Machine Learning-Based Prediction of Complex Combination Phases in High-Entropy Alloys

2.1. Dataset Construction

2.1.1. Data Collection and Feature Extraction

The data for this study were sourced from the existing literature [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64]. The dataset underwent a manual refinement process which included removing duplicate entries, addressing missing and inconsistent phase data, and limiting the scope to alloys synthesized via arc melting. Alloys produced by different synthesis methods such as powder metallurgy, mechanical alloying, laser melting deposition and additive manufacturing were excluded to avoid the influence of production techniques on the results [9]. After this refinement process, 329 entries remained including 96 BCC, 55 FCC, 52 BCC+FCC, 67 BCC+IM, 46 FCC+IM and 13 BCC+FCC+IM phases. When working with imbalanced datasets, as in this study, careful data handling is crucial to maintain predictive accuracy. To address this, a Boolean vector approach was introduced. In cases where multiple phase combinations need to predicted, a Boolean vector can represent each phase as a separate binary outcome. This enables the model to focus on learning which elements or conditions are associated with each phase combination individually rather than struggling to learn from an uneven distribution across multiple overlapping classes [65]. The corresponding phases were encoded with three entries corresponding to {BCC, FCC, IM} phases as shown in Table 1. A value of 0 indicates the absence and 1 indicates the presence of a specific phase. The logical values in the target variable indicated the set of phases present in each HEA sample. For instance, [1,0,0] represents BCC, [1,1,0] represents BCC+FCC and [1,1,1] represents BCC+FCC+IM.

The dataset comprises numerical features derived from the chemical composition of the alloys and their corresponding phases. Seven features were extracted from the chemical composition of each entry using the equations presented in Table 2. These features were ${∆S}_{mix}$, $∆H_{mix}$, $T_m$, $\Omega$, $\delta$, $VEC$ and $∆\mathsf{\chi }$. These calculated features provide essential information for the ML models to understand the underlying relationships in the alloy compositions enabling the prediction of phase structures with greater accuracy.

2.1.1. Data Normalization

Preprocessing data in ML is an essential step to enhance data quality and ensure their suitability for ML algorithms. One common technique is normalization which standardizes the range of feature data, ensuring that each feature contributes approximately equally to the final prediction as described by the following formula:

$$x_{new}={\displaystyle \frac{x_i-\overline{x}}{\sigma }}$$

(1)

where $x_{new}$ is the standardized value of the feature, $x_i$ is the original value of the feature, $x_i$ is the mean value of the feature and $\sigma$ is the standard deviation of the feature. The purpose of normalization is to produce dimensionless numerical features so that each data point is on the same numerical scale. This process is crucial for improving the accuracy of the ML model by ensuring that no single feature dominates the learning process due to its scale.

2.2. ML Algorithms

ML algorithms have become a power tool for predicting complex material properties including the phase structure of HEAs. By leveraging large datasets and sophisticated algorithms, researchers can identify patterns and relationships that traditional methods may overlook. In this study, four ML algorithms were employed to predict the phases of HEAs: SVM, KNN, RF and NN. These algorithms were implemented using the scikit-learn library and were selected based on their frequent application in similar studies and their demonstrated effectiveness in handling classification tasks in material science. Each algorithm offers unique strengths: SVM excels in distinguishing non-linear relationships, KNN is simple yet effective for pattern recognition, RF is robust against overfitting and interpretable due to feature importance analysis, and NN is well-suited for capturing complex, non-linear interactions within the dataset.

