Input Variable Effects on TBM Penetration Rate: Parametric and Machine Learning Models

In this study, linear and nonlinear parametric models (M1–M6) were jointly evaluated alongside machine learning (ML)-based approaches to achieve reliable and interpretable prediction of the penetration rate (ROP) of tunnel boring machines (TBMs). The analyses incorporate key geomechanical and structural variables, including the brittleness index (BI), uniaxial compressive strength (UCS), mean spacing of weakness planes (DPW), the angle between the tunnel axis and weakness planes (α), and Brazilian tensile strength (BTS). The coefficients of the parametric models were optimized using the Differential Evolution (DE) algorithm. Variable effects were systematically examined through Jacobian-based elasticity analysis under both original and standardized data scenarios. The results indicate that the M6 model, which explicitly incorporates interaction terms, delivers superior predictive accuracy and a more balanced, physically meaningful representation of variable contributions compared to widely used parametric formulations reported in the literature. While the dominant influence of BI and UCS on ROP is consistently preserved across all models, the indirect contributions of variables such as DPW and BTS are more clearly revealed in M6 owing to its interaction-based structure. Model performance improves systematically with increasing complexity, with the coefficient of determination (R²) rising from 0.62 for M1 to 0.69 for M6. Relative to the linear model, M6 achieves a 9.07% reduction in RMSE and a 10.48% increase in R², while providing additional improvements of 2.47% in RMSE and 2.37% in R² compared with the closest competing model. ML-based variable importance analyses are largely consistent with the parametric findings, highlighting BI and α in tree-based models, and UCS and α in SVM and GAM frameworks. Notably, the GAM exhibits the highest predictive performance under both data scenarios. Overall, the integrated use of parametric and ML approaches establishes a robust hybrid modeling framework that enables highly accurate and engineering-interpretable prediction of TBM penetration rate.