# Comparison of Deep Transfer Learning Models for the Quantification of Photoelastic Images
## 3 Results and Discussion
### Results of Transfer Learning Models  
Figure 7 shows the validation results of the predicted load against the applied load for the six transfer learning models tested in this study. The graph also includes the R2 value, which indicates how well the model explains the variability in the data and was calculated according to Equation (11). The coefficient of determination ranges between 0 and 1, with values closer to 1 indicating a perfect fit. \[R^{2}=\frac{\sum_{i=1}^{n}(\dot{y}_{i}-\overline{y})^{2}}{\sum(y_{i}-\overline {y})^{2}} \tag{11}\]  
where:  
\(n=\) the total number of observations. \(\dot{y}_{i}=\) the predicted value. \(\overline{y}=\) the mean of the observed values. \(y_{i}=\) the actual observed values. Figure 6: Evaluation results of photoelastic responses: (**a**) G2 method; (**b**) [PERSON] and [PERSON]’s method. Figure 7: _Cont._  
All six transfer learning models displayed coefficients of determination approaching 0.9, demonstrating significantly higher accuracy compared with the results from [PERSON]'s method and [PERSON] and [PERSON]'s method, as shown in Figure 6. Among these, the VGG16 and VGG19 models showed exceptional performance, with coefficients of determination of 0.984 and 0.988, respectively. InceptionV3, ResNet152V2, and DenseNet201 also showed high coefficients of determination, with values of 0.932, 0.957, and 0.946, respectively. However, we observed a noticeable increase in the deviation of the predicted data when the axial load levels ranged between 100 and 400 N. The validation data in this study are structured in the same six steps as the training data, increasing by 100 N from a minimum of 0 N to a maximum of 500 N. Applying a photoelastic response image at a load level between 200 and 300 N, such as 270 N, to the InceptionV3 model as validation data would result in a significantly larger error compared with the VGG models. The coefficient of determination for EfficientNetB7 was the lowest among the six models tried, 0.886, and even the predicted results at 0 N axial load showed a much larger deviation compared with the other models. Figure 8 visualizes the learning process, showing the loss function values for training and validation data against epochs, which represent the total number of iterations over  
Figure 7: Evaluation results of validation data for transfer learning models: (**a**) VGG16; (**b**) VGG19; (**c**) InceptionV3; (**d**) ResNet152V2; (**e**) DenseNet201; (**f**) EfficientNetB7. the entire dataset. The loss function used here is the MSE, commonly used for regression problems, and is calculated according to Equation (12). \[MSE=\frac{1}{n}{\sum_{i=1}^{n}}(y_{i}-\hat{y}_{i})^{2} \tag{12}\]  
By observing the trend of loss values against the number of epochs, one can assess the learning status, such as the presence of overfitting or underfitting and the stability of the learning. Except for EfficientNetB7, the models generally showed a decreasing trend in both training and validation loss as the number of epochs increased, indicating a favorable learning status. In contrast, EfficientNetB7 exhibited significant fluctuations in validation loss, suggesting the possibility of overfitting and an unstable learning state. Table 3 summarizes the performance metrics of the transfer learning models, allowing for a clearer comparison. Equations (13) and (14) calculate the MAE and RMSE. The mean absolute error intuitively represents the average deviation of the model's predictions, while the root mean square error is more sensitive to larger errors due to squaring the deviations. Figure 8: Examine the loss of training and validation data in accordance with Epoch: (**a**) VGG16; (**b**) VGG19; (**c**) InceptionV3; (**d**) ResNet152V2; (**e**) DenseNet201; (**f**) EfficientNetB7. Both metrics have the same units as the prediction errors and are simple to compute, making them widely used as performance evaluation metrics for regression models.