• Support Vector Machine (SVM)

The SVM aims to find the hyperplane that best separates different classes of data points. This separation is achieved by minimizing the loss function which involves finding the hyperplane that maximizes the margin between the classes. It can handle nonlinear boundaries by using the Kernel function [68,69]. the Kernel function transforms the data into a higher dimensional space where separation might be possible. The key hyperparameters used for the SVM in this study are penalty (C), the Kernel function and the Kernel coefficient (gamma). When C is too large, the model may overfit the training data, leading to poor generalization to new data. When C is too small, the model might not capture outliers well resulting in high training set errors and limiting to fit the training data. Different Kernel functions serve different purposes and can capture various types of relationship in the data. The Radial Basis Function (RBF), Polynomials, and Sigmoid Kernels are Kernel functions used for the SVM in this study. Gamma affects the sensitivity to differences in feature vectors. If gamma is set too high, the model might overfit as it would become sensitive to individual data points. If gamma is chosen too low, the model may not capture the underlying patterns in the data.

• K-Nearest Neighbor (KNN)

The KNN makes predictions based on the minimum distance between data points and a simple majority vote from the nearest neighbors. Its hyperparameters are the number of neighbors to consider for prediction, the weight function, the algorithm for computing the nearest neighbors and a distance metric [29,70]. Smaller values of nearest neighbors can make the model sensitive to noise, while larger values can make it biased. For this study, two common weight function options are uniform (equal weights) and distance (inverse of distance as weight). The algorithms used for efficiently computing the nearest neighbors are Auto, ball tree, KD tree and brute force. Euclidean and Manhattan are used to measure the distance between data points.

• Random Forest (RF)

RF is an ensemble technique that combines multiple decision trees to make predictions. Each tree relies on values from a random vector sampled independently and from the same distribution for all trees. The majority predicted class across trees become the final predicted class. The hyperparameters for tuning this model include the number of trees, the maximum feature, the maximum tree depth, the minimum sample split, the minimum sample leaf and bootstrap sampling. The number of trees determine the decision tree in the forest. Increasing the number of trees can improve model performance, but it also increases computation time. The maximum feature specifies the maximum number of features considered for splitting a node in each decision tree. This parameter can impact the diversity and randomness of the trees, reducing overfitting. The maximum tree depth defines the maximum number of levels a decision tree can have. Restricting tree depth can help prevent overfitting and enhance generalization. The minimum sample split sets the minimum number of data points required in a node before it is eligible for further splitting. The minimum sample leaf specifies the minimum number of data points that a leaf node must contain. If bootstrap sampling is set to true, sampling is performed with replacements when constructing individual decision trees [68].

• Neural Network (NN)

The NN is a type of feedforward neural network with multiple layers including an input layer, one or more hidden layers and an output layer. The key hyperparameter associated with training the NN are number of the hidden layers, size, activation functions, the loss function, the optimizer and the learning rate. The number of hidden layers and the number of neurons in each hidden layer are crucial parameters which define the depth and width of the neural network. More layers and neurons allow the network to capture more complex patterns but also increase the risk of overfitting and the computational complexity [71]. Activation functions like rectified linear unit (ReLU), sigmoid or tanh are used to introduce nonlinearities in the model, allowing it to learn more complex relationships. Different activation functions have different properties and are suitable for different types of data. The loss function is a measure of how well the neural network is performing. The optimizer is an algorithm that adjust the weights of the network to minimize the loss. Common optimizers include stochastic gradient descent (SGD), adaptive moment estimation (Adam), etc. The learning rate is a hyperparameter that controls how to change the model in response to the estimated error when the model weights are updated. A small value can make the training process very slow while a large value can cause the model to converge too quickly to the optimal solution or even diverge.

Table 3 summarizes the hyperparameter options and their ranges for the SVM, KNN, RF and NN used for the phase prediction of HEAs in this study. Each hyperparameter is crucial for tuning each model to achieve optimal performance. The dataset was randomly divided into two subsets at an 80:20 ratio for training and testing. This means that 80% of the data were used to train the models while the remaining 20% were reserved for testing the model performance. This split helps ensure that the models are trained on a comprehensive portion of the data while also providing an unbiased evaluation of their performance on unseen data. To prevent overfitting where a model performs well on the training data but poorly on new data, a cross-validation (CV) technique was employed during the model training process. In five-fold CV, the training data are divided into five subsets. The model is trained on four of these subsets and validated on the remaining subset. This process is repeated five times, each time with a different subset used for validation [72]. The results are then averaged to provide a more accurate estimate of the model’s performance. This method ensures that the model generalizes well to new data, improving its robustness and reliability.

2.3. Evaluation of Model Performance

During model training, five-fold CV was applied to assess the mean CV accuracy of each model, providing an initial measure of model reliability and generalizability. For testing with an unseen dataset, multiple evaluation metrics were calculated to provide a comprehensive performance assessment. These metrics included accuracy, precision, recall and F1-score. Accuracy measures the ratio of correctly predicted instances to total instances. Precision calculates the proportion of true positives among all predicted positives. Recall assesses the model’s ability to identify all true positives. F1-score combines precision and recall to find their harmonic mean for a balanced assessment. These metrics are defined as follows [73]:

$$Accuracy={\displaystyle \frac{TN+TP}{TP+TN+FN+FP}}\times 100\%$$

(2)

$$Precision={\displaystyle \frac{TP}{FP+TP}}\times 100\%$$

(3)

$$Recall={\displaystyle \frac{TP}{FN+TP}}\times 100\%$$

(4)

Where a true negative ($TN$) is an instance that is actually negative and correctly predicted as negative, a true positive ($TP$) is an instance that is actually positive and correctly predicted as positive, a false positive ($FP$) is an instance that is actually negative but incorrectly predicted as positive and a false negative ($FN$) is an instance that is actually positive but incorrectly predicted as negative. Additionally, the Receiver Operating Characteristic (ROC) curve and the Precision-Recall (PR) curve were analyzed to provide a better understanding of model performance. The ROC curve provides insight into the trade-off between the true positive and false positive rate, while the PR curve highlights the balance between precision and recall, which is especially valuable for assessing models on imbalanced datasets.

The complex combination phase prediction accuracy offers a rigorous evaluation standard for applications requiring precise phase identification in HEAs. This metric along-side the other performance indicators, provides a comprehensive overview of each model’s suitability for predicting HEA phase compositions in both individual and multi-phase scenarios.

3. Results and Discussion

3.1. Data analysis

The pair plot generated using the Seaborn library, as depicted in Figure 2, showcases the relationships between pairs of features. Each scatter plot point corresponds to a specific phase in the dataset, with its position representing the values of the two features for that phase. The diagonal cells contain Kernel density plots illustrating the distribution of each individual features. The pair plot reveals that certain feature pairs, such as $VEC$ and ${∆H}_{mix}$ exhibit noticeable trends, where specific phase categories cluster within distinct regions. For instance, FCC phases tend to align with higher VEC values, while BCC phases correlate with lower $VEC$ values and more negative ${∆H}_{mix}$. These relationships indicate that some features play a critical role in determining phase stability and composition, offering valuable insights for phase prediction.

The radar plot in Figure 3 provides a comparative visualization of the normalized feature values for each phase category. The plot highlights distinct patterns across the phases, with BCC phases exhibiting lower $\delta$ values and higher $T_m$ while FCC phases show relatively higher $\delta$ values. IM phases, on the other hand, demonstrate a unique combination of high $\delta$ and more negative ${∆H}_{mix}$, reflecting their ordered structures and complex bonding characteristics. These differences emphasize the significance of these features in distinguishing between phase categories

Figure 4 shows a heat map illustrating the relationships or correlations between feature in the dataset. Each cell in the matrix represents the correlation coefficient between two features, ranging from -1 to 1. A value of 1 indicates a strong positive correlation, -1 indicates a strong negative correlation and 0 indicates no correlation between features. The values in the heatmap are correlation coefficients which measure the strength and direction of the linear relationship between two features. The most commonly used correlation coefficient is the Pearson correlation coefficient calculated as follows:

$$r_{xy}={\displaystyle \frac{\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\sqrt{\sum {\left(x_i-\overline{x}\right)}^2}\sum {\left(y_i-\overline{y}\right)}^2}}$$

(5)

where $r_{xy}$ is the Pearson correlation coefficient between feature $x$ and $y$. $x_i$ and $y_i$ are the individual sample points indexed with $i$. $\overline{x}$ and $\overline{y}$ are the means of the $x$ and $y$ features, respectively.

The correlation values in heat map of two independent features range from -0.75 to 0.58. Specifically, $VEC$ and $T_m$ exhibit a negative correlation implying that as $VEC$ increases, $T_m$ tends to decrease. Higher $VEC$ values favor FCC structures whereas lower $VEC$ values stabilize BCC structures, since FCC phases often have lower $T_m\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}$. Conversely, the electronegativity difference and $T_m$ show a positive correlation, meaning that as $∆\mathsf{\chi }$ increases, so does the melting temperature. Larger $∆\mathsf{\chi }$ values can promote the formation of complex phases or IM phases which often have higher $T_m$ than SS phases [74]. Overall, there are no strong positive or negative correlations between the features. Thus, all the features can be retained.

The 3D scatter plot was created to illustrate the phase distribution across three axes: $\mathsf{\delta }$, $∆H_{mix}$ and $VEC$. The relationship between these features and the corresponding phases is displayed in Figure 5. From the plot, it is apparent that IM phases have higher $\delta$ and lower $∆H_{mix}$ values compared to solid solution phases (BCC and FCC). IM phases often involve the bonding of atoms with significantly different atomic sizes, creating lattice distortions and stains within the crystal structure leading to distinctive and ordered structures. Gibbs free energy $(∆G$) determines the thermodynamic stability of phases. A lower $∆H_{mix}$ results in a lower $∆G$, which can favor the formation of intermetallic compounds. When examining solid solution phases. A higher $VEC$ tends to result in the formation of FCC phases, while a lower $VEC$ tends to lead to the formation of BCC phases. Lower $\delta$ values generally encourage the formation of SS phases, whereas significant atomic size disparities are more likely to result in the formation of IM phases. The results are consistent with earlier research [10,12,26].

This study utilizes the RF classifier to rank the relevant features and illustrate their importance as shown in Figure 6. The results are consistent with earlier research [30,74] which also identified $VEC$ and $T_m$ as significant predictors of phases in HEAs.

3.2. Model Performance for Phase Prediction

In this study, four ML algorithms (SVM, KNN, RF and NN) were employed to predict phase compositions in HEAs based on a structured dataset. To ensure model generalizability and avoid overfitting, hyperparameters were optimized using grid search combined with layered cross-validation. This process involved training the models on four sets of data and validating of them on the fifth, with grid search identifying the best parameter combinations to enhance model performance. Table 4 provides a comprehensive evaluation of these model for each phase (BCC, FCC and IM) using metrics such as mean CV accuracy, test accuracy, precision, recall and F1-score. A summary of the CV and test accuracies in Figure 7 shows that all the models achieve an average CV accuracy above 81%, confirming their viability for phase prediction in HEAs. The following section discusses each model’s performance by phase.

1.

SVM

• BCC Phases: The SVM achieves a mean CV accuracy of 90.89% and a test accuracy of 92.42%. Precision, recall and F1-score are all above 90%, with precision at 93.15 being the highest. This suggests that the SVM is highly reliable for correctly identifying BCC phases with only minor classifications.
• FCC Phases: The SVM performs exceptionally well for the FCC phases with test accuracy and all other metrics at 96.97%. This indicates that the SVM effectively classifies the FCC phase with near-perfect accuracy, likely due to clear distinguishing features in the dataset for this phase.
• IM Phases: For IM phases, the SVM shows a lower performance compared to for BCC and FCC phases, with a mean CV accuracy of 79.81% and a test accuracy of 85.24%. Precision and recall are both around 81-82% indicating moderate performance.

2.

KNN

• BCC Phases: The KNN shows excellent performance for BCC phases with a high mean CV accuracy of 92.79% and an impressive test accuracy of 98.48%. Both precision and recall are very high at 97.83% and 98.48%, respectively, indicating that the KNN is particularly effective in distinguish BCC phases.
• FCC Phases: The KNN’s performance for FCC phases is similarly strong, with a mean CV accuracy of 93.18%, a perfect test accuracy and other metrics at 96.97%. This suggests that the KNN can reliably identify FCC phases, similar to SVM.
• IM Phases: the KNN also performs well for IM phases with a mean CV accuracy of 83.27% and a test accuracy of 86.36%. The precision and F1-score are around 85-87%, indicating a balanced performance with a slight improvement over the SVM.

3.

RF

• BCC Phases: The RF achieves a mean CV accuracy of 89.36% and a strong test accuracy of 95.45% along with high precision (94.64%) and recall (96.50%). These metrics suggest that RF is highly effective in identifying BCC phases although it slightly lags behind the KNN in precision.
• FCC Phases: The RF performs well for FCC phases with a mean CV accuracy of 91.26% and a test accuracy of 93.94%. The precision, recall and F1-scores are all close to 94%, indicating a solid performance but slightly lower than the performance of the SVM and KNN. The RF is still reliable for FCC phases but the slightly lower metrics suggest that FCC features are well-learned but not perfectly.
• IM Phases: The RF’s performance for IM phases is moderate with a mean CV accuracy of 81.36% and a test accuracy of 81.81%. Precision and recall are around 80-83%, which are the lowest among the models. This lower performance for IM phases could indicate that RF struggles to differentiate IM from other phases, possibly due to the complex feature space of IM phases or insufficient training samples.

4.

NN

• BCC Phases: The NN has the highest mean CV accuracy for BCC phases at 94% and a high test accuracy of 95.45%. Precision, recall and F1-score are all above 94%, showing that the NN performs comparably to the KNN and RF for BCC phases. This suggests that the NN has a strong predictive capability for BCC phases and handles the features well.
• FCC Phases: The NN shows a strong performance for FCC phases with all metrics at 96.97%. This indicates that the NN is reliable and effective for FCC phase classification, similar to the SVM and KNN.
• IM Phases: The NN achieves the highest mean CV accuracy and test accuracy for IM phases at 86% and 87.88% respectively. Precision and F1-scores are also high at around 87-88%, indicating that the NN is the most effective model for distinguishing IM phases among the four.

Overall, the results indicate that the NN and KNN consistently perform best for BCC and FCC phases followed closely by RF. The SVM also performs well but is slightly behind in term of recall and F1-score for BCC phase. IM phase classification has the lowest performance across models, which is likely due to the complex or overlapping features associated with IM phases. The NN’s superior performance in this category suggests that it is particularly adept at handling complex, non-linear patterns in the feature space. This finding highlights that both The KNN and NN are highly effective for BCC and FCC phase predictions, whereas The NN is the most robust for distinguishing IM phases, making it a promising model for complex HEA compositions where phase combinations may be intricate.

To further assess the performance of the NN model which achieved the highest CV accuracy, additional evaluation methods, namely, the Precision-Recall (PR) and Receiver Operating Characteristic (ROC) curve were compiled in this study. These curves help analyze classifier performance and provide visual insights into the model’s predictive abilities. The PR curve illustrated in Figure 8(a) is used to visualize the classifier’s predictive accuracy. A larger area under the curve (AUC) indicates better predictive accuracy. For the NN algorithm, the area for the BCC (PR AUC: 99%) and FCC (PR AUC: 100%) phases shows an excellent performance in predicting BCC and FCC phases. The area of the IM (PR AUC: 82%) phase has a reasonable performance but it is lower than the others. This suggests that the model has some difficulty in consistently identifying IM phase instances correctly compared to BCC and FCC phases. The discrepancy could be due to the inherent differences in the dataset’s feature representation or the phase overlap in feature space. Similarly, the ROC curve shown in Figure 8(b) is another metric used to evaluate the performance of the ML model. The AUC of the ROC curve indicates the classifier’s ability to distinguish between positive and negative classes. The high AUC scores for BCC (ROC AUC: 99%) and FCC (ROC AUC: 100%) phases show that the model is very effective in identifying these two phases. However, the slightly lower AUC for the IM (ROC AUC: 89%) phase suggests that the model has more difficulty in distinguishing this phase which may be due to fewer training samples for the IM phase in the dataset.

3.3. Complex Combination Phase Prediction Results

The previous sections detailed each model’s performance for individual phases, but in practical applications, accurate prediction of complex phase combinations in HEAs is crucial. Therefore, a strict evaluation criterion was applied, where only predictions with entirely correct combinations were scored as accurate. Partially or totally incorrect predictions were excluded from the accuracy calculation. This approach was chosen due to the multi-phase nature of the task where accurate prediction of all phase components in each alloy is necessary for the prediction to be considered valid. As illustrated in Figure 9, the complex combination phase prediction accuracy across the models ranges from 75.75% to 84.85%, with notable variations in performance. This strict evaluation criterion reveals that the test accuracy for each model is slightly lower than the mean CV accuracy previously reported. This reduction is expected as the stringent criterion applied during testing requires the model to predict all phases in each combination accurately rather than evaluating them individually. Under this challenging evaluation, the NN and KNN achieved the highest prediction accuracy reaching 84.85%. This demonstrates the NN’s and the KNN’s robustness in handling complex phase combinations which may involve non-linear interactions among features. In contrast, the SVM and RF achieved moderate prediction accuracies of 77.27% and 75.75%, respectively, indicating that they may be less reliable for complex combinations where all phases must be identified simultaneously. This difference could be attributed to the feature space complexity and the overlapping characteristics of certain phases.

While this study demonstrates the efficacy of ML in predicting complex phase combination in HEAs, certain limitations should be acknowledged. First, the dataset used in this study, though refined and diverse, remains relatively small for certain phase combinations such as BCC+FCC+IM. The limited sample size for these phases may have affected the model’s ability to generalize effectively, particularly for underrepresented combinations. Expanding the dataset through additional experimental data or synthetic data augmentation could further improve model performance and robustness. Second, although this study employed well-established algorithms like SVM, KNN, RF and NN, the exploration of other advanced ML models, such as ensemble approaches (e.g., gradient boosting, XGboost) or deep learning architectures tailored to materials data, may uncover addition insights or improve prediction accuracy for more complex phase structures. Future studies could explore these alternative approaches. Another limitation lies in the feature space, which is restricted to features derived from chemical composition. Incorporating additional features, such as processing parameter (e.g., cooling rate or synthesis method) or microstructural descriptors, could further enhance predictive capabilities by capturing more intricate dependencies.

4. Conclusion

This study presents a machine learning-based approach for predicting phase structures in HEAs. Four ML algorithms including SVM, KNN, RF and NN were evaluated to determine their efficacy in phase prediction. The feature importance analysis highlighted $VEC$ and $T_m$ as crucial factors in distinguishing the phase structures of HEAs. Among the algorithms studied, the NN and KNN showed superior performance, achieving a complex combination phase prediction accuracy of 84.85% on the test data. Although the prediction accuracy for BCC and FCC phases was high, the accuracy for IM phases was comparatively lower, suggesting potential limitations in capturing the complexities of IM phases. This result highlights the opportunity for further research focused on improving IM phase prediction accuracy which could enhance the model’s overall predictive capabilities for HEAs. The findings confirm the high effectiveness of machine learning in accurately predicting HEA phase structure and underscore its potential application in materials science for the design and development of HEAs with tailored properties