Comparative analysis of machine learning approaches for predicting and interpreting Cz-sapphire growth

• Performed comprehensive analysis of Cz-sapphire growth using machine learning methods. • SVM, XGBoost, and Gradient Boosting excel in predicting nonlinear relationships. • Crystal rotation rate has shown most critical factor for interface deflection. • Absorption coefficient influences temperature gradients at Interface. • Optimizing reflector length and angle improved local heat distribution. This study evaluates four machine learning (ML) “white-box” methods − Decision Trees, Linear Regression, Python Symbolic Regression (PySR), and Sure Independence Screening and Sparsity Operation (SISSO) − and five “gray-box” methods − Gradient Boosting, XGBoost, Support Vector Machines (SVM), Gaussian Processes, and Random Forests − for analyzing data from the Cz-sapphire crystal growth process. The objective is to develop a model that achieves a balance between high predictive accuracy and interpretability in this small-data domain. Twelve input variables − including process parameters, sapphire optical properties, and furnace geometry parameters − were analyzed in relation to five output variables: heating power, interface deflection, temperature gradients averaged at the solid/liquid interface and symmetry axis, and v/G. Using 500 data tuples from CFD simulations, the results highlighted significant performance differences across the models. SVM demonstrated superior performance in predicting temperature gradients, XGBoost excelled in interface deflection predictions, and Gradient Boosting was most effective for v/G. SISSO, known for its high interpretability, performed best in predicting heating power, particularly in cases where nonlinear noise was less pronounced. The best performing models for each output generated explicit equations that relate inputs to outputs, feature importance plots, and 3D plots illustrating relationships within the 17-dimensional space. These findings offer theoretical insights for optimizing the Cz-sapphire crystal growth process and design.