks to handle such equations, along with their underlying causes.

    1 Preliminary 6 2 Exploring the Solution of Singular Differential Equation with Non-local Boundary Conditions by Calculating 13 2.1 Primary Idea about Observing The Behavior of v(x) And w(x) Near Boundary Points . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Observing The Behavior of v(x) And w(x) in The Interior Domain 17 2.3 Observing The Behavior of v(x) And w(x) Near Boundary Points 24 2.4 Approximation of Boundary Condition u(0) And u(1) . . . . . . 27 2.5 Approximation of Derivative of Solution u(x) . . . . . . . . . . 31 3 Deep Learning Method 34 3.1 How to Approximate The Solution And What Methods We Give 35 3.1.1 Solving Ordinary Differential Equation by Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Implementation Process for Two Methods . . . . . . . . 36 3.2 Addressing Challenges: Tips and Techniques . . . . . . . . . . . 38 3.2.1 Unstable Solutions Caused by Ill-Conditioning . . . . . . 38 3.2.2 A Challenging Problem Coming From Non-Local Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Optimization Process with Annealing Scheme . . . . . . 41 3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . 42 4 Summary 48

    [1] Bochkarev, A. D. (1985). Solution of a problem with nonlocal boundary
    conditions for an equation of hyperbolic type with a coefficient that has a
    first-order singularity in the interior of the domain. (Russian). Differential
    equations and mathematical physics, 58–62, Kuĭbyshev; Gos. Ped. Inst.
    [2] Chew, W. C., Kong, J. A. (1982). Microstrip capacitance for a circular disk
    through matched asymptotic expansions. SIAM J. Appl. Math. 42, no. 2,
    302–317.
    [3] Diederik P Kingma and Jimmy Ba. (2014). Adam: A method for stochastic
    optimization. arXiv:1412.6980.
    [4] Dong C. Liu, Jorge Nocedal. (2014). On the limited memory BFGS
    method for large scale optimization. Mathematical Programming volume
    45, pages503–528.
    [5] M. Raissi, P. Perdikaris, G.E. Karniadakis. (2019). Physics-informed neural
    networks: A deep learning framework for solving forward and inverse
    problems involving nonlinear partial differential equations. Journal of Computational
    Physics Volume 378, Pages 686-707.
    [6] Olde Daalhuis, A. B., Chapman, S. J., King, J. R., Ockendon, J. R., Tew, R.
    H. (1995). Stokes phenomenon and matched asymptotic expansions. SIAM
    J. Appl. Math. 55, no. 6, 1469–1483.
    [7] Peskoff, A., Eisenberg, R. S., Cole, J. D. (1976). Matched asymptotic expansions
    of the Green’s function for the electric potential in an infinite
    cylindrical cell. SIAM J. Appl. Math. 30, no. 2, 222–239.
    [8] Schult, Daniel A. (2000). Matched asymptotic expansions and the closure
    problem for combustion waves. SIAM J. Appl. Math. 60, no. 1, 136–155.
    [9] Shampine, L.F., and J. Kierzenka. (2001). A BVP Solver based on residual
    control and the MATLAB PSE. ACM Trans. Math. Softw. Vol. 27, Number
    3, pp. 299–316.
    [10] Sifan Wang, Yujun Teng, Paris Perdikaris. (2020). Understanding and
    mitigating gradient pathologies in physics-informed neural networks. A
    PREPRINT.
    [11] Wollkind, David J. (1977). Singular perturbation techniques: a comparison
    of the method of matched asymptotic expansions with that of multiple
    scales. SIAM Rev. 19, no. 3, 502–516.
    [12] Yang Song, Stefano Ermon. (2019). Generative Modeling by Estimating
    Gradients of the Data Distribution. Advances in Neural Information Processing.
    Systems 32.
    [13] Yeonjong Shin1, Jer ome Darbon, and George Em Karniadakis. (2020).
    On the convergence of physics informed neural networks for linear secondorder
    elliptic and parabolic type PDEs. arXiv:2004.01806.
    [14] Zengji Dua, Lingju Kong. (2010). Asymptotic solutions of singularly perturbed
    second-order differential equations and application to multi-point
    boundary value problems. Applied Mathematics Letters 23, 980–983.
    [15] Sukesh Roy. (2020). NVUDIA/modulusL A PyTorch based deep-learning
    toolkit for developing DL models for physical systems.
    Retrieved from https://github.com/NVIDIA/modulus.
    [16] Maximum principle.
    Retrieved from https://people.bath.ac.uk/mw2319/ma40203/sec-1d.html.
    [17] Xianjin Chen, Chiun-Chang Lee, Masashi Mizuno. (2024). Unified asymptotic
    analysis and numerical simulations of singularly perturbed linear differential
    equations under various nonlocal boundary effects. to appear in
    Communications in Mathematical Sciences (40 pages).

    QR CODE