🔬 condensed matter

Physics-Informed Symbolic Regression for Predicting the Glass Transition Temperature of Alkali Borate Glasses

This paper demonstrates that physics-informed symbolic regression, utilizing the Rigid Unit Packing Fraction descriptor, successfully derives an interpretable closed-form model for predicting the glass transition temperature of alkali borate glasses with high accuracy while explicitly capturing the physical interplay between structural rigidity, dissociation energy, and network packing.

Original authors: Leonardo dos Santos Vitoria, Marcio Luis Ferreira Nascimento, Susana de Souza Lalic, Daniel Roberto Cassar

Published 2026-08-18

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CC BY 4.0

Original authors: Leonardo dos Santos Vitoria, Marcio Luis Ferreira Nascimento, Susana de Souza Lalic, Daniel Roberto Cassar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Glass is a material that defies the strict rules of crystals. While a crystal must arrange its atoms in a perfect, repeating pattern, glass is a frozen liquid, a chaotic tangle of atoms that has cooled too quickly to find order. This lack of rigid structure gives glass its incredible versatility, allowing scientists to mix and match chemical ingredients to create materials for everything from windowpanes to radiation shields. One of the most critical properties of any glass is its glass transition temperature, the specific point where the material shifts from a hard, brittle solid into a soft, pliable state. Knowing this temperature is essential for manufacturing, yet predicting it for new mixtures has long been a challenge. The complexity arises because the atoms in glass do not just sit still; they form a network that changes shape and strength depending on exactly which elements are added and in what amounts.

For decades, researchers have relied on powerful computer programs, known as machine learning, to predict these properties. These programs are excellent at finding patterns in vast amounts of data, but they operate like black boxes. They can tell you the answer, but they cannot explain why they reached that conclusion, leaving scientists unable to understand the physical laws driving the result. A new study by a team of researchers in Brazil and at the Brazilian Center for Research in Energy and Materials seeks to change this by using a different approach called physics-informed symbolic regression. Instead of hiding the logic inside a complex algorithm, this method searches for a simple, readable mathematical sentence that describes how the glass behaves. The researchers focused on a family of glasses made by mixing boron oxide with various alkali metals, such as lithium, sodium, and potassium. These materials are famous for a strange behavior where adding more metal initially strengthens the glass, but eventually weakens it, a phenomenon known as the boron anomaly.

The team gathered hundreds of experimental records of glass compositions and their corresponding transition temperatures from existing scientific databases. They then fed this data into a computer program designed to hunt for a clear equation. The program was not allowed to use just any numbers; it was forced to use only variables that represent real physical things, such as the energy required to break the bonds between atoms and how tightly the atoms are packed together. The goal was to find a formula that was not only accurate but also made sense to a human physicist. After a rigorous search, the computer produced a single, compact equation. This equation successfully predicted the transition temperature for the known mixtures with a high degree of precision, missing the experimental values by only about 15 to 16 degrees on the Kelvin scale. More importantly, the equation revealed that the strength of the chemical bonds and the way the atoms pack together are the primary drivers of the glass's behavior, confirming that the model was capturing real physics rather than just memorizing data.

To test if their discovery was truly robust, the researchers asked the model to predict the behavior of glasses containing rubidium and cesium, two heavy metals that were not part of the original training data. This is a difficult test, as the model had never seen these elements before. The equation worked remarkably well, accurately predicting the transition temperatures for these new materials. This success suggests that the physical rules the model learned are universal for this family of glasses, not just specific to the examples it studied. The researchers also used the model to estimate the energy required to break the bonds in these new rubidium and cesium glasses. These estimated values followed a logical order, decreasing as the atoms got larger, which matched what scientists expect based on fundamental chemistry. This consistency provided strong evidence that the model had found a genuine physical relationship.

The study also highlighted a flaw in older methods of calculating how tightly atoms are packed. Traditional calculations often overestimated the rigidity of the glass structure, leading to inaccurate predictions. The new model used a more refined way of measuring this packing, which corrected these errors and produced much more realistic results. Furthermore, the team used a statistical technique to map out the uncertainty of their predictions. They found that the model was most uncertain in the specific range of chemical mixtures where the boron anomaly occurs. This makes sense, as this is the region where the glass structure is changing most dramatically, making it the hardest part to predict. By identifying exactly where the model is unsure, the researchers provided a clearer picture of the limits of their understanding.

In the end, this work demonstrates that it is possible to build computer models that are both accurate and transparent. By forcing the mathematics to speak the language of physics, the researchers created a tool that does not just guess the answer but explains the story of the glass. They showed that the transition temperature is a direct result of how the network of atoms connects and how much energy is needed to pull those connections apart. This approach offers a promising path forward for designing new materials, allowing scientists to understand the "why" behind the "what" without relying on opaque, unexplainable algorithms. The findings confirm that for these specific glasses, the interplay between bond strength and atomic packing is the key to unlocking their thermal behavior, providing a clear, interpretable rule for future discoveries.

Technical Summary: Physics-Informed Symbolic Regression for Predicting the Glass Transition Temperature of Alkali Borate Glasses

Problem Statement
The glass transition temperature ( $T_g$ ) of alkali borate glasses ( $x\text{M}_2\text{O} \cdot (100-x)\text{B}_2\text{O}_3$ , where M = Li, Na, K, Rb, Cs) is a critical property that is strongly composition-dependent. Predicting $T_g$ from first principles is challenging due to the structural complexity of the boron network, particularly the "boron anomaly" where coordination changes occur. While machine learning (ML) models, such as GlassNet, have demonstrated high predictive accuracy using large datasets, they typically function as "black boxes," lacking the internal interpretability required to reveal the physical mechanisms governing these properties. Existing models often fail to provide explicit mathematical relationships that connect composition to structure and thermodynamics in a human-understandable format.

