# Prediction of Boiling Heat Transfer Coefficient for Micro-Fin Using Mini-Channel
## 3 Databases and Methods
### Method
Figure 7: Prediction using micro-fins and consideration of physical mechanisms (Case 3, Present). \[We=\frac{G^{2}d}{\rho_{m}\sigma^{\prime}},\ \ \rho_{m}=\frac{1}{\left(\frac{x}{ \rho_{V}}+\frac{1-x}{\rho_{L}}\right)} \tag{6}\]  
\[Re_{V}=\frac{Gxd}{\mu_{V}} \tag{7}\]  
\[Re_{L}=\frac{G(1-x)d}{\mu_{L}} \tag{8}\]  
Figure 9 illustrates the structure of the learning process through fine-tuning. Fine-tuning involves using the weights of a pre-trained model as the initial values (in this paper, based on the mini-channel data) and constructing a new model by re-training the entire model using the micro-fin data. In the training process, a prediction function \(f(x)\) is learned, where \(x\) represents the input data, and \(f(x)\) returns the predicted heat transfer coefficient and the corresponding uncertainty. In the prediction process, based on the target input \(x^{*}\), the heat transfer coefficient and the uncertainty are output. There are three steps in the training phase. In the first step, a DNN performs preliminary training using standardized mini-channel data. The training data consists of physical conditions, experimental conditions, and the corresponding heat transfer coefficients. Equation (9) shows the method for standardizing the mini-channel training data \(X_{c}=\{x_{c1},\ldots,x_{cn}\}\) and the micro-fin training data \(X\!f=\{x_{\text{f1}},\ldots,x_{\text{fn^{\prime}}}\}\). The training data uses 30 different physical quantities, and because their values vary significantly, standardization of the data is performed as preprocessing to improve the model's accuracy. Here, \(x_{\text{i}}\) represents the \(i\)th sample of the training dataset. \(x_{\text{i}}\) comprises 30 conditions (15 physical conditions, ten experimental conditions, and five experimental values). Let \(x_{ij}\) represent the value of the \(j\)th condition of the \(i\)th sample. For each \(i\) and \(j\), we calculate the value. For each \(i\) and \(j\), Mean(\(\cdot\)) and Std(\(\cdot\)) represent the mean and standard deviation of \(\cdot\). Here, let us denote the standardized mini-channel data as \(\hat{X}_{c}=\{\hat{x}_{c1},\hat{x}_{c2},\ldots,\hat{x}_{cn}\}\)  
Figure 9: Model structure of training phase using pre-training and fine-tuning, with consideration of physical mechanisms (Case 3, Present). and \(\hat{x}_{i}=\{x_{c,i,1},\hat{x}_{c,i,2},\ldots,\hat{x}_{c,i,30}\}\), respectively. Similarly, denote the micro-fin data as \(\hat{X}_{f}=\left\{\hat{x}_{f1},\hat{x}_{f2},\ldots,\hat{x}_{fu^{\prime}}\right\}\) and \(\hat{x}_{fi}=\left\{\hat{x}_{fi,1},\hat{x}_{fi,2},\ldots,\hat{x}_{fi,30}\right\}\), respectively. \[\hat{x}_{ij}\ \leftarrow\ \frac{x_{ij}-Mean\big{(}\{x_{1j},\ldots,x_{nj}\} \big{)}}{Std\big{(}\{x_{1j},\ldots,x_{nj}\}\big{)}} \tag{9}\]  
In the second step of the training phase, before fine-tuning based on micro-fin data, the weight \(w_{1,1}^{(\mathcal{L})}\), \(w_{2,1}^{(\mathcal{L})},\ldots,w_{L-1,1}^{(\mathcal{L})}\) and \(\mathfrak{b}_{1}^{(\mathcal{L})}\) are initialized. The weights are connected to the final layer of the DNN, which was trained based on mini-channel data. After that, the DNN model, with the weights connected to the final layer initialized, is retrained (fine-tuning) using micro-fin data based on the \(L^{L-1}\). In the third step of the training phase, GPR is trained based on the output of layer \(L^{L-1}\) of the fine-tuning model and the corresponding heat transfer coefficient \(y\). We do not use the final output value of the fine-tuning model but instead use \(y^{L-1}\). In the prediction phase, the samples needed to determine the heat transfer coefficient are first standardized based on Equation (10). \[Bo =\frac{(\rho_{L}-\rho_{V})gd^{2}}{\sigma} \tag{4}\]0  
Continuing, the GPR outputs the predicted heat transfer coefficient and the corresponding uncertainty for the target sample based on \(y^{L-1}\). The training and prediction processes are repeated q times, where q is a predefined hyperparameter. Finally, the results obtained from the repeated iterations (q times) are compared.