臺灣博碩士論文加值系統

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-dependent
cases, 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i
Abstract ii
Contents iii
Figure lists iv
Table lists v
1 Introduction 1
2 Methodology 3
2.1 Latin hypercube sampling 3
2.2 Automatic differentiation 3
2.3 PINNs 3
2.3.1 Continuous-time model 4
2.3.2 Discrete-time model 5
2.4 Optimizer 6
2.5 Network structures 8
2.5.1 Fully-connected neural network 8
2.5.2 Radial basis function neural network 8
3 Elliptic basis function neural networks 11
4 Numerical results 12
4.1 Poisson equations 12
4.2 Oscillatory solutions 15
4.3 Singularly perturbed equations 18
4.4 Time-dependent equations 23
5 Conclusion 27
References 28

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