Methodology
The authors employed Physics-Informed Symbolic Regression (SR) to derive an interpretable, closed-form expression for $T_g$ . The approach combined evolutionary search algorithms (using the PySR library) with a curated set of physically meaningful descriptors rather than raw compositional data alone.

Data Curation: A dataset of 309 compositions containing Li, Na, and K was assembled from the SciGlass database and literature. To prevent data leakage, chemically equivalent compositions were grouped, and median $T_g$ values were assigned. A separate set of 142 compositions containing Rb and Cs was reserved exclusively for extrapolation testing.
Feature Engineering: Input features were derived from fundamental physicochemical properties, including molar fractions, atomic radii, packing fractions, and dissociation energies. Crucially, the study utilized the Rigid Unit Packing Fraction (RUPF) proposed by Shi et al., contrasting it with the conventional Atomic Packing Fraction (APF).
Model Induction: The SR algorithm searched for mathematical expressions linking input features to $T_g$ . Constraints included a parsimony factor to favor compact equations and a restricted set of operators. The resulting raw expressions were manually revised to ensure dimensional consistency and physical interpretability.
Parameter Estimation & Extrapolation: Since dissociation energies ( $g_{\text{M}_2\text{O}}$ ) for Rb and Cs were unavailable, they were treated as adjustable parameters. These were fitted by minimizing the Mean Squared Error (MSE) between experimental and predicted $T_g$ values for the Rb and Cs systems.
Validation & Uncertainty: The model was validated through:
1. Statistical metrics (MAE, RMSE) on test and extrapolation sets.
2. Cross-property validation using the revised Makishima–Mackenzie (MM) model to predict Young's modulus ( $E$ ) using the fitted dissociation energies.
3. Monte Carlo (MC) uncertainty quantification to propagate errors from dissociation energy variability to the final $T_g$ prediction.

Key Contributions and Results

Derivation of an Interpretable Model: The study successfully induced a closed-form expression for $T_g$ (Eq. 10) that explicitly incorporates dissociation energies and packing fractions. The final equation highlights the dominance of dissociation energy terms, suggesting that local bonding strength is a primary driver of $T_g$ .
$T_g = \frac{23}{11} \frac{T_g^{\text{B}_2\text{O}_3}(x=0)}{g_{\text{BO}_3}} \left[ \frac{g_{\text{BO}_4}}{12} - \left( \sum g_i X_i - \frac{g_{\text{BO}_3}}{3} \right) \Delta_M \right]$
This model captures the non-monotonic behavior of $T_g$ (the boron anomaly) by accounting for the transition from three-fold ( $\text{BO}_3$ ) to four-fold ( $\text{BO}_4$ ) boron coordination and the subsequent formation of non-bridging oxygens.
Superiority of RUPF over APF: The study demonstrates that models based on the Rigid Unit Packing Fraction (RUPF) significantly outperform those using the conventional Atomic Packing Fraction (APF).
- RUPF Performance: Achieved an overall Root-Mean-Square Error (RMSE) of 15.61 K and Mean Absolute Error (MAE) of 12.61 K.
- APF Performance: Resulted in an RMSE of 89.86 K, indicating systematic overestimation of structural rigidity and $T_g$ due to the APF's inability to accurately represent the compactness of $\text{BO}_3$ units at intermediate compositions.
Successful Extrapolation: The model, trained only on Li, Na, and K systems, successfully extrapolated to Rb and Cs systems with maintained accuracy (RMSE ranging from 12.48 K to 17.22 K). The fitted dissociation energies for Rb and Cs followed the physically expected trend ( $g_{\text{Li}_2\text{O}} > g_{\text{Na}_2\text{O}} > g_{\text{K}_2\text{O}} \gtrsim g_{\text{Rb}_2\text{O}} \gtrsim g_{\text{Cs}_2\text{O}}$ ), confirming the model's physical consistency.
Cross-Property Validation: The dissociation energies fitted for $T_g$ prediction were used to calculate Young's modulus via the revised MM model. The predicted moduli for Rb and Cs systems showed good agreement with experimental and theoretical reference values (e.g., RMSE of 1.26 GPa for Cs systems), validating that the inferred parameters are physically consistent beyond the specific property they were trained to predict.
Uncertainty Quantification: Monte Carlo analysis revealed that prediction uncertainty is not uniform; it peaks in compositional regions associated with the boron anomaly (around 30 mol% alkali). This directly links model limitations to known structural transitions where the network's energetic balance is most sensitive to coordination changes.

Significance and Claims
The paper posits that physics-informed symbolic regression offers a transparent and interpretable alternative to black-box ML models for property prediction in glass systems. By integrating physicochemical descriptors (specifically dissociation energy and RUPF) into the regression process, the authors achieved a model that is not only statistically accurate but also physically grounded.

The significance of this work lies in its ability to:

Recover explicit structure-property relationships that align with established physical theories (e.g., topological constraint theory and the boron anomaly).
Demonstrate that fitting parameters for one property ( $T_g$ ) can yield physically consistent values for other properties (Young's modulus), suggesting a transferable framework for estimating scarce physical data (like dissociation energies) using abundant data (like $T_g$ ).
Provide a method to quantify prediction uncertainty, identifying specific compositional regimes where structural transitions introduce higher variability.

The authors conclude that while the model is robust within the studied alkali borate family, transferability to chemically distinct glass families should not be assumed without further validation, as different systems may be governed by different balances of bonding, packing, and network topology.

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