Estimating the settlement of board piles socketed in rock using explainable and symbolic regression techniques for optimized deep foundation performance and sustainable design

Accurate prediction of settlement in bored piles socketed in rock remains a significant geotechnical challenge due to the complex nonlinear interaction between pile geometry, material properties, and loading conditions. This study develops an explainable data-driven framework based on symbolic regression to achieve high predictive accuracy while maintaining engineering interpretability. A comprehensive database containing pile geometry ratios, in-situ resistance indices, rock strength parameters, and applied loads was used to construct four predictive models: Genetic Programming (GP), Evolutionary Polynomial Regression (EPR), Group Method of Data Handling Neural Network (GMDH-NN), and Response Surface Methodology (RSM). Model performance was assessed using SSE, MAE, MSE, RMSE, prediction error percentage, accuracy, and coefficient of determination (R²) for both training and validation datasets. The results indicate strong predictive performance for all models, with training R² values ranging from 0.80 to 0.88. Validation performance improved significantly, where GP achieved R² = 0.92, GMDH-NN achieved R² = 0.94, and EPR provided the best overall performance with R² = 0.95, minimum prediction errors, and approximately 90% accuracy. Although GMDH-NN produced the lowest training errors, slight overfitting tendencies were observed, while EPR demonstrated superior robustness and generalization capability for unseen data. Taylor diagram analysis further confirmed the strong agreement between predicted and observed settlements. Sensitivity analysis revealed that pile geometry and applied load are the dominant settlement-controlling parameters, whereas material strength properties exert moderate influence. The study demonstrates that symbolic regression models can provide accurate, transparent, and practically applicable tools for settlement prediction and optimization of deep foundation systems in rock conditions.