## 3 Results and Discussion  
### Performance of the DNN-Bernoulli Model for Synthetic Shapes  
The history of 5-fold cross validation results for synthetic shapes is plotted in Figure 9. The horizontal axis corresponds to the number of epochs and the vertical axis corresponds to the mean absolute error (MAE) between the predicted and target value of \(\mathbf{y}\). Since dropout is activated when training but deactivated when evaluating on the validation set, MAE on the validation set is smaller than on the training set. The scatter plot of the performance of the trained model on the test set is illustrated in Figure 10. The horizontal and vertical axes correspond to the true and predicted values of \(\mathbf{y}\) on the test set, respectively. MAE on the training set, validation set and test set are 0.0769, 0.0705 and 0.0487, respectively. The relative MAE divided by the mean ground-truth value of \(\mathbf{y}\  
To demonstrate the improvement of the present DNN-Bernoulli model over the original Bernoulli equation, we define the relative error of flow rate and pressure distribution on a certain shape as follows:  
\[E_{Q}=\left|\frac{Q-\hat{Q}}{Q}\right|\times 100\% \tag{12}\] \[E_{P}=\frac{1}{n}\frac{\sum_{i=1}^{n}\left|P_{i}-\hat{P}_{i} \right|}{P_{s}}\times 100\% \tag{13}\]  
where \(Q\) and \(P_{i}\) are the flow rate and pressure distribution obtained by the N-S solver, respectively, while \(\hat{Q}\) and \(\hat{P}_{i}\) are the corresponding values obtained by the DNN-Bernoulli or Bernoulli model. The comparison of the error range between the DNN-Bernoulli and Bernoulli model on the test set is listed in Table 4. We can see that the error range of both flow rate and pressure distribution of the DNN-Bernoulli model is significantly lower than that of the original Bernoulli model.  
Figures 11-14 show the contour of the surface pressure on the vocal folds in three representative glottal shapes predicted by the DNN-Bernoulli, N-S and Bernoulli models. A common phenomenon can be observed from all these figures, that is, the pressure distribution predicted by the DNN-Bernoulli model is much closer to that obtained by the N-S solver, which shows the improvement of the present DNN-Bernoulli model. Note that compared with the Bernoulli model, the additional CPU time required for the present DNN-Bernoulli model during prediction is almost negligible. For the prediction of a newly generated synthetic glottal shape, the average CPU time required for the DNN-Bernoulli model is just around 1 s on a single CPU, while that required for the N-S model is around 2 h per CPU on a parallel computer with 32 CPUs.  
\begin{table}
\begin{tabular}{c c c} \hline  & \(E_{Q}\) & \(E_{P}\) \\ \hline Bernoulli & 0.27–48.7 & 0.2–19.16 \\ DNN-Bernoulli & 0.01–8.94 & 0.01–8.53 \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of error range between Bernoulli and DNN-Bernoulli prediction.  
Figure 10: Performance of the trained model on the test set (synthetic shape).  
Figure 11: Surface pressure contour and distribution (convergent).  
Figure 12: Surface pressure contour and distribution (divergent).  
Figure 13: Surface pressure contour and distribution (convergent-divergent).