# A Novel Aerodynamic Modeling Method Based on Data for Tiltrotor evtol
## 4 Results and Discussion
### Network Training Method and Results  
The process of constructing a data-driven model involves a crucial step in the selection of appropriate hyperparameters, which necessitates the adoption of appropriate criteria for evaluating the predictive efficacy of different hyperparameter combinations. In this study, relative error and coefficient of determination are used as performance metrics to evaluate the network. The relative error is defined as  
\[error=\frac{||y_{pre}-y_{t}||_{F}}{||y_{t}||_{F}}*100\%\]  
where \(y_{pre}\) is the predicted output of the network, and \(y_{t}\) is the true value, yielding an aggregate relative error over the entire performance of the network. Specifically, for the lift coefficient (\(C_{L}\)), the individual errors for each aerodynamic coefficient are described as follows:  
\[error_{L}=\frac{||C_{L_{pr}}-C_{L_{t}}||_{2}}{||C_{L_{t}}||_{2}}*100\%\]  
Meanwhile, the coefficient of determination is defined as  
\[R^{2}=1-\frac{SS_{T}}{SS_{R}}\]  
\[SS_{T}=\sum_{t=1}^{k}\bigl{(}C_{L_{t}}-C_{L_{pr}}\bigr{)}^{2}\]  
Figure 5: XV-15 tilt angle–velocity flight envelope. \[SS_{R}=\sum_{t=1}^{k}\Big{(}C_{L_{t}}-\overline{C_{L_{t}}}\Big{)}^{2}\]  
where \(C_{L_{t}}\) is the true lift coefficient, and \(C_{L_{pr}}\) is the network predicted lift coefficient. The relative error can indicate how well a trained neural network behaves on new data that were not used in training. It can be thought of as a curve fitting, where the relative error represents the average deviation from the fitted curve to the wind tunnel test data. The coefficient of determination is used to assess how well a regression model explains the relationship between a dependent variable and an independent variable. It reflects how well the model fits the data it was trained on. An acceptable relative error of 5% [11] or less is based on established evtol aerodynamic models. A coefficient of determination closer to 1 indicates a stronger relationship between the variables, meaning the model explains most of the variation in the dependent variable. For an MLP neural network, key hyperparameters that influence prediction performance include the number of hidden layers, neurons per layer, activation functions, loss function types, training epochs, and batch size. This article utilizes the random search method in Python to randomly sample hyperparameter values from a defined range and compare the above evaluation criteria to select the optimal hyperparameter. Note that this article uses the average of the errors and determination coefficients for each aerodynamic coefficient as the evaluation criterion, due to their varying relative errors and determination coefficients under different hyperparameters. The results of training the MLP neural network with varying hyperparameter values for predicting lift coefficient errors are tabulated below as Table 2.  
In summary, by iteratively simulating the neural network and evaluating the relative error and coefficient of determination across different network hyperparameter settings, selecting the hyperparameter combination with the minimum relative error and the maximum coefficient of determination, the optimal values for the number of layers, neurons, and other hyperparameters for the MLP neural network are derived, as shown in the accompanying Table 3. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Hyperparameters**} & \multicolumn{8}{c}{**MLP**} \\ \hline Number of Epochs & 100 & 100 & 100 & 50 & 150 & 100 & 100 & 100 & 100 \\ \hline Batch Size & 10 & 20 & 30 & 20 & 20 & 20 & 20 & 20 \\ \hline Number of & & & & & & & & \\ Neurons/hidden Layers & 64/2 & 64/2 & 64/2 & 64/2 & 32/2 & 96/2 & 64/1 & 64/3 \\ \hline Relative error & 1.625\% & 1.361\% & 1.455\% & 1.413\% & 1.327\% & 1.314\% & 1.391\% & 1.861\% & 1.937\% \\ \hline Coefficient of & & & & & & & & & \\ Determination & 0.9921 & 0.9967 & 0.9925 & 0.9981 & 0.9986 & 0.9972 & 0.9975 & 0.9907 & 0.9943 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Different combinations of MLP training parameters.