Interpretable Machine Learning for High-Strength High-Entropy Alloy Design

Abstract: High-entropy alloys (HEAs) are metallic materials with solid solutions stabilized by high mixing entropy. Some exhibit excellent strength, often accompanied by additional properties such as magnetic, invar, corrosion, or cryogenic response. This has spurred efforts to discover new HEAs, but the vast compositional search space has made these efforts challenging. Here we present a framework to predict and optimize the yield strength of face-centered cubic (FCC) HEAs, using CoCrFeMnNi-based alloys as a case study due to abundant available data. Our novel Residual Hybrid Learning Model (RELM) integrates Random Forest and Gradient Boosting, enhanced by material attribute data, to handle sparse, skewed datasets for real-world alloys. A hybrid Generative Adversarial Network-Variational Autoencoder model explores new alloy compositions beyond existing datasets. By incorporating processing parameters, which determine the microstructure and thus strength, RELM achieves an R$^2$ score of 0.915, surpassing traditional models. SHapley Additive Explanations (SHAP) and Partial Dependencies enhance interpretability, revealing composition-processing-property relationships, as validated by experiments, including X-ray diffraction, SEM analysis, and tensile testing. The model discovered two novel Co${20}$Cr${16}$Fe${20}$Mn${16}$Ni${24}$Al$_4$ and Co${24}$Cr${12}$Fe${12}$Mn${16}$Ni${28}$Al$_4$Si$_4$ HEAs with a maximum possible yield strength of 842 and 937 MPa, significantly exceeding previously reported values for these alloy systems. This study pioneers interpretable machine learning in alloy design, providing a rigorous, data-driven approach to discovering, processing, and optimizing real-world materials. The findings highlight the critical role of both compositional and post-fabrication processing parameters in advancing the understanding of composition-processing-property relationships.

• The paper presents an interpretable machine learning framework integrating hybrid predictive (RELM) and generative (GAN-VAE) models to navigate the complex design space for high-strength high-entropy alloys.
• The RELM model achieved a high R² of 0.915 for predicting yield strength by incorporating both compositional and processing features, demonstrating improved performance over traditional models.
• Interpretability techniques like SHAP and PDP elucidated key composition-processing linkages, leading to the discovery and experimental validation of novel HEA compositions with high yield strengths of 842 MPa and 937 MPa.

This work presents a framework employing interpretable ML for predicting and optimizing the yield strength (YS) of face-centered cubic (FCC) high-entropy alloys (HEAs), using the CoCrFeMnNi system as a case study. The primary challenge addressed is navigating the vast compositional and processing parameter space inherent in HEA design to discover alloys with superior mechanical properties.

Methodology: Residual Hybrid Learning and Generative Models

The core of the framework is a novel Residual Hybrid Learning Model (RELM), designed to handle the sparse and often skewed datasets typical of real-world materials science research. RELM integrates Random Forest (RF) and Gradient Boosting (GB) algorithms. The rationale for this hybrid approach is to leverage the strengths of both methods: RF's robustness to overfitting and ability to handle high-dimensional data, and GB's capacity to sequentially correct errors and capture complex dependencies. The "residual" aspect implies that one model potentially learns from the errors or residuals of the other, aiming for improved predictive accuracy compared to standalone models.

Crucially, the model incorporates not only compositional information but also processing parameters, acknowledging their significant influence on microstructure and, consequently, mechanical properties like YS. Features include elemental concentrations (Co, Cr, Fe, Mn, Ni, Al, Si, etc.) and processing details such as annealing temperature, annealing time, and resulting grain size. Calculated material attributes like Valence Electron Concentration (VEC), mixing enthalpy ($\Delta H_{mix}$), and atomic size mismatch ($\delta$) are also included as features to provide physically grounded information to the model.

To explore novel compositional spaces beyond the existing dataset, a hybrid Generative Adversarial Network-Variational Autoencoder (GAN-VAE) model is utilized. This generative component allows for the suggestion of new alloy compositions predicted to possess high YS, guided by the patterns learned by the RELM model.

Implementation and Evaluation

The framework was implemented using a dataset compiled for CoCrFeMnNi-based HEAs, chosen due to the relative abundance of available experimental data covering variations in both composition and processing conditions.

Data Handling: The dataset, comprising compositional details, processing parameters, and measured YS, likely required careful preprocessing to handle missing values and normalize features, given the typical heterogeneity of materials data aggregated from various sources.

Model Training: The RELM model was trained on this dataset. The RF component likely involved training an ensemble of decision trees on different subsets of the data and features, while the GB component sequentially built trees to minimize a loss function (e.g., mean squared error for regression) based on the prediction errors of the previous trees. The integration mechanism between RF and GB in RELM is a key aspect; it might involve using RF predictions as features for GB, averaging predictions, or a more complex ensembling strategy.

