Predicting glass structure by physics-informed machine learning

Oxide glasses are crucial in both traditional and high-tech applications, where properties are tailored by composition. The vast compositional design space demands accelerated, predictive modeling. Prior machine learning models trained directly on composition-property mappings often fail to extrapolate beyond the training domain because they do not embed physical or chemical laws. In contrast, glass properties correlate more linearly with short-range order (SRO) structural motifs than with composition, suggesting a structure-mediated path to better extrapolation. However, experimental SRO data (e.g., from solid-state NMR) are scarce and labor-intensive to obtain. Statistical mechanical models can predict SRO from composition by using reaction enthalpies and entropic considerations, but with simplified interactions they can show systematic errors in multicomponent systems. This work hypothesizes that combining statistical mechanics (to encode thermodynamics and non-linear composition–structure relations) with data-driven ML (to correct systematic biases) will yield a physics-informed model that improves both interpolation and extrapolation of glass structure from composition.

Prior ML efforts in glass science have accurately interpolated properties like glass transition temperature, density, elastic moduli, and others when trained within a given compositional space, but they struggle with extrapolation. Structure-property relations (e.g., via topological constraint theory) often exhibit near-linear behavior, making them attractive intermediates for predictive models. NMR spectroscopy provides detailed SRO information but becomes more complex as components increase. Statistical mechanical treatments have been developed to predict SRO in binary and ternary oxide glasses based on modifier–former reactions, using enthalpy parameters trained on simpler systems and leveraging entropy via non-central hypergeometric distributions. These approaches extrapolate well in some cases (e.g., Na/Li/Cs–SiO₂–B₂O₃) but can show systematic deviations in more complex systems (e.g., Na₂O–Al₂O₃–SiO₂; Na₂O/CaO–P₂O₅–SiO₂), often correctable with a small number of additional parameters. This motivates a combined, multi-fidelity approach where physics-based statistical predictions guide a neural network to learn residual corrections.

Three approaches were developed and evaluated: (1) a statistical mechanics (SM) model trained on binary oxide glasses; (2) a composition-to-structure multilayer perceptron neural network (MLP-NN) using only composition as input; and (3) a combined SM+MLP-NN model that uses both composition and the SM-predicted structural outputs as inputs. Statistical mechanics model: The SM framework, originally proposed by Mauro and implemented in StatMechGlass (Python), computes probabilities of modifier–structural-unit reactions based on enthalpy (Hi) and entropy contributions. Probabilities are modeled with a Wallenius non-central hypergeometric distribution, reflecting composition-dependent entropy and fixed interaction energies assumed independent of composition. The fictive temperature Tf sets the entropic scaling; for fair comparison to ML, a fixed representative value Tr = 700 K was used. Reaction schemes include silicate Qn evolution and the boron anomaly in borosilicates, using a critical concentration parameter to toggle between BIII→BIV and BIII→BII pathways. Structural fractions are updated iteratively with concentration step ω via balance equations that account for formation and consumption of structural units. Enthalpies are fitted to binary systems (e.g., Na₂O–SiO₂, Na₂O–B₂O₃), then transferred to multi-component glasses without further fitting (or with minimal additional parameters when necessary). MLP-NN model: The neural network maps composition (molar % oxides) to measured structural fractions (e.g., Qn distributions) across ~400 glass compositions including common formers (SiO₂, P₂O₅, B₂O₃, Al₂O₃) and modifiers (Na₂O, Li₂O, K₂O, Cs₂O, CaO, MgO). Hyperparameters (hidden layers and neurons) were optimized via grid search with 10-fold cross-validation and SciPy’s basin-hopping to minimize validation MSE. The optimal architecture uses two hidden layers with 13 and 16 neurons, respectively, balancing underfitting and overfitting. A random 10% of data was held out as a test set. Combined SM+MLP-NN model: Inputs consist of concatenated composition vectors and SM-predicted structural fractions, enabling the NN to learn residual corrections to the physics-based baseline. Both ML models were trained only on composition as the fundamental input (the SM predictions themselves are composition-derived). Data layout and training: Approximately 400 labeled compositions with measured structures (from literature, using measured compositions) were used. The SM model was trained only on binary datasets and tested on multicomponent systems (extrapolation). The ML models used random train/test splits across the full dataset (interpolation). Additional experiments varied the fraction of training data (50–100%) while maintaining a 0.9/0.1 split to assess data efficiency. Evaluation metrics: Performance was evaluated using RMSE (%) between predicted and measured structural fractions across training and test sets, with additional system-specific evaluations for Na₂O–SiO₂ and blind extrapolation to Na₂O–P₂O₅–SiO₂ (excluded from training). Implementation: Statistical mechanics calculations used the StatMechGlass Python package. Neural network training and optimization used standard Python scientific libraries with cross-validation and basin-hopping optimization.

