Harnessing Metaheuristics and Probabilistic Machine Learning for Uncertainty-Aware Bearing Capacity Estimation of Shallow Foundations

In response to the limitations of traditional deterministic machine learning approaches in geotechnical engineering, this study introduces a probabilistic model to quantify uncertainty in bearing capacity predictions for cohesionless soils. Using a dataset from existing literature, the Probabilistic Gradient Boosting Machine (PGBM) was employed to address the gap in uncertainty estimation. Four optimization algorithms—Particle Swarm Optimization (PSO), Slime Mould Algorithm (SMA), Gray Wolf Optimization (GWO), and Runge–Kutta Optimization (RUN)—were applied to fine-tune the model, with GWO-PGBM showing the lowest error and uncertainty. A novel model selection method, emphasizing both accuracy and reduced overfitting and uncertainty, was developed to enhance robustness. The GWO-PGBM model achieved a 36% lower error compared to the best-performing model in the literature with an R² of 0.97. This approach offers a powerful tool for geotechnical applications, ensuring high predictive accuracy and critical uncertainty quantification for informed decision-making.