Performance: The RELM model, incorporating both compositional and processing features, achieved a coefficient of determination (R²) score of 0.915 in predicting yield strength. This performance reportedly surpasses that of traditional ML models (presumably standard RF or GB alone) applied to the same task, highlighting the benefit of the hybrid approach and the inclusion of processing data.

import numpy as np from sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor from sklearn.model_selection import train_test_split from sklearn.metrics import r2_score class RELM: def __init__(self, rf_params={}, gb_params={}): self.rf = RandomForestRegressor(**rf_params) self.gb = GradientBoostingRegressor(**gb_params) # Potentially other components or weighting factors def fit(self, X_train, y_train): # Option 1: Sequential fitting (GB learns RF residuals) self.rf.fit(X_train, y_train) rf_preds = self.rf.predict(X_train) residuals = y_train - rf_preds self.gb.fit(X_train, residuals) # GB learns to predict the error of RF # Option 2: Feature augmentation (RF preds as input to GB) # self.rf.fit(X_train, y_train) # rf_preds_train = self.rf.predict(X_train) # X_train_augmented = np.hstack((X_train, rf_preds_train.reshape(-1, 1))) # self.gb.fit(X_train_augmented, y_train) # Option 3: Simple Averaging/Stacking (requires meta-learner) # Fit both independently, combine predictions later def predict(self, X_test): # Corresponding prediction logic based on fitting strategy # Option 1: rf_preds_test = self.rf.predict(X_test) gb_residual_preds = self.gb.predict(X_test) final_preds = rf_preds_test + gb_residual_preds return final_preds # Option 2: # rf_preds_test = self.rf.predict(X_test) # X_test_augmented = np.hstack((X_test, rf_preds_test.reshape(-1, 1))) # final_preds = self.gb.predict(X_test_augmented) # return final_preds

Interpretability and Validation

A significant emphasis is placed on interpretability using SHapley Additive Explanations (SHAP) and Partial Dependence Plots (PDP).

SHAP Analysis: SHAP values were calculated to quantify the contribution of each input feature (elemental concentrations, processing parameters like annealing temperature and time, grain size) to the prediction of YS for individual alloys and across the dataset. This allows for ranking feature importance and understanding the directionality of their influence (e.g., does increasing Ni content generally increase or decrease predicted YS?). The analysis revealed the critical interplay between composition and processing in determining strength.

Partial Dependence Plots (PDP): PDPs were generated to visualize the marginal effect of one or two features on the predicted YS while averaging out the effects of other features. For instance, a PDP could illustrate how predicted YS changes as annealing temperature varies, holding other factors constant on average. This helps elucidate specific composition-processing-property relationships identified by the model.

Experimental Validation: The insights derived from SHAP and PDP analyses regarding influential factors and optimal conditions were validated through targeted experiments. This involved synthesizing specific HEA compositions (including model-discovered ones), subjecting them to predicted optimal processing conditions, and characterizing their microstructure (XRD for phase identification, SEM for morphology and grain size) and mechanical properties (tensile testing for YS). The experimental results reportedly confirmed the model's predictions and the identified relationships.

Key Discoveries and Contributions

The interpretable ML framework successfully identified complex relationships governing the YS of CoCrFeMnNi-based HEAs. A key finding reaffirmed by the model and SHAP analysis is the substantial impact of post-fabrication processing parameters, particularly annealing conditions and the resultant grain size, which can modulate the strength achieved for a given composition according to Hall-Petch principles and phase stability considerations.

Guided by the GAN-VAE exploration and RELM predictions, the study identified and subsequently synthesized two novel HEA compositions: Co$_{20}$Cr$_{16}$Fe$_{20}$Mn$_{16}$Ni$_{24}$Al$_4$ and Co$_{24}$Cr$_{12}$Fe$_{12}$Mn$_{16}$Ni$_{28}$Al$_4$Si$_4$. Experimental validation confirmed high yield strengths of 842 MPa and 937 MPa, respectively. These values represent significant improvements over previously reported YS for similar FCC HEAs within this system, demonstrating the framework's efficacy in discovering high-performance materials.

The work establishes a data-driven approach that integrates predictive modeling (RELM), generative design (GAN-VAE), and interpretability techniques (SHAP, PDP) to accelerate HEA discovery. It underscores the necessity of incorporating processing variables into materials informatics models for accurate property prediction and optimization.

Conclusion

This research provides a detailed framework combining hybrid machine learning models (RELM) and generative techniques (GAN-VAE) with interpretability methods (SHAP, PDP) for the design of high-strength HEAs. By integrating compositional and processing data for CoCrFeMnNi-based alloys, the model achieved high predictive accuracy (R² = 0.915) for yield strength. The interpretability analysis elucidated key composition-processing-property linkages, and the framework guided the discovery and experimental validation of novel HEAs with exceptionally high yield strengths, demonstrating a promising pathway for accelerated, data-informed materials development.