Machine Learning-Based Optimization of Clausius-Mossotti Equation for Dielectric Constant Calculation

The Clausius-Mossotti (C–M) equation is one of the most fundamental theoretical models to calculate the dielectric constant of materials. As was proposed over a century ago, it has been widely applied in dielectric material research due to its relatively simple mathematical form and clear physical interpretation. The equation, derived based on the Lorentz effective field approximation and the assumption of constant polarization, establishes a relationship among the macroscopic dielectric constant and the microscopic polarizability and unit cell volume. However, with the growing understanding of dielectric materials, it becomes clear that dielectric constant is affected by a range of additional factors, such as bond length, packing density and symmetry. These factors are not captured by the conventional C–M equation, indicating the limitations of the equation. Recently, machine learning (ML) methods show a promise in improving the prediction accuracy of dielectric properties. The ML models can outperform classical equations in terms of accuracy, but they often lack interpretability, which hinders their widespread use in material design and optimization. To address these issues, this study proposed a novel bidirectional embedding approach with domain knowledge and machine learning. The ML-based correction term was integrated into the classical C–M equation, enhancing its predictive accuracy, while retaining physical interpretability. This work could explore a potential of combining physical formulas with ML methods to create more robust and explainable models for material design.

The data of single-phase microwave dielectric ceramics were collected from published literatures and the Materials Project database. The dataset included materials whose dielectric constants were measured by the Hakki-Coleman method in a frequencies range from 5 to 18 GHz. A machine learning model identified molecular dielectric polarizability per volume (ppv), average bond length (blm) and unit cell volume (va) as the most significant contributors to the dielectric constant in a previous study. To enhance the accuracy of the C–M equation, correction terms were introduced based on the three key features. Symbolic regression, combined with genetic algorithms, was used to derive mathematical expressions for the correction terms. The symbolic regression technique utilized evolutionary algorithms to generate and evolve potential expressions. The structures and parameters of expressions were optimized to minimize prediction errors. Hyperparameters were optimized by a grid search method to identify the best-performing models. The process was carried out iteratively. A total of 16,000 candidate mathematical expressions were generated and compared. Finally, the Pareto front analysis was used to select the optimal expression that balances accuracy and complexity.

The revised C–M equation results in significant improvements in its predictive performance. For the Shannon polarizability dataset, the R² value of the revised C–M equation is increased by 42.29%, and the RMSE value is decreased by 14.72%. The correction term compensates for the inaccuracies of the original equation, particularly in cases where the molecular dielectric polarizability per volume of Shannon value (ppv_s) is either overestimated or underestimated. For low-polarizability materials with ppv_s values less than 0.2087, the Shannon database tends to underestimate the dielectric constant, and the correction term compensates for this error. Conversely, for high-polarizability materials, the dielectric constant is often overestimated, and the correction term reduces the overestimation. The analysis of the relationship between the correction term (Δppv_s) and features reveals that the correction term is most sensitive to ppv_s, followed by blm, with va having the least impact. This finding indicates that the dielectric polarizability plays a dominant role in determining the accuracy of the revised C–M equation. The influence of the features on the correction term is further explored through partial derivatives. The first-order partial derivatives show that the correction term’s contribution from ppv_s is much greater than that from blm or va. The second-order partial derivatives reveal a non-linear relationship between the correction term and ppv_s and blm, while the relationship with va is linear. The results indicate that the compensatory effect of the correction term gradually diminishes, and the suppression effect becomes more pronounced as ppv_s increases. This behavior is consistent with the correction mechanism of “elevating the underrated, reducing the overrated”.

This study introduced a bidirectional embedding approach that could integrate machine learning with domain knowledge to enhance both the accuracy and interpretability of the Clausius-Mossotti equation. The revised equation achieved a higher prediction accuracy with a 42.29% increase in R² and a 14.72% decrease in RMSE via incorporating the correction term derived from symbolic regression. The revised equation retained the physical meaning of the original one. The research could highlight a potential of bidirectional embedding approach to create models that could balance both accuracy and interpretability, offering a promising perspective on optimizing classical equations and material design.