## 5 Airfoil Noise Prediction Model Validation  
To verify the feasibility and superiority of ResNet-18 in airfoil noise prediction, a comparative analysis was conducted between ResNet-18, Random Forest (RF) [24], Long Short-Term Memory (LSTM) [25] networks, traditional CNN, and ResNet-50. Mean Squared Error was used as the loss function to quantify the error between actual and predicted values. Additionally, the coefficient of determination (\(R^{2}\)) was adopted as a key evaluation metric, as it quantifies the correlation between predicted and actual values in regression models. The formulas for MSE and \(R^{2}\) are:  
\[E_{MSE}=\frac{1}{m}{\sum_{j=1}^{m}{\left({{{\hat{y}}_{i}}-{{\hat{y}}_{i}}} \right)^{2}}} \tag{26}\]  
\[R^{2}=1-\frac{{\sum_{\delta=1}^{n}{\left({{{y}_{i}}-{{\hat{y}}_{i}}}\right)^{2 }}}}{{\sum_{\delta=1}^{n}{\left({{{y}_{i}}-{{\overline{{\hat{y}}}_{i}}}} \right)^{2}}}} \tag{27}\]  
After training, the performance of the five models on the test set is summarized in Table 3. While all models performed well in airfoil noise prediction, ResNet-18 and ResNet-50 exhibited the best \(R^{2}\) and MSE values. Among these, ResNet-18 achieved the highest \(R^{2}\) and the lowest MSE, demonstrating the most accurate prediction performance. Table 3. shows the computation times of these models, where the random forest is the fastest, ResNet-50 is the slowest, and ResNet-18 has a speed in between; however, the differences in computation time between the models are actually very small. Considering the computational accuracy and speed of each model, the ResNet-18 model is the best choice. The relationship between predicted and actual values for ResNet-18 on the test set is depicted in Figure 16. Ideally, if predicted values perfectly match actual values, data points would align along the diagonal. In the figure, the data points closely approximate the diagonal, confirming the high prediction accuracy of ResNet-18. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & \(\mathbf{R^{2}}\) & **MSE** &
\begin{tabular}{c} **The Average Time Required to Compute** \\ **the Noise of a Single Airfoil.** \\ \end{tabular} \\ \hline Random Forest & 0.98641 & 1.9847 & 0.0406 \\ LSTM & 0.99896 & 0.1349 & 0.0437s \\ Normal CNN & 0.99961 & 0.0858 & 0.0496s \\ ResNet-50 & 0.99962 & 0.0388 & 0.0881s \\ ResNet-18 & 0.99972 & 0.0282 & 0.0457s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The prediction effect of each model under the test set is compared. Figure 16: Scatter plot comparing the predicted and actual noise levels for airfoil configurations. As shown in Figure 17, the overlap between predicted and actual noise values for the test set at a frequency of 1000 Hz is highly pronounced, indicating excellent prediction accuracy. Similarly, Figure 18 illustrates the relationship between predicted and actual noise frequency spectra across 31 frequency points ranging from 20 Hz to 20,000 Hz for random test samples. The predicted and actual values are nearly identical at all frequency points, further validating the high precision of the ResNet-18 model. Figure 17: The comparison between the predicted value and the real value of noise at 1000 Hz frequency. Figure 18: _Cont._  
To evaluate the efficiency of the neural network in airfoil noise prediction, the computational time of ResNet-18 was compared with that of BPM theory, The computational time of the BPM model considers the parameters obtained using RFOIL. As summarized in Table 1, the accuracy of the proposed method is \(\sim 10\%\). Figure 18: The comparison between the predicted value of noise spectrum and the real value. (**a**) Random airfoil 1 and its noise prediction results. (**b**) Random airfoil 2 and its noise prediction results. (**c**) Random airfoil 3 and its noise prediction results. (**d**) Random airfoil 4 and its noise prediction results. (**e**) Random airfoil 5 and its noise prediction results.