## 3 Results
### The Performance of ROM for Nanofluid Temperature Field Prediction  
After training with 1800 data points, the temperature field predictions of the outlet cross section with random fin size and position in the validation dataset are executed. The prediction performance of the proposed model is evaluated by the MAE loss, computed according to Equation (5). Three representative results are chosen with the rectangular height, width, and adjacent fin angle [\(h\), \(w\), \(\alpha\)] = [12 mm, 5 mm, 45\({}^{\circ}\)], [12 mm, 6 mm, 18\({}^{\circ}\)], [10 mm, 21 mm, 52\({}^{\circ}\)], as presented, respectively, in Figure 9a-c. The ROM prediction, numerical simulation, and MAE loss are presented respectively in the first, second, and third columns. As can be seen, the CNN-based ROM learns the mechanism of heat transfer from heated absorber tube to the working nanofluid. The temperature is highest in the lower part of the absorber wall, which is reasonable with our heat flux setting. Fromthe lower part to the center and top part of the absorber tube, the temperature decreases. Furthermore, a great temperature gradient occurs on the contact surface of the solid tube and the working fluid. From the results in the third column, the largest relative error is about 1.9%, which only appears on the middle fin's contact surface in the third case. Most areas in the cross section show an error value less than 0.2%, leading to the high average accuracy value of 99.935%. All the results above demonstrate that the proposed CNN-based ROM has excellent performance for field prediction of nanofluid-filled absorber tubes with various rectangular fin sizes and positions.  
Additionally, we carry out the model analysis on learning rate, a hyperparameter controlling how quickly the model is updated to the solution. A relatively large learning rate ensures the training within a reasonable time, while a relatively small learning rate helps the convergence of the solution. Therefore, a larger learning rate is usually applied in the early stage of training to accelerate the computation. On the contrary, a relatively small learning rate is employed at the end of training to make the training converge in the desired point. In this study, we set the learning rate multiplied by 0.1 per 10 epochs, to achieve a satisfactory learning rate at both the beginning and end of training. Figure 10 shows the good performance of the strategy of changeable learning rate (abbreviated as \"Ir\" in the figure) in the field prediction of the nanofluid-filled absorber tube. All MSE loss curves  
Figure 9: Temperature field predictions of three representatives with various fin heights, widths, and positions: (**a**) [\(h\), \(w\), \(\alpha\)] = [12 mm, 5 mm, 45\({}^{\circ}\)]; (**b**) [\(h\), \(w\), \(\alpha\)] = [12 mm, 6 mm, 18\({}^{\circ}\)]; (**c**) [\(h\), \(w\), \(\alpha\)] = [10 mm, 21 mm, 52\({}^{\circ}\)].  
with various initial learning rates sharply decrease before 10 epochs; the greater learning rate value shows higher convergence speed. Then after 10 epochs, we multiply the learning rate by 0.1 for every 10 epochs; after that, the curves start to oscillate and converge to the similar level. For the learning rate values of \(5\times 10^{-7}\), \(2\times 10^{-7}\), and \(5\times 10^{-8}\), the average accuracies in the validation set are 0.9967, 0.9966, and 0.9951, respectively. Therefore, the \(5\times 10^{-7}\) initial learning rate gives the highest accuracy and the fastest calculation speed, and the strategy of 90% reduction in learning rate per 10 epochs is applied in this study.