- Across the full dataset, RMSE values were: SM (Approach 1) = 6.6%, MLP-NN (Approach 2) = 6.8%, and combined SM+MLP-NN (Approach 3) = 4.2%, representing a 35–40% RMSE reduction for the combined model relative to the individual models. - The combined model better captures non-linear composition–structure transitions (e.g., Qn evolution in Na₂O–SiO₂), whereas the pure MLP-NN tends toward overly linear predictions between data points. - Data efficiency: For the combined model, RMSE decreases and plateaus when using more than ~80% of the dataset, indicating the model leverages thermodynamic information from SM predictions and does not require substantially more data for further gains. - Extrapolation test (blind set): For Na₂O–P₂O₅–SiO₂ glasses excluded from training, RMSEs were SM = 5.9%, ML = 4.0%, and SM+ML = 2.1%. The combined model significantly outperformed both physics-only and data-only models, successfully extrapolating to a previously unseen glass family. - In Na₂O–SiO₂, both ML models extrapolate reasonably in high-modifier regions, though inaccuracies at very high modifier contents are noted (regions where glass formation is unlikely). The combined model reproduces the non-linear Q³ distribution vs. Na₂O content, and analysis with isolated inputs shows that SM inputs capture most non-linear features while composition-only inputs do not. - SM extrapolates well from binaries to some ternaries (e.g., M₂O–SiO₂–B₂O₃) but shows systematic deviations in others (e.g., Na₂O–Al₂O₃–SiO₂; Na₂O/CaO–P₂O₅–SiO₂), consistent with oversimplified interaction sets; such systematic errors are learnable and correctable by the combined model.

The results support the hypothesis that embedding physical knowledge via statistical mechanics improves both the accuracy and extrapolative capability of ML models for glass structure prediction. By supplying SM-predicted structural distributions as additional inputs, the neural network learns residual corrections rather than reconstructing complex non-linear composition–structure relations from scratch. This multi-fidelity fusion of low-fidelity physics-based outputs with high-fidelity experimental data yields superior performance over either approach alone. The combined model not only reduces RMSE substantially but also captures non-linear structural transitions, enabling more faithful predictions in systems like Na₂O–SiO₂ and successful extrapolation to Na₂O–P₂O₅–SiO₂ that were not included in training. Unlike physics-informed loss constraints, the present approach uses a frozen, independently developed SM model to provide a physically grounded baseline. This strategy addresses a core limitation of purely data-driven models (poor extrapolation) and of purely physics-based models (oversimplified interactions), offering a path toward robust composition–structure–property pipelines where structure mediates improved property predictions beyond the training range.

This work introduces and validates a physics-informed machine learning framework for predicting SRO structural distributions in oxide glasses from composition. By combining a statistical mechanical model with an MLP-NN, the approach achieves substantially lower RMSE and markedly improved extrapolation compared to SM or ML alone. The model accurately reproduces non-linear composition–structure relations in Na₂O–SiO₂ and successfully predicts structures in a blind Na₂O–P₂O₅–SiO₂ set. The methodology enables scalable composition–structure–property modeling, leveraging physical priors while learning data-driven corrections. Future work includes extending to glass families with additional oxide components, integrating estimated fictive temperatures when available, expanding high-fidelity structural datasets, and coupling the structure predictor with property models to drive accelerated glass design across broader compositional spaces.

- The fictive temperature was fixed to an approximate representative value (700 K) to enable fair comparisons and broad applicability, which may introduce deviations for systems with substantially different Tf. - The statistical mechanics model simplifies interaction sets and assumes composition-independent interaction energies; transfer from binary to multi-component systems can yield systematic errors requiring additional parameters. - Experimental SRO datasets are limited and noisy, particularly for multicomponent glasses; NMR-based structure determination becomes challenging as compositional complexity increases. - ML training uses a relatively small dataset (~400 compositions); while the combined model shows data efficiency, broader datasets could further improve robustness. - Prediction inaccuracies appear at very high modifier contents where glass formation is unlikely; applicability in such regions is limited. - Extrapolation was demonstrated for a specific blind ternary system (Na₂O–P₂O₅–SiO₂); broader validation across other families requires additional experimental data.