當今，數學模型被廣泛的應用在設計越來越複雜的系統并判定這些系統的性能。這些模型旨在通過求解複雜的數學方程來重現物理過程。它們利用可用的計算資源來解出複雜的數學方程，往往在模型的進行運算可能花費數小時甚至數天的時間來計算出所需的數量。主要於模型的參數通常是不確定的，但它們是由數據中被推導出來，透過模型進行概率性數值遞迴分析計算。然而，對於複雜且高度可靠的模型和系統來說，通過數學模型來傳播參數不確定性的代價非常高昂。易於加速求解的過程，從而評估函數的替代模型被用來取代原本的數學模型。另一方面，替代模型的使用引入了額外的模型不確定性，其來源包括：（朱）訓練數據集的變異性，（朲）隨機模型參數，（朳）模型結構，導致錯誤估計了所需的數量。因而，本論文提出了幾個框架來量化模型不確定性的來源。在提出的第一個框架中，模型的不確定性來源於訓練數據集的可變性，即訓練數據輸入域的採樣算法。採樣算法通常傾向於輸入空間的高概率區域採樣，而忽略了低概率區域。這導致了培訓替代模型的重要培訓數據點的遺漏，從而減少了模型的泛化屬性。因此，這種方法的基本原理是從原始的訓練數據集中生成隨機的樣本，用這種引導數據來訓練一組代理模型，從而推斷出從哪裡獲得數據集的總體。隨後，從而展示該框架的有效性，依據計算一個穩健的置信區間，量化上述數量的不確定性的來源，前饋人工神經網絡（杆杆札杁李李）被作為代替模型用來測試了該框架的各種分析性能及其對一個坐落在塞拉菲爾德核廢料站的昂貴的放射性廢物管理的不確定性量化模型（杓杩杴来
杉杯杮来杘杣杨条杮杧来材杬条杮杴木杓杉杘杅材朩）的量化。第二個框架提出了一種通過模型參數的隨機波動來量化不確定性的方法。具體地說，當一個特定的代理模型被相同的訓練數據反復精化時，從而產生不同的模型。通常情況下，根據某些性能指標選擇最好的模型并捨棄其餘部份是一種常見的做法。但這種做法有幾個問題，最明顯的是會浪費計算資源。此外，由於測試數據中存在的隨機不可預期的變量，每個模型的性能指標都有偏差。因此在一個測試中表現良好的模型可能對另外的數據集產生非所預期的結果。因而，基於隨機波動模型參數而確定最佳表現模型存在不確定性。了量化模型的不確定性，這個框架中提出的方法結合了基於貝葉斯統計和模型平均技術的整合模型。與前述的方法一樣，不同的分析例子的適用性也被測試過。此外，本論文採用該前饋人工神經網絡對替代模型杓杉杘杅材和某核電站的故障診斷進行了分析。第三個框架提出了一種量化源自模型結構的模型不確定性的方法。通過把一個旨在于指定空間內定位全局最優化模型結構的多目標優化問題考慮在內，該框架對先前的框架進行了擴展。再次，對該方法的適用性進行了幾個分析實例的測試，並對核電站的故障診斷進行。

I would like to express my sincere gratitude towards my supervisors Dr Edoardo Patelli and Professor Rong-Jiung Sheu for giving me the opportunity to take part in the rst dual PhD program between the University of Liverpool and National Tsing-Hua University Taiwan. This has been an amazing lifetime experience for me as I have had the opportunity to work in dierent multi-cultural research groups, broaden my network, and expand the visibility of my research. In particular, Edoardo Patellis guidance and support have been crucial to make my work visible on a global scale and for developing key academic partnership by giving me the opportunity to attend a variety of international conferences and workshops. Also, his personality and sense of humour made my research unique and entertaining. I am very grateful to him. Professor Rong-Jiun Sheu has been more than just a supervisor, he has given me fraternal support, constant guidance, and has taken me on a fantastic journey upon my arrival
in Taiwan, integrating me eortlessly into his research group. I am very grateful to him for having taken me on board. I also acknowledge Matteo Broggi's help, as his understanding of computational analysis and his programming abilities considerably helped me in gaining familiarity with Matlab and OpenCossan. I would like to thank the National Nuclear Laboratory (NNL) for providing a realistic case study to demonstrate the applicability of the proposed approaches developed in this thesis. I am also very grateful to my colleagues at the Institute of Risk and Uncertainty and friends, who proof-read and signicantly improved the presentation of this thesis. I am also grateful to Karen who helped me translate my abstract into traditional mandarin anguage. Special thanks to my family, who have always motivated me to push harder during dicult times. Finally, I give God the glory for giving me the wisdom, knowledge and understanding to complete the research work presented in this thesis.

