## 3 Results
### The Performance of ROM for Heat Flux Prediction  
The results of Section 3.1 show that the proposed ROM enables the reconstruction of the temperature field of the whole cross section of the nanofluid-filled absorber tube, which allows us to clearly see the temperature distribution in the nanofluid area and tube area, in order to alarm us if there is a locality with high temperature. According to our literature review, no researchers evaluate heat transfer performance directly from the physical field. Therefore, the second ROM is proposed to predict the scalar performance parameter of the nanofluid-filled absorber tube. Useful heat production, as one of the most important physical quantities for PTC absorber tube, measures the heat captured by the working fluid passing through the absorber tube. It can be calculated using the energy balance in the flow volume, equal to the heat variation of the inlet and outlet cross section:  
\[Q_{u}=(q_{out}-q_{in})\times t \tag{14}\]  
where \(t\) presents the time, and the expression of the heat flux in a cross section is defined as:  
\[q=\iint\limits_{A}cTu\cdot dA \tag{15}\]  
where \(A\) represents the cross-sectional area, \(c\) represents the specific heat capacity of the working fluid, \(u\) is the flow velocity on the cross section, and \(T\) is the temperature. The temperature, velocity of the inlet section, and the fluid cross-sectional area for each type of finned absorber tube are fixed. Therefore, to obtain \(Q_{u}\), \(q_{out}\) is the only scalar that needs to be predicted. Thus, the second ROM is designed to predict the outlet heat flux \(q_{out}\). In the following three figures, from Figures 11-13, we present the comparison of the predicted outlet heat flux \(q_{out}\) and the true value. Furthermore, the size of the datasets has been investigated. Figure 10: Accuracy evolution of temperature field on training and validation dataset with various learning rates. Figure 11: Coefficient of determination R\({}^{2}\) of 81 samples in test dataset with four sizes of datasets: (a) 500 data points; (b) 1000 data points; (c) 1500 data points; (d) 2000 data points. Figure 12: Error distribution density maps of 81 samples in test dataset with four dataset sizes: (a) 500 data points; (b) 1000 data points; (c) 1500 data points; (d) 2000 data points. It is well known that the size of datasets plays an important role in training: a small training dataset size generally leads to a poor approximation, while a big dataset size requires more training time, and sometimes it is expensive to generate raw data or the data are just limited. In our study, numerical simulation in OpenFOAM is applied to generate training data. Thereby, repeatedly solving large numbers of partial differential equations consumes huge computational resources. Therefore, we attempt to find the most appropriate dataset scale to avoid the waste of the computational resources. Figure 11a-d display the comparison of outlet heat flux \(q_{out}\) between ROM prediction and numerical simulation with four sizes of dataset: 500, 1000, 1500, and 2000 data points, respectively. The 81 red points present sampled results; the blue line illustrates the regression line. All four regression lines almost coincide with the line y = x, which means that the predicted results with all four sizes of datasets agree well with the CFD results. Furthermore, we evaluate our ROM with the coefficient of determination R\({}^{2}\), which measures how well the proposed model predicts the outlet heat flux. All four R\({}^{2}\) values are higher than 0.999, and the differences are very small; the peak value occurs with the dataset of 1000 and 1500. The error distribution density maps of four size datasets with 81 samples are presented in Figure 12, indicating that the ROM trained with 1000 data points shows the lowest number of outliers.