## 4 Results and Discussion
### Influence of Horizontal Scale of Fluctuation (\(\delta_{h}\)) in the Random Field  
The development of the mean and standard deviation of \(N_{ch}\) and \(N_{ch}\) with the variation of \(\delta_{h}\) is shown in Figure 11. It is indicated that the mean value and standard deviation of \(N_{ch}\) and \(N_{ch}\) fluctuated slightly with the increase of \(\delta_{h}\) from 30 to 60 m (e.g., case Ani-6, 7, 3, and 8 given in Table 1). The samples of the four cases are composed to a mixed dataset in training a 2D-CNN model for the prediction of uniaxial bearing capacity factors of the skirted foundation considering the variation of \(\delta_{h}\). There are 1680, 240 and 480 samples for the training, validation, and prediction datasets, respectively.  
Figure 12 displays the comparison of \(N_{ch}\) and \(N_{ch}\) predicted by the trained 2D-CNN model and calculated by RFEM. As shown in Figure 12a,b, the predicted and calculated uniaxial horizontal and moment bearing capacity factors are located around the solid line, which is predefined as the predicted value equal to the actual value, with the minimum R\({}^{2}\) value at 0.9802 and maximum RMSE value at 0.1204, which indicates a satisfactory prediction performance of the surrogate model. It is observed from Figure 12c that the distribution of the predicted uniaxial horizontal bearing capacity factors is highly consistent with the RFEM-calculated results, in which the relative errors of the mean, minimum, and maximum values are 0.15%, 0.22%, and 0.69%, respectively. Similarly, the relative errors of the three evaluating values are 0.23%, 0.42%, and 0.13% for the uniaxial moment bearing capacity factors predicted by the 2D-CNN model, as presented in Figure 12d, which indicates that the typical distribution of \(N_{ch}\) and \(N_{ch}\) are well predicted by the surrogate model on account of the variation of \(\delta_{h}\).  
Figure 11: Variation of the bearing capacity factors (\(N_{ch}\) and \(N_{ch}\) ) statistics with \(\delta_{h}\): (**a**) Mean; (**b**) Standard deviation.  
Figure 10: Distribution of the relative errors of the predicted bearing capacity factors under mixed COV: (**a**) PDF of \(N_{ch}\); (**b**) PDF of \(N_{ch}\).  
In order to visualize the accuracy of the CNN-predicted uniaxial bearing capacity factors, the probability distribution of the relative errors between the predicted and calculated results is illustrated in Figure 13. It is obviously that the relative errors are all restricted in 8%, with a CI of 99.2% and 97.3% for the margin of 5% error in the prediction of uniaxial horizontal and moment bearing capacity factors, respectively. It indicates that the architecture of the 2D-CNN model proposed in Figure 5 has the potential to predict the uniaxial bearing capacity factors of the skirted foundation within an acceptable error range when \(\delta_{h}\) of the soil random field extends from 30 to 60 m.  
Figure 12: Comparison of the predicted and actual uniaxial bearing capacity factors under mixed \(\delta_{h}\): (**a**) Scatter of \(N_{CH}\); (**b**) Scatter of \(N_{CM}\); (**c**) CDFs of \(N_{CH}\); (**d**) CDFs of \(N_{CM}\).  
Figure 13: Distribution of the relative errors of the predicted bearing capacity factors under mixed \(\delta_{h}\): (**a**) PDF of \(N_{CH}\); (**b**) PDF of \(N_{CM}\).