| 姓名 |
林靖旻(Jing-Min Lin)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
一種計算曲率的類神經網路方法
(A Neural Network Approach for Computations of Curvature)
|
| 相關論文 | |
| 檔案 |
 [Endnote RIS 格式]
[Bibtex 格式]
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| 摘要(中) |
本文旨在討論使用類神經網路逼近由離散點構成的曲線或曲面的方法，並期望通
過這種逼近結果來估計目標函數的曲率。在這個研究裡，我們發現一般架構的神經網
路在逼近週期函數的過程中，受限於輸入的資料以及架構的侷限，並不能很好的逼近
週期函數。為此，我們提出一種想法，將輸入資料映射至圓的參數方程式，再送入神
經網路進行訓練，使得神經網路具備週期函數的特徵，藉此能夠更好的逼近週期函數
以及極座標系的曲線。此外，基於這個想法，我們還嘗試將輸入資料映射至球的參數
方程式，以此來逼近球座標系的曲面。而在最後，我們測試並且紀錄了不同架構的神
經網路逼近目標函數的表現，以及再計算曲率後的結果，以期在未來透過神經網路處
理這類問題時，能夠更精準的使用適合的神經網路。 |
| 摘要(英) |
This thesis aims to discuss the methods of using neural networks to approximate curves
or surfaces composed of discrete points and estimate the curvature of the target function
through this approximation. In this study, it was found that normal shallow neural
network structures are limited in their ability to approximate periodic functions due to
constraints imposed by the input data and network structure. To address this limitation,
a new idea is proposed, which involves mapping the input data to the parameter equation
of a circle and training the neural network using this transformed data. This approach
allows the neural network to possess the characteristics of periodic functions, enabling
better approximation of periodic functions and the curves in polar coordinate system.
Additionally, based on this idea, we try to map the input data to the parameter equation
of a sphere to approximate surfaces in a spherical coordinate system. Finally, we tested
and trained different network structures for their performance in approximating the target
function and calculating the curvature. The aim is to have more precise utilization of
suitable neural networks in the future when dealing with similar problems through neural
network processing. |
| 關鍵字(中) |
★ 類神經網路
★ 曲率 |
關鍵字(英) |
★ Neural Network
★ Curvature |
| 論文目次 |
摘要 v
Abstract vi
Contents vii
1 Introduction 1
2 Method 3
2.1 Construct Neural Network .................................................... 3
2.2 Approximation of Curve ...................................................... 5
3 Extend to Surface 8
3.1 Ellipsoid ....................................................................... 9
3.2 Spherical Harmonics .......................................................... 10
4 Numerical Results 12
5 Conclusion 15
Bibliography 16 |
| 參考文獻 |
Bibliography
[1] H. L. França and C. M. Oishi, “A machine learning strategy for computing interface curvature in front-tracking methods,” Journal of Computational Physics, vol. 450, p. 110 860,
2022.
[2] W.-F. Hu, T.-S. Lin, and M.-C. Lai, “A discontinuity capturing shallow neural network for
elliptic interface problems,” Journal of Computational Physics, vol. 469, p. 111 576, 2022. |
| 指導教授 |
胡偉帆(Wei-Fan Hu)
|
審核日期 |
2023-8-15 |
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