Oxide glasses, widely used in various applications, require accelerated design of compositions and processing techniques due to the vast number of possible combinations. Predictive composition-property models, especially those based on machine learning, are valuable but often fail to extrapolate beyond their training sets. This limitation arises because machine learning models, unlike physics-based models, are solely data-driven and may violate physical or chemical laws. The relationship between short-range order (SRO) structure and properties is more direct and informative than the composition-property relationship, making SRO-property models easier to generalize. However, obtaining reliable SRO data is challenging. Statistical mechanical modeling offers promise in predicting SRO structure but may oversimplify interactions in multicomponent glasses. This study proposes a combined model leveraging the extrapolation ability of statistical mechanics and the accuracy of machine learning to address these challenges.

Existing machine learning models in glass science are primarily trained on composition-property data and struggle with extrapolation beyond the training set. While structure-property relationships are often more linear and thus easier to extrapolate, obtaining sufficient and reliable structural data is a major hurdle. Statistical mechanics modeling provides a promising alternative, but its simplification of interactions can lead to systematic errors in predicting the structure of complex glasses. Previous work has shown that combining physics-based models with machine learning can improve prediction accuracy and extrapolation capabilities, motivating the current study.

The study compares three approaches: (1) a purely statistical mechanical model trained on binary oxide glasses; (2) a multilayer perceptron neural network (MLP-NN) model using only composition as input; and (3) a combined model using both composition and statistical mechanics results as input. The statistical mechanical model employs a Wallenius non-central hypergeometric distribution function to calculate the probability of modifier-structure unit interactions based on enthalpy and entropy considerations. For the MLP-NN models, hyperparameter optimization (number of neurons and layers) was performed using 10-fold cross-validation and SciPy’s basin-hopping algorithm to minimize the mean squared error (MSE). The combined model leverages the statistical mechanics model as an additional input layer, allowing the neural network to correct systematic errors from the statistical mechanics predictions. The models are evaluated by comparing predictions to experimental data from various glass systems, including Na₂O-SiO₂ and Na₂O-P₂O₅-SiO₂ glasses. The Na₂O-P₂O₅-SiO₂ system serves as a test for extrapolation capability as it was not included in the training datasets.

The statistical mechanical model performed reasonably well for binary and some ternary glasses, showing some extrapolation capability, but exhibited significant deviation for others. The MLP-NN model demonstrated good interpolation but poor extrapolation. The combined model significantly outperformed both individual methods, with a 35-40% reduction in root-mean-square error (RMSE). The combined model captured the non-linearity in the composition-structure relationship of Na₂O-SiO₂ glasses significantly better than the MLP-NN model alone, and also displayed excellent predictive power when extrapolating to the unseen Na₂O-P₂O₅-SiO₂ glass system. Analysis of the Na₂O-SiO₂ system showed that the combined model could accurately predict Q³ fractions across different Na₂O concentrations, even when either the chemical composition or the statistical mechanics information was partially withheld. The combined model also offered significantly improved RMSE compared to the other approaches in predicting the structure of the held-out Na₂O-P₂O₅-SiO₂ glass data.

The results demonstrate that integrating statistical mechanics with machine learning creates a powerful tool for predicting glass structure. The combined model successfully overcomes the limitations of purely data-driven models by incorporating physical constraints derived from statistical mechanics. This approach provides a baseline structural prediction and only requires the neural network to learn corrections, leading to superior predictive performance in complex systems. The multi-fidelity approach of combining high-fidelity experimental data and low-fidelity predictions from statistical mechanics shows promise for future research to improve the accuracy and extrapolative power of composition-property models by considering structural features. This approach could accelerate the design of new glasses with desired properties.

This study successfully demonstrated that a physics-informed machine learning approach, combining statistical mechanics and a multilayer perceptron neural network, significantly improves the prediction and extrapolation of non-linear composition-structure relations in oxide glasses compared to individual methods. This combined model offers a powerful tool for materials design, allowing for accurate predictions even outside the training data range. Future work could extend this approach to other glass systems with additional oxide components and integrate it with composition-property models for a comprehensive structure-property prediction framework.

The current study focuses on a specific set of oxide glasses. The generalizability of the model to other glass systems with different network formers or modifiers requires further investigation. The accuracy of the statistical mechanical model depends on the accuracy of the enthalpy parameters, which were obtained from fitting to experimental data from binary and some ternary glasses. Also, the choice of neural network architecture and hyperparameters could influence model performance; however, a thorough optimization process was implemented in this study to find the optimal configuration. Further work could explore other machine learning architectures and techniques.