## 5 Conclusions  
In summary, this study combines the random field theory and convolutional neural network to develop an effective machine-learning-based model for predicting the uniaxial bearing capacity factors of the skirted foundation in spatial variable soil under pure horizontal and moment loading. The spectral representation method is adopted to simulate the non-stationary random field of soil undrained shear strength with various coefficients of variation and scale of fluctuations, which is acted as the input of the proposed 2D-CNN model. Meanwhile, the uniaxial horizontal and moment bearing capacity factors are defined as the output of the surrogate model. The predicted values are compared with the results calculated by the random finite element method to present the prediction performance of the surrogate model. Several conclusions are summarized as follows:  
1. The proposed 2D-CNN model can replace the time-consuming RFEM in predicting the uniaxial bearing capacity factors of the skirted foundation in spatially variable soil with reasonable accuracy;
2. There are three 2D-CNN models with the same architecture that are trained to deal with the prediction of skirted foundation bearing capacity considering the variation of COV, \(\delta_{h}\) and \(\delta_{v}\) in the soil random field, respectively. The minimum R\({}^{2}\) value and maximum RMSE value for the three surrogate models are 0.9781 and 0.1204, indicating satisfactory prediction performance of the proposed model;
3. The confidence interval of the relative error is more than 96.3% with a margin of 5% for the predicted bearing capacity factor with the variation of COV, while the minimum confidence intervals are 97.3% and 97.9% for the relative errors that are located within 5% on account of the variation of \(\delta_{h}\) and \(\delta_{v}\), respectively.  
The proposed 2D-CNN model is an effective surrogate model for the prediction of uniaxial bearing capacity factors of skirted foundation. However, it is noted that the bearing capacity prediction of the skirted foundation subjected to the combination of horizontal, vertical and moment loads have not been illustrated. Further study is recommended to develop a reasonable machine-learning model to predict the skirted foundation bearing capacity and failure envelope under different loadings.  
Figure 15: Distribution of the relative errors of the predicted bearing capacity factors under mixed \(\delta_{v}\): (**a**) PDF of \(N_{ch}\); (**b**) PDF of \(N_{ch}\).  
**Author Contributions:** Conceptualization, Y.W. and H.Z.; methodology, H.C.; software, Z.L. and H.Z.; validation, Y.W. and H.Z.; formal analysis, H.Z. and H.C.; investigation, Z.L., H.C. and Y.W.; resources, H.C.; data curation, Z.L. and H.C.; writing-original draft preparation, H.Z.; writing-review and editing, H.C., Y.W. and Z.L.; visualization, H.Z.; supervision, H.Z.; project administration, Y.W. and H.C.; funding acquisition, H.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the science foundation of China Three Gorges Corporation grant number 202203004.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors acknowledge the financial support of the science foundation of China Three Gorges Corporation (Grant No. 202203004).  
The authors declare no conflict of interest.