As a critical reinforcing material, high-modulus glass has experienced a rapidly growing demand in aerospace, wind power and other high-tech industries. This trend makes the development of novel high-performance, high-modulus glass formulations an urgent and significant task for materials science.

Machine learning has emerged as a transformative new research paradigm following traditional trial-and-error experiments and computer simulations. It offers remarkable advantages in efficiency, accuracy, and low energy consumption, and has been widely used to predict various glass properties.

However, purely data-driven machine learning models face severe performance bottlenecks when predicting glass Young's modulus (YM), showing significantly lower accuracy than models for other glass properties. This limitation mainly stems from two factors: poor data consistency in public databases due to the sensitivity of YM to preparation-related thermal history and testing conditions, and the lack of effective strategies to eliminate abnormal samples, which leads to model overfitting and limited generalization and extrapolation capabilities.

Liu and co-workers showed that pure data-driven models face three major contradictions in materials science, and proposed using data governance and domain knowledge embedding to improve model performance. Data quality is decisive for model performance, and combining anomaly detection with a divide-and-conquer strategy can yield accurate and interpretable models.

The Makishima-Mackenzie (MM) model is a classic semi-empirical model for predicting the YM of oxide glass with clear physical significance, but it exhibits a well-documented systematic underestimation of actual YM values. Residual modeling has been proven to be an effective approach to address the deficiencies of physical models. Therefore, this study aims to develop a physics-informed machine learning (PIML) model by integrating the MM model with machine learning to overcome the limitations of pure data-driven models.

The original dataset was constructed from the SciGlass database, which contained 2861 samples, with 15 core oxides as features and YM and density as target variables. Basic data cleaning was first performed to remove samples with incomplete compositions, missing values, zero values, negative values, and duplicate formulations.

Five pure data-driven models, namely Gradient Boosting Machine (GBM), Random Forest (RF), eXtreme Gradient Boosting (XGB), Artificial Neural Network (ANN) and Support Vector Regression (SVR), were trained on the original dataset. All models were optimized using Bayesian optimization and five-fold cross-validation, with the dataset split into training and test sets in an 8:2 ratio.

Subsequently, the MM model was used to calculate the theoretical YM values for all the samples. Based on the physical principle that the MM model systematically underestimates glass YM, samples with negative residuals (measured YM minus MM-calculated YM) were eliminated. This procedure yielded a high-quality dataset of 2463 samples constrained by physical principles.

The GBM model, which exhibited the best performance among the three pure data-driven models, was selected for constructing the PIML model. Instead of directly predicting YM, the PIML model used oxide compositions as input features and the MM model residuals as the prediction target, and was trained on the high-quality dataset. The final YM prediction was obtained by adding the predicted residual to the MM-calculated value.

Model performance was evaluated using three comprehensive metrics: the coefficient of determination (R²), mean squared error (MSE), and median absolute error (MedAE). SHapley Additive exPlanations (SHAP) analysis and physicochemical-descriptor modeling were employed to reveal the error mechanism of the MM model. Extrapolation capability was tested using 29 samples with Li₂O, Y₂O₃, La₂O₃ and ZrO₂ contents outside the training set range.

The pure data-driven GBM model achieved a test-set R² of 0.910 on the original dataset, representing the best performance among the three models. However, its performance was still substantially limited by the abnormal samples present in the dataset.

In contrast, the developed GBM-PIML model trained on the physically constrained high-quality dataset achieved a markedly improved test set R² of 0.962. Compared with the pure GBM model, its MedAE was reduced by approximately 40% and its MSE was reduced by approximately 70%. These results demonstrated that the physical-constraint-based data cleaning effectively removed misleading abnormal samples, and the residual learning strategy reduced the learning difficulty of the model by allowing the model to learn the systematic error pattern of the MM model rather than the entire complex composition-YM relation.

SHAP analysis revealed that increases in Y₂O₃, La₂O₃, MgO, and Li₂O contents significantly increased the MM model residuals. This result was attributed to the fact that the MM model treated glass as a homogeneous system and ignored two key effects: the accumulation of non-bridging oxygen by high-field-strength modifiers and the interaction between rare earth metals and aluminum ions.

A physicochemical descriptor model was further constructed using four descriptors (Ma, ACFS, Re/Al, IMO/Si) that characterize internal glass structure and oxide interactions and achieved a test set R² of 0.932. SHAP analysis of this model confirmed that a lower degree of network polymerization, higher average cation field strength, higher Re/Al ratio and higher IMO/Si ratio all contributed to larger MM model residuals, which was consistent with the underlying physical mechanism.

Extrapolation tests showed that the GBM-PIML model exhibited significantly lower prediction errors than the pure GBM model in the high-modulus region. Most importantly, its error did not increase sharply as YM increased, indicating strong extrapolation stability.

This study constructs a residual-learning-based PIML model by integrating the MM physics-based model with the GBM machine-learning algorithm. The model achieves a test set R² of 0.962, a MedAE of 0.77, and an MSE of 7.04, and demonstrates superior prediction accuracy, robustness and generalization capability compared to conventional purely data-driven models trained on the same original dataset.

The combination of SHAP analysis and physicochemical descriptor modeling provides deep insight into the error mechanism of the MM model. The results reveal that the systematic error of the MM model mainly originates from its neglect of the effects of high-field-strength ions and oxide interactions on glass YM.

Extrapolation tests confirm that the incorporation of physical prior knowledge significantly enhances the model's extrapolation ability, especially in the high-modulus region. The proposed PIML model provides a new paradigm for the rapid development of high-modulus glass formulations with high prediction accuracy, interpretability, and strong extrapolation capability.