# Machine Learning Applications and Uncertainty Quantification Analysis for Refload Tests
## 2 Methods and Data Preparation
### UQ Analysis  
The best estimate plus uncertainty (BEPU) approach is practical to propagate the physical models' uncertainty and their system states, resulting in a reliable numerical analysis [31; 32; 33; 34]. The results of simulations can be strongly influenced by the selection of physical models and their uncertainty ranges. Therefore, in the validation process of the system code models, UQ analysis can be advantageous to validate the physical models and enhance the simulation results. In this process, the propagation of parameters under their uncertainty ranges and corresponding system states is performed to achieve the specific requirements of model validations.  
The UQ analysis for FLECHT SEASET refloods tests was established using the PA-PIRUS framework and SPACE [25], revealing a slight improvement in the simulation and high computation costs. Subsequently, STARU [26] was developed to examine and enhance the UQ for nonindependent and high-dimension parameters, resulting in a highly non-linear relationship between the physical models and their system states. The acceptance probability \(\beta\) and step size \(\varepsilon\) can be justified in STARU, as can the adjusted multipliers and their uncertainty ranges based on parameters and system responses. This effectively improves the validation process and increases the system code's reliability. STARU will be used in this study to examine the UQ analysis.  
In this implementation, the STARU sampling algorithm (with \(\beta\) = 0.95 and \(\varepsilon\) = 0.01) and the absolute relative difference (ARD) method for accuracy evaluation were adopted. The cladding temperatures and quenching times were deployed as the primary responses for our investigations. The selected physical models and their uncertainty ranges for this UQ analysis are displayed in Table 2, C1 to C42, corresponding to the adjusted parameters in the UQ analysis. Note that the uncertainty bands were recommended by the DA results, and the obtained samples included uniform and continuous samples. The massive samples for training and testing the machine learning models based on these configurations were generated. It is crucial to confirm that these datasets were of high quality and high fidelity to achieve the efficiency of training and validating the machine learning models.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**C**} & \multirow{2}{*}{**Type**} & \multicolumn{2}{c}{**Uncertainty Band**} \\  & & & **(Min to Max)** \\ \hline
1 & & Form loss coefficient for forward flow corresponding to elevation & 0.01–0.24 \\
2 & & Form loss coefficient for reverse flow corresponding to elevation & 0.98–1.55 \\ \hline
3 & & Interfacial friction coefficient between vapor-droplet & 0.99–1.80 \\
4 & & Continuous liquid–vapor interfacial friction factor for bubbly flow & 1.43–2.04 \\
5 & & Continuous liquid–vapor interfacial friction factor for slug flow & 0.35–1.07 \\
6 & & Continuous liquid–vapor interfacial friction factor for annular flow & 0.46–1.05 \\
7 & & Continuous liquid–vapor interfacial friction factor for horizontally & 0.08–0.48 \\  & & & Continuous liquid–vapor interfacial friction factor for inverted & 0.82–1.41 \\
8 & & annular flow & 0.72–1.04 \\
9 & & Continuous liquid–vapor interfacial friction factor for inverted slug flow & 0.72–1.04 \\ \hline
10 & & Droplet entrainment factor & 0.15–0.40 \\
11 & & Droplet de-entrainment factor & 0.22–1.20 \\ \hline
12 & & Single-phase liquid flow regime & 1.45–2.08 \\
13 & & Bubbly flow regime & 0.01–0.52 \\
14 & & slug flow regime & 1.61–2.06 \\
15 & & Bubbly-annular flow regime & 1.08–1.51 \\
16 & & Cap/ slug-annular transition flow regime & 0.59–1.09 \\
17 & & Annular flow regime & 0.62–1.05 \\
18 & & Stratified flow regime & 0.85–1.45 \\
19 & & Stratified–bubbly transition flow regime & 0.46–0.87 \\
20 & & Cap/ slug–stratified transition flow regime & 0.86–1.36 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The selected physical model and uncertainty band for UQ.