1 Introduction... 1
1.1 Context...1
1.2 General Framework for Uncertainty Quantication in this Thesis ...3
1.3 Problem Statement... 4
1.4 Objectives of the Thesis... 4
1.5 Original Contributions... 5
1.6 Numerical Implementation...5
1.7 Outline of Thesis... 6
2 Modelling and Quantication of Parameter Uncertainties... 8
2.1 Modelling Aleatory Uncertainty... 8
2.1.1 Probability Theory... 8
2.1.1.1 Data to Cumulative Distribution Function... 9
2.2 Quantication of Parameter Uncertainties... 10
2.2.1 Reliability Analysis of Systems... 10
2.2.1.1 Estimation of the Failure Probability by means of
Simulation ...11
2.2.2 Sensitivity Analysis of Systems...13
2.2.2.1 Estimating Sobol' Indices by means of Monte Carlo
Simulation ... 15
2.3 Chapter Summary... 16
3 Classical Surrogate Models...17
3.1 State of the Art... 17
3.2 Background to Neural Networks... 18
3.2.1 Articial Neural Network ... 19
3.2.1.1 The Articial Neuron...19
3.2.2 Deep Learning with Articial Neural Networks ... 22
3.2.2.1 Convolution Neural Networks . . . . . . . . . . . . . 22
3.2.2.2 Recurrent Neural Networks . . . . . . . . . . . . . . 23
3.2.2.3 Innite Impulse Response-Locally Recurrent Neural
Network...24
3.2.3 Uncertainty in Articial Neural Network Computation... 25
3.2.3.1 Uncertainty from Sampling Variability in Training Data
Set...25
3.2.3.2 Uncertainty from ANN Weight Parameters ... 25
3.2.3.3 Uncertainty from the Model Structure ... 25
3.3 Chapter Summary... 26
4 Robust Surrogate Models - Variability in Training Data.... 28
4.1 Background to the Bootstrap Technique ... 28
4.2 Succiant Theory of Reliability and Sensitivity Analyses ... 29
4.2.1 Reliability Analysis... 29
4.2.2 Sensitivity Analysis... 29
4.2.3 Modelling of Articial Neural Network for Reliability and Sensitivity Analysis ... 30
4.2.4 Variability in Training Data Set...31
4.2.5 Adaptive Bootstrap Algorithm for Surrogate Models...31
4.2.5.1 Criterion for Selecting the Number of Bootstrap Models
to be Constructed ...34
4.3 Case Study... 36
4.3.1 Case Study 1: The Four Branch Function ... 36
4.3.1.1 Analysis...36
4.3.1.2 Results ... 37
4.3.2 Case Study 2: The Ishigami Function ... 39
4.3.2.1 Analysis...40
4.3.2.2 Results...40
4.4 Chapter Summary ... 41
5 Robust Surrogate Models - Random Model Parameters... 43
5.1 Background Theory of Proposed Approach...44
5.1.1 Bayesian Model Selection for Identical Trained Articial Neural Networks...44
5.1.2 Robust Articial Neural Network Prediction... 46
5.1.3 Condence Interval for Robust Estimate ... 47
5.1.3.1 Criterion for Selecting the Number of Identical Networks
to be Constructed . . . . . . . . . . . . . . . . 48
5.1.4 Adaptive Procedure for Robust Articial Neural Network Training... 48
5.2 Case Study... 51
5.2.1 Case Study 1: The 2-D Non-Linear Function ... 51
5.2.2 Case Study 2: The Rosenbrock Function... 54
5.3 Chapter Summary... 56
6 Robust Surrogate Models - Model Structure and Random Model
Parameters... 58
6.1 Background to Problem ...58
6.2 Proposed Approach... 59
6.2.1 Formulation of Optimization Problem for Optimal ANN Architecture and Training Sample Size Selection ... 59
6.2.1.1 Searching Optimal ANN Solutions with Evolutionary
