The increasing demand for next-generation soft-magnetic materials in power generation and distribution necessitates the development of advanced materials. Fe-based metallic glasses (MGs) are promising candidates due to their low cost, low core loss, and high saturation flux density (Bs). However, conventional design methods rely heavily on "trial and error", making it challenging to balance Bs and thermal stability. This is because the glass-forming elements are often anti-magnetic, creating a complex interplay between glass formation and magnetic coupling. Machine learning (ML) offers a potential solution by enabling the system to learn from data and predict material properties based on intrinsic elemental characteristics. While ML has been applied to various materials, its application to the design of soft-magnetic Fe-based MGs remains limited, with existing models often acting as "black boxes" without revealing underlying physical mechanisms. This study aims to address this gap by developing an interpretable ML model to identify key parameters governing the properties of Fe-based MGs and to unveil the physical origins of these relationships, thereby facilitating efficient alloy design.

Existing literature extensively explores Fe-based MGs and their composites, such as FINEMET, NANOPERM, and HITPERM, highlighting their successful commercialization despite reliance on extensive "trial and error" experimentation. A significant challenge lies in the trade-off between soft-magnetic properties and thermal stability; improving one often deteriorates the other. Previous ML studies on MGs have focused primarily on predicting glass formation behavior using techniques like support vector machines (SVM). However, these studies haven't addressed the specific challenge of designing soft-magnetic Fe-based MGs with a balanced combination of high Bs and thermal stability, nor have they provided insights into the underlying physical mechanisms. Recent efforts to improve the interpretability of ML models by incorporating physical features into the ML process provide a foundation for this research.

The researchers compiled a dataset of 252 Fe-based MGs from the literature, including composition data and corresponding properties (Bs and Tx). The XGBoost algorithm was employed to develop a predictive model. The input features included 30 parameters, such as valence electron concentration (VEC), VEC without Fe (VEC1), electronegativity (χ), averaged atomic radius difference (δ), and melting point (Tm), representing intrinsic elemental properties. A 5-fold cross-validation was used to evaluate the model's performance. Model parameters (test size and max depth) were optimized to achieve R² > 0.92, prioritizing Bs prediction accuracy, using a composite R² metric (R² = 0.6R²Bs + 0.4R²Tx). Feature importance was assessed by iteratively removing the least important feature and retraining the model until the top ten most significant features for each target property were identified. Data scaling was employed using a simplified method to normalize different ranges of values (e.g., temperature and atomic radius). The XGBoost model operates on an objective function that minimizes a combination of loss function and regularization to prevent overfitting, leveraging techniques such as shrinkage and descriptor subsampling. The detailed definition of features such as δ, Tm, χ, VEC, and VEC1 are mathematically defined in the paper using standard formulas incorporating atomic radii, melting points, electronegativity and number of valence electrons for the constituent elements. The core XGBoost algorithm is mathematically described, including the definition of the objective function (including loss and regularization terms), additive training procedure and splitting criteria for the tree-based model.

The optimized XGBoost model achieved high prediction accuracy, with R² values of 0.934 and 0.947 for Bs and Tx, respectively. Feature importance analysis revealed that VEC1 was the most important feature for predicting Bs, while δ and VEC were the most important features for Tx. Further analysis revealed a strong linear correlation between Bs and VEC1 for Fe-based MGs without Co and Ni, represented by the equation Bs = 2.32 - 0.998 × VEC1. This equation is consistent with magnetic valence theory and charge transfer model, explaining the influence of valence electrons on Bs. The crystallization temperature (Tx) showed a positive correlation with δ and a negative correlation with VEC, leading to the empirical relationship Tx = 1518.5 + 27.1 × δ - 123.7 × VEC. This is explained by considering both topological and kinetic aspects of crystallization: δ reflects the contribution of atomic size mismatch in stabilizing the amorphous phase, while VEC signifies the influence of chemical short-range order on nucleation. Several Fe-based MG ribbons were fabricated to validate the model. The experimental results closely matched the model's predictions, with less than 10% error in Bs and 5% error in Tx. Specifically, Fe73.8B15.79Si6.9Ta0.75Zr2.76 achieved Bs = 1.34 T and Tx = 865 K, significantly outperforming existing materials in terms of both soft-magnetic properties and thermal stability. The results demonstrate that the developed ML model and the resulting empirical formulas can effectively guide the design of Fe-based MGs with desirable properties.

The high accuracy and interpretability of the XGBoost model demonstrate its effectiveness in designing Fe-based MGs. The identified key features (VEC1, δ, and VEC) provide valuable insights into the fundamental relationships between elemental properties and macroscopic material behavior. The derived empirical equations for Bs and Tx offer practical guidelines for alloy design, enabling the efficient selection of elements and their concentrations to achieve desired properties. The successful experimental validation confirms the model's predictive power and its potential to accelerate materials discovery. The model's interpretability avoids the limitations of "black box" ML approaches, allowing for physical understanding of the predictions. The limitations of excluding Co and Ni in the analysis of Bs are noted and should be considered in future research.

This research successfully developed an interpretable XGBoost ML model for designing Fe-based MGs with enhanced soft-magnetic properties and thermal stability. The model's high prediction accuracy, coupled with the identification of key features and the derivation of empirical equations, provides a powerful tool for guiding materials design. Future research could focus on extending the model to include more alloying elements, exploring the influence of Co and Ni, and investigating other MG systems.

The study focused primarily on Fe-based MGs without Co and Ni, limiting the generalizability to systems containing these elements. The empirical equations derived for Bs and Tx might require further refinement with a larger and more diverse dataset. The accuracy of the model relies on the quality and comprehensiveness of the input data. Finally, the simplified data scaling method might be replaced with more sophisticated methods.