fig4

Figure 4. Comparison of experimental vs. predicted HV across different models. (A) Physical baseline model, (B) RF, (C) SVR, (D) XGBoost, (E) ANN and (F) the proposed Hybrid PIML model. The dashed line indicates perfect prediction (y = x). The Hybrid PIML model demonstrates significantly tighter scattering around the diagonal, validating its superior accuracy. HV: Vickers hardness; RF: random forest; SVR: support vector regression; XGBoost: eXtreme Gradient Boosting; ANN: artificial neural network; Hybrid PIML: hybrid physics-informed machine learning; R²: coefficient of determination; RMSE: root mean square error; MLP: multi-layer perceptron.