Algorithm ... 62
6.2.1.2 Encoding the Chromosome ...62
6.2.1.3 Procedures Taken to Search for Optimal Network Architectures... 63
6.2.1.4 Ensemble of Optimal Networks... 64
6.2.2 Bayesian Model Selection for Matrix Consisting of Identical ANNs ...64
6.2.2.1 Robust Prediction from Articial Neural Networks.... 66
6.2.2.2 Condence Interval for Robust Neural Network Prediction
... 67
6.2.3 Model Averaging for the Ensemble of Robust Neural Networks... 68
6.2.3.1 Combination of the Robust Networks Condence Intervals
... 69
6.2.3.2 Criterion for Selecting the Number of Identical Networks
to be Constructed ... 69
6.2.3.3 Adaptive Procedure for Optimized Robust ANN Training
... 69
6.3 Case Study... 72
6.3.1 Case Study 1... 72
6.3.1.1 Analysis ... 72
6.3.1.2 Results ... 72
6.3.2 Case Study 2 ... 76
6.3.2.1 Analysis... 77
6.3.2.2 Results...77
6.4 Chapter Summary ... 80
7 Uncertainty Quantication of the Site Ion eXchange Plant Using
Robust Articial Neural Networks 81
7.1 Overview ... 81
7.1.1 Computational Model of the SIXEP ...82
7.1.1.1 Modelling the Uncertainty in the Model Input Parameters
... 85
7.2 Reliability Analysis of the SIXEP Using Robust Articial Neural Networks... 85
7.2.1 Analysis... 86
7.2.2 Results... 87
7.3 Sensitivity Analysis of the SIXEP Using the Proposed Approaches... 88
7.3.1 Analysis ... 89
7.3.2 Results ... 89
7.4 Conclusion... 92
8 Fault Diagnosis of a Nuclear Power Plant Using Robust Articial
Neural Networks... 93
8.1 Background to Problem ... 93
8.1.1 Case Study: The Indian Pressurized HeavyWater Reactor (PHWR)... 94
8.1.2 Aim and Objectives of Chapter ...95
8.1.3 Data Set for Training Predictive Model ...95
8.2 Fault Diagnostic by Robust ANN...98
8.2.1 Case 1 .. 98
8.2.2 Predicting Blind Case Data ...103
8.2.3 Case 2 ...104
8.2.3.1 Adapting the IIR-LRNN to the Multi-Objective Optimization
Framework... 104
8.2.4 Case 3 ... 107
8.2.5 Proposed Framework for Combining Chapter 4 and 6 Approaches...107
8.3 Chapter Summary ...110
9 Uncertainty Analysis of Spectral Correction Schemes in High-Energy
Environments 111
9.1 Problem Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.1.1 Neutron Detectors for Measuring High-Energy Neutrons . . . 112
9.1.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.1.3 Materials and Method . . . . . . . . . . . . . . . . . . . . . . 113
9.1.3.1 Bonner Spheres and Neutron Dose Meters . . . . . . 113
9.1.3.2 Neutron Spectra and Dose Correction Factors . . . . 115
9.1.4 Results and Discussion ... 117
9.1.4.1 Characterising the Neutron Field ... 117
9.1.4.2 General Trends in Spectral Correction Factors ...118
9.1.4.3 Neutron Calibration Sources and Spectral Correction
Factors ... 121
9.1.4.4 Neutron dose meters and spectral correction factors ... 122
9.1.5 Validation of the Proposed Models ... 124
9.2 Uncertainty Analyses of the Spectral Correction Schemes ... 127
9.2.1 Proposed Approach for Quantifying Uncertainty in Spectral
Correction Schemes... 128
9.3 Chapter Summary ... 136
10 Conclusions and Recommendations ...138
10.1 Summary ... 138
10.1.1 Research Contributions... 139
10.1.2 Applications...140
10.1.3 Future Work ...141

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