# 臺灣博碩士論文加值系統

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 我們應用機器學習方法解偏微分方程。我們藉由徑向基底函數神經網路表達解。這篇文章提出新的網路架構改進徑向基底函數神經網路的表達能力。我們將高斯徑向基底函數轉成橢圓基底函數。此新的網路結構稱為橢圓基底函數神經網路。我們應用此網路解偏微分方程。在靜態偏微分方程方面，我們解邊界層問題和震盪問題。在動態偏微分方程方面，我們解Schrödinger 方程和Allen–Cahn 方程。在數值結果上，我們展現橢圓基底函數神經網路可以捕捉解的局部結構，表現的比一般的徑向基底函數神經網路好。
 We apply machine learning techniques to solve partial differential equations. We use radical basis function neural networks (RBFNNs) to express the solutions. This thesis proposes the new network architecture to improve the expressive ability of RBFNNs. We turn the Gaussian radial basis function into the elliptic basis function. This new network architecture is called an elliptic basis function neural network (EBFNN). We apply networks to solve the equations. For time-independent cases, we solve boundary layer problems and oscillated problems. For time-dependentcases, we solve the Schrödinger equation and the Allen–Cahn equation. In the numerical results, we show that the EBFNNs capture the local structures of the solutions, better than the regular RBFNNs.
 摘要iAbstract iiContents iiiFigure lists ivTable lists v1 Introduction 12 Methodology 32.1 Latin hypercube sampling 32.2 Automatic differentiation 32.3 PINNs 32.3.1 Continuous-time model 42.3.2 Discrete-time model 52.4 Optimizer 62.5 Network structures 82.5.1 Fully-connected neural network 82.5.2 Radial basis function neural network 83 Elliptic basis function neural networks 114 Numerical results 124.1 Poisson equations 124.2 Oscillatory solutions 154.3 Singularly perturbed equations 184.4 Time-dependent equations 235 Conclusion 27References 28
 [1] K. Hornik, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359–366, 1989.[2] G. E. K. Maziar Raissi, Paris Perdikaris, “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.[3] B. Y. Weinan E, “The deep ritz method: A deep learning-based numerical algorithm for solving variational problems,” Communications in Mathematics and Statistics, vol. 6, 09 2017.[4] M. Stein, “Large sample properties of simulations using latin hypercube sampling,” Naval Research Logistics Quarterly, vol. 29, pp. 143–151, 1987.[5] A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” 2015.[6] W.F. Hu, T.S. Lin, and M.C. Lai, “A discontinuity capturing shallow neural network for ellipticinterface problems,” ArXiv, vol. abs/2106.05587, 2021.[7] J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Computation, vol. 3, no. 2, pp. 246–257, 1991.[8] D. Pepper, “Meshless methods for PDEs,” Scholarpedia, vol. 5, no. 5, p. 9838, 2010.[9] W. Chen, Z.J. Fu, and C. Chen, Different Formulations of the Kansa Method: Domain Discretization, pp. 29–50. 11 2014.[10] P. R. Amuthan A. Ramabathiran, “Sparse, physics-based,and partially interpretable neural networks for pdes,” Journal of Computational Physics, vol. 445, p. 110600, 2021.[11] P.W. Hsieh, Y.T. Shih, S.Y. Yang, and C.S. You, “A novel technique for constructing difference schemes for systems of singularly perturbed equations,” Communications in Computational Physics, vol. 19, 05 2016.[12] R. J. LeVeque, “Finite difference methods for ordinary and partial differential equations,” pp. 43–45.
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 1 用李茲法結合淺層神經網路解偏微分方程 2 利用有物理根據的淺層神經網路解曲面上的偏微分方程

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