## 4 Conclusions  
The stiffness coefficient is an important parameter to be considered in designing circular foundations. The current determination methods for the stiffness coefficient were either time-consuming or inaccurate. In this study, three-dimensional elastic simulation data were collected via a literature search to construct a hybrid model PSO-DT to predict the stiffness coefficient quickly and accurately. Extra finite-element simulation data were used for independent validation of the model. Furthermore, variable importance and PDP were used to analyze the sensitivity of different features. The specific conclusions were as follow:  
1. For \(K_{\mathrm{v}}\), \(K_{\mathrm{h}}\), \(K_{\mathrm{m}}\), and \(K_{\mathrm{c}}\), the R and Evar value of the DT model, as tuned by PSO on the training and testing sets, were both higher than 0.95, and the MSE and MAE were less than 0.15, indicating that the model had a high level of robustness.
2. Feature importance showed that \(T/2R\) had the most significant effect on the four stiffness coefficients, with an average importance value of about 0.58. Furthermore, there were differences in the way features affected output variables. When a \(T/2R\) less than 0.5 interacted with \(G_{\mathrm{sand}}/G_{\mathrm{clay}}\) and \(Z_{\mathrm{D}}/2R\) in a coupled way, the effect on the stiffness coefficient was greater.
3. Independent verification results showed that the R values between the true and predicted stiffness coefficients (\(K_{\mathrm{v}}\), \(K_{\mathrm{h}}\), \(K_{\mathrm{m}}\), and \(K_{\mathrm{c}}\)) were 0.97, 0.99, 0.99, and 0.95, respectively, indicating that the model has a high level of generalization.  
In the future, many other variables should be considered in ML modeling, such as the water table, the difference in the soil profile, etc. Moreover, other important parameters for foundation design, such as the lower and upper bounds of plasticity, should be included. Finally, other advanced ML algorithms should be used, and more advanced feature engineering and data pre-processing techniques should be attempted.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13042653/s1](https://www.mdpi.com/article/10.3390/app13042653/s1)]([https://www.mdpi.com/article/10.3390/app13042653/s1](https://www.mdpi.com/article/10.3390/app13042653/s1)), Figure S1: Performance of model using DT default parameters on (a) original and (b) normalized datasets.  
Conceptualization, C.Q. and M.W.; methodology, C.Q. and J.Z.; software, M.W.; validation, C.Q. and C.M.; formal analysis, C.M.; investigation, J.Z.; resources, C.Q.; data curation, C.M.; writing--original draft preparation, C.Q. and J.Z.; writing--review and editing, C.Q. and M.W.; visualization, J.Z.; supervision, C.Q.; project administration, C.M.; funding acquisition, C.Q. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University (No. HESS-2120).  
The data presented in this study are available upon request from the corresponding author.  
The authors declare no conflict of interest.