臺灣博碩士論文加值系統

神經網路的架構是機器學習過程中很重要的工具之一。而DeepONet 神經
網路是機器學習的一個新方法，我們主要關心DeepONet 的結構。許多機器學習的問題並不能一開始就立刻訓練出如預期結果的函數。實際上，大部分情況我們無從得知滿足條件的解到底存不存在，或者我們到底是不是訓練出這個問題的真解。一般來說，如果得到的結果不理想，我們就改變一些參數、增加或減少一些layors，然後重新訓練，直到訓練出預期的結果為止。
　　在本論文中，我們以微分算子為例。我們主要使用一些數值分析的方法，比如Sepctral Theorem 以及Lagrange 插值多項式來構造一個算子的近似函數，使這個近似函數構造的過程滿足神經網路的結構。如此一來，我們不僅說明了滿足此神經網路問題的函數解真實存在，同時也說明了利用傳統的數值方法來構造函數也能有不錯的近似效果。
　　在本論文的最後，我們利用實際的例子來做模擬。我們也探討了當給定的sensors 增加或減少，對於整體近似的效果的影響，以及當sensors 的離散程度改變時，對近似函數的影響如何。

The architecture of neural networks is one of the most important tools in the process of machine learning. The DeepONet neural networks are a new approach to machine learning, and we are mainly interested in the structure of DeepONet. Many machine learning problems cannot be trained immediately at the beginning to produce functions with the expected results. In fact, in most cases we have no way to know whether the solution satisfying the condition exists or not, or whether we have trained the true solution of the problem or not. If the result is not satisfactory, we change some parameters, add or subtract some layers, and retrain until we get
the expected result.
In this thesis, we take differential operators as an example. We mainly use numerical analysis methods such as Spectral Theorem and Lagrange interpolation polynomials to construct an approximation function for the operator so that the process of constructing the approximation function satisfies the structure of the neural network. In this way, we not only show the existence of functional solutions to this neural network problem, but also show that it is possible to construct functions using the conventional numerical methods with good approximation results. At the end of this thesis, we will demonstrate some practical examples. We also investigate the effect of increasing or decreasing the number of sensors on the overall approximation effect, and how the approximation function is affected when the dispersion of sensors changes.

摘要 i
Abstract ii
誌謝 iv
Content v
List of Figures vii

1 Introduction 1
2 Interpolating Polynomials 3
2.1 Lagrange Interpolation 3
2.2 Hermite Interpolation 6
3 Methods of Numerical Differentiation 9
3.1 The Finite Difference Method 9
3.1.1 Truncation errors 11
3.1.2 Example and Numerical Results 12
3.2 The Richardson Extrapolation 15
3.3 The Spectral Method 21
3.4 Comparison of different methods 24
4 Neural Network 27
4.1 The Universal Approximation Theorem 27
4.2 Construction of Approximations 28
4.3 Numerical Results 33
5 Conclusion and Future Work 38
Bibliography 40

[1] Annette M. Burden, Richard L. Burden, and J. Douglas Faires. Numerical Analysis, 10th ed., pages 109–110, 134–135. Cengage, 2016.
[2] Catherine F. Higham and Desmond J. Higham. Deep learning: An introduction for applied mathematicians. SIAM Review, 61(3):860–891, jan 2019.
[3] Randall L. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equation: steady-state and time0pedendent problems, pages 3–7. Society for industrial and Applied Mathematics, 2007.
[4] Lu Lu, Pengzhan Jin, Guofei Pang, Handy Zang, and George Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3:218–229, 03 2021.
[5] Jengnan Tzeng. Linear regression to minimize the total error of the numerical differentiation. East Asian Journal on Applied Mathematics, 7:810–826, 11 2018.