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Article

Machine Learning-Assisted Hardness Prediction of Dispersion-Strengthened Tungsten Alloy

by
Shaowu Dai
1,2,
Chong Chen
2,3,*,
Cong Zhang
4,
Shizhong Wei
2,*,
Beibei Han
2,3,
Changji Wang
2,3,
Kunming Pan
2,3,
Liujie Xu
2,
Feng Mao
2,3 and
Hua Yu
2,3
1
School of Materials Science and Engineering, Henan University of Science and Technology, Luoyang 471000, China
2
National Joint Engineering Research Center for Abrasion Control and Molding of Metal Materials, Henan University of Science and Technology, Luoyang 471000, China
3
Intelligent Manufacturing Fundamental Research Center, Longmen Laboratory, Luoyang 471000, China
4
Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, Beijing 100083, China
*
Authors to whom correspondence should be addressed.
Metals 2025, 15(3), 294; https://doi.org/10.3390/met15030294
Submission received: 27 January 2025 / Revised: 25 February 2025 / Accepted: 6 March 2025 / Published: 7 March 2025
Browse Figures
Figure 1
<p>An illustration of the machine learning process for dispersion-strengthened tungsten alloys.</p> "> Figure 2
<p>Diagram illustrating the 10-fold cross-validation used in this work. The blue rectangle represents the training set, and the green rectangle represents the test set.</p> "> Figure 3
<p>A Pearson correlation coefficient diagram of the features affecting the hardness of the dispersion-strengthened tungsten alloy.</p> "> Figure 4
<p>A comparison of R<sup>2</sup> and MAE for the seven regression models.</p> "> Figure 5
<p>SHAP feature importance rank for target variable hardness.</p> "> Figure 6
<p>Predicted hardness vs. actual hardness for RF algorithm model: (<b>a</b>) training set; (<b>b</b>) test set.</p> "> Figure 7
<p>Predicted hardness vs. actual hardness for SVR algorithm model: (<b>a</b>) training set; (<b>b</b>) test set.</p> "> Figure 8
<p>Predicted hardness vs. actual hardness for XGB model: (<b>a</b>) training set; (<b>b</b>) test set.</p> "> Figure 9
<p>Comparison of model metrics of the training sets and test sets for the RF, SVR and XGBoost models: (<b>a</b>) R<sup>2</sup>; