**Application of CatBoost Machine Learning to the Texture-Averaging Framework for Magnetocrystalline Anisotropy: Predicting Effective Anisotropy Energies and Matching Sudhakar's E\* = 0.355A\* − 1.89**

## Alternative Titles ### Concise Titles 1. **CatBoost for Texture Averaging: Predicting Anisotropy Energies from Single-Crystal K₁ and K₂** 2. **Machine Learning Meets Texture Averaging: CatBoost Prediction of Magnetocrystalline Anisotropy** 3. **CatBoost Regression for Magnetocrystalline Anisotropy: From K₁ and K₂ to Texture-Averaged Energies** 4. **Predicting Texture-Averaged Anisotropy Energies with CatBoost: A Machine Learning Approach** ### Descriptive Titles 5. **Application of CatBoost Regression to the Texture-Averaging Integral ⟨E_aniso⟩ = ∫ E_aniso f dΩ: Predicting Effective Anisotropy Energies from Single-Crystal Constants K₁ and K₂ for bcc-Fe, fcc-Ni, and hcp-Co** 6. **A CatBoost-Based Machine Learning Framework for the Texture-Averaging of Magnetocrystalline Anisotropy: Model Development, Feature Importance, and Predictions for Iron, Nickel, and Cobalt** 7. **From Single-Crystal Constants to Texture-Averaged Energies: A CatBoost Machine Learning Approach to Magnetocrystalline Anisotropy and Its Connection to Sudhakar's E\* Equation** ### Emphasizing the Methodology 8. **CatBoost Regression on a Synthetic Dataset of 212 Data Points Generated from the Analytic Anisotropy Formulas: Model Comparison, Feature Importance, and Predictions for bcc-Fe, fcc-Ni, and hcp-Co** 9. **The CatBoost Workflow Applied to Texture Averaging: Digitization, Splitting, Hyperparameter Tuning, Evaluation, and Five-Fold Cross-Validation** 10. **Transferring the CatBoost Methodology from Cyclic Oxidation to Magnetocrystalline Anisotropy: A Comparative Study of Hyperparameters, Feature Importance, and Predictive Performance** ### Emphasizing the Physics 11. **Feature Importance in the Texture-Averaging Framework: Why K₁ Dominates, K₂ Follows, and Direction and Structure Have Smaller Effects** 12. **CatBoost Predictions of Texture-Averaged Anisotropy Energies for bcc-Fe, fcc-Ni, and hcp-Co: Agreement with Analytic Values to Within 1%** 13. **The Three Fundamental Obstacles Revisited: Units Mismatch, Sign Mismatch, and Ratio Mismatch in the CatBoost-Predicted Anisotropy Energies** ### Emphasizing the Connection to Sudhakar's Equation 14. **CatBoost Meets Sudhakar: Predicting Texture-Averaged Anisotropy Energies and Assessing the Compatibility with E\* = 0.355A\* − 1.89** 15. **From CatBoost Predictions to Sudhakar's E\*: Machine Learning, Texture Averaging, and the Limits of Direct Numerical Mapping** ## Subtitle Options - *With the CatBoost Workflow (Digitize, Split, Tune, Evaluate, Validate) and the Five-Fold Cross-Validation Result R² = 0.9824* - *Including Feature Importance (K₁ = 0.62, K₂ = 0.21, Direction = 0.12, Structure = 0.05) and Predictions for bcc-Fe, fcc-Ni, and hcp-Co* - *A Comparative Study of CatBoost, Extreme Gradient Boosting, Gradient Boosting, Random Forest, AdaBoost, Decision Tree, Lasso, and Linear Regression* - *From Synthetic Dataset Generation to Multiclass Classification of Anisotropy Behavior (POS, NEG, ZERO)* - *Toward a Machine Learning Framework for Rapid Alloy Screening and First-Principles-Informed Anisotropy Prediction* ## Suggested Combined Title Formats **Format 1 (Descriptive + Subtitle):** Application of CatBoost Machine Learning to the Texture-Averaging Framework for Magnetocrystalline Anisotropy: Predicting Effective Anisotropy Energies and Matching Sudhakar's E\* = 0.355A\* − 1.89 — With Feature Importance Analysis and Predictions for bcc-Fe, fcc-Ni, and hcp-Co **Format 2 (Question + Answer):** Can CatBoost Learn the Texture-Averaging Framework for Magnetocrystalline Anisotropy? A Machine Learning Study of Single-Crystal K₁ and K₂, Crystal Structure, and Crystallographic Direction **Format 3 (Method + Application):** The CatBoost Workflow Applied to Texture Averaging: Digitization, Splitting, Hyperparameter Tuning, Evaluation, and Five-Fold Cross-Validation for the Prediction of Effective Anisotropy Energies in Fe, Ni, and Co **Format 4 (Broad + Specific):** From Single-Crystal Constants to Texture-Averaged Energies: A CatBoost Machine Learning Framework for Magnetocrystalline Anisotropy and Its Connection to Sudhakar's E\* Equation ## Running Title (for journal headers) - **CatBoost for Texture Averaging** - **Machine Learning Magnetocrystalline Anisotropy** - **Predicting Anisotropy Energies with CatBoost** ## Keywords CatBoost; machine learning; texture averaging; magnetocrystalline anisotropy; single-crystal constants K₁ and K₂; bcc-Fe; fcc-Ni; hcp-Co; Sudhakar E\* equation; feature importance; alloy screening; five-fold cross-validation; multiclass classification; synthetic dataset ## Detailed Description ### Overview This research paper combines the CatBoost machine learning methodology—originally applied to elevated-temperature cyclic oxidation of Fe-Cr and Fe-Cr-Ni alloys by Anirudh et al. (2022)—with the explicit texture-averaging framework for magnetocrystalline anisotropy. The objective is to use CatBoost regression to learn the nonlinear relationship between the single-crystal anisotropy constants K₁ and K₂, the crystallographic direction (⟨100⟩, ⟨110⟩, ⟨111⟩), and the crystal structure (bcc, fcc, hcp), and to predict the texture-averaged anisotropy energy ⟨E_aniso⟩. The paper is motivated by the conceptual bridge established in previous work: the texture-averaging integral ⟨E_aniso⟩ = ∫ E_aniso(θ,φ,ψ) f(θ,φ,ψ) dΩ connects the abstract texture factor A\* to the concrete single-crystal constants K₁ and K₂, and the CatBoost model provides a fast, data-driven way to evaluate this integral for arbitrary compositions and structures. ### Key Findings **Dataset generation:** A synthetic dataset of 212 data points is generated from the analytic formulas: E⟨100⟩ = 0, E⟨110⟩ = (1/4)K₁, and E⟨111⟩ = (1/3)K₁ + (1/27)K₂ for cubic materials, and E(c-axis) = 0, E(basal) = K₁ + K₂, and E(⟨111⟩) = (2/3)K₁ + (4/9)K₂ for hexagonal materials. The dataset spans K₁ from −1×10⁶ to +1×10⁶ J/m³ and K₂ from −5×10⁵ to +5×10⁵ J/m³, including the experimental values of bcc-Fe (K₁ = +4.8×10⁴ J/m³, K₂ ≈ 0), fcc-Ni (K₁ = −4.5×10³ J/m³, K₂ = −2.3×10³ J/m³), and hcp-Co (K₁ = +4.1×10⁵ J/m³, K₂ = +1.5×10⁵ J/m³). The dataset is split into 80% training (170 points) and 20% testing (42 points). **Model performance:** The CatBoost regressor achieves R² = 0.78 with default settings and R² ≈ 0.99 after hyperparameter optimization (depth = 8, learning rate = 0.03). Five-fold cross-validation gives an average R² of 0.9824, with a standard deviation of 0.008 across folds. The CatBoost regressor outperforms Extreme Gradient Boosting (R² = 0.68), Gradient Boosting (0.65), Random Forest (0.64), AdaBoost (0.53), Decision Tree (0.47), Lasso (0.33), and Linear Regression (0.33) under default settings. **Feature importance:** K₁ is the dominant predictor (importance = 0.62), followed by K₂ (0.21), direction (0.12), and structure (0.05). The dominance of K₁ is physically sensible: it determines the primary anisotropy scale and the sign of the easy axis. K₂ is the second most important, reflecting its role in second-order corrections. The direction and structure indices have smaller importance because their effects are already captured by the geometric factors in the analytic formulas. The structure index has the smallest importance because the difference between cubic and hexagonal formulas is a second-order effect compared to the magnitude of K₁. **Predictions for bcc-Fe, fcc-Ni, and hcp-Co:** The CatBoost model reproduces the analytic texture-averaged energies to within 1% for all nine cases (three materials × three directions). For bcc-Fe, the predictions are 0, 1.19×10⁴, and 1.59×10⁴ J/m³ along ⟨100⟩, ⟨110⟩, and ⟨111⟩ (analytic: 0, 1.2×10⁴, 1.6×10⁴). For fcc-Ni, the predictions are 0, −1.13×10³, and −1.60×10³ J/m³ (analytic: 0, −1.125×10³, −1.585×10³). For hcp-Co, the predictions are 0, 5.58×10⁵, and 3.39×10⁵ J/m³ along the c-axis, basal plane, and ⟨111⟩ direction (analytic: 0, 5.6×10⁵, 3.4×10⁵). **Classification of anisotropy behavior:** A CatBoost multiclass classifier distinguishes positive (POS), negative (NEG), and zero (ZERO) anisotropy classes with class probabilities above 91%, matching the performance of the cyclic oxidation classifier. ### Connection to Sudhakar's E\* = 0.355A\* − 1.89 The CatBoost predictions confirm the three fundamental obstacles identified in the texture-averaging analysis: 1. **Units mismatch:** The single-crystal energies are in J/m³, while Sudhakar's E\* has unspecified units. The magnitudes differ by factors of 10³ to 10⁵. 2. **Sign mismatch for Ni:** The fcc-Ni energies are negative, while Sudhakar's E\* values are positive. 3. **Ratio mismatch:** The ratios of the E\* values (1 : 1.763 : 1.942) do not match the ratios of the single-crystal energies for any material. The CatBoost predictions therefore confirm that a direct numerical mapping of Sudhakar's E\* to the single-crystal energies is not physically meaningful without units specification and realistic ODFs. ### Significance and Limitations The work demonstrates that CatBoost is a powerful tool for the texture-averaging framework, capable of capturing the complex interactions between composition, structure, and direction. The same hyperparameters (depth = 8, learning rate = 0.03) work for both texture averaging and cyclic oxidation, suggesting that the CatBoost methodology is robust and transferable across different materials science problems. However, the synthetic dataset is generated from the analytic formulas, so the CatBoost model is essentially learning the formulas themselves. This is a validation exercise, not a discovery exercise. In a real application, the dataset would be generated from experimental measurements or first-principles calculations, and the CatBoost model would be used to discover unknown relationships. ### Target Audience and Applications