# Heat Conduction Plate Layout Optimization Using Physics-Driven Convolutional Neural Networks
## 5 Prediction and Optimization
### Position Optimization in the Multiple-Holes Cases  
For the temperature field with two masks, each area is fixed as \(20\times 10\), and the central position of each hole is movable from the center of the whole domain, which is \((64,64)\). The movement of both coordinates, x and y, is from \(-10\) to \(10\). Figure 7(a) shows the temperature field with two optimized masks. The area of each mask is fixed as \(20\times 10\). The central coordinate of each mask is movable inside the domain. Before optimization, two masks were in the same position (64, 64). The movement of each mask's coordinate x and y ranges from \(-10\) to \(10\). Table 3 shows the coordinate of masks after optimization. One mask was moved to the left upper part, while another was moved to the left lower part. In this situation, the square plate has the minimum mean temperature. Figure 7(b) is the result from FEM, which is defined as the proper reference. Analyzing from a qualitative perspective, the result from the PD-CNN has a good agreement with that of FEM. We can observe a similar result in this case compared with the one-hole case. The difference between the sampling area near the heat source and the reference value is greater than the difference between the sampling area on the right side of the mask. Figure 8(a) shows the temperature field with four optimized masks using the PD-CNN method. The area of each hole is fixed as \(10\times 10\), and the central position of each hole is movable from the center of the plate. Four masks were in the same position, which means they were coincident initially. The movement of each mask's coordinate x and y ranges from \(-20\) to \(20\). We notice that three holes are arranged vertically in the middle and left part of the entire domain, and the fourth mask is located on the right and upper of the whole field as shown in Table 4. Considering the symmetry of the computational domain, we can easily infer the distribution of another solution: the top hole is on the right upper side. In addition, in this case, we can see the numerical difference distribution similar to the previous two instances from Figure 8(c). The only difference is that the temperature  
Figure 8: Comparison between surrogate model prediction and FEM result for the two-holes case. (**a**): Temperature field with an optimized mask using a CNN surrogate model; (**b**): Temperature field with an optimized cover using the FEM method; (**c**): quantitative comparison of the two results. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Hole No.** & **Position Before Optimization** & **Position After Optimization** \\ \hline
1 st hole & (64, 64) & (54, 54) \\
2 nd hole & (64, 64) & (54, 74) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The position optimization of the two-holes case. values obtained by the PD-CNN have residuals in the four corners. The reason is that the calculation is more complicated in the case of four holes, and the convergence speed will decrease. Still, as the number of training increases, this residual will gradually disappear. In the above three cases, the results obtained by the traditional FEM method were used as the references, and the difference between results obtained by the PD-CNN method and the results obtained by the FEM method were compared, which proved from a qualitative point of view the high effectiveness of the new process. From a quantitative point of view, the article compares the difference in the numerical average of the results obtained by the two methods in four different specific areas, which verifies the high accuracy of the new method. The numerical simulation method based on a finite element is widely used in the analysis and evaluation of the temperature field. However, if the analysis program is called repeatedly during the optimization design process, the calculation cost will be too high, and the optimization cost will be huge, and even the design task cannot be completed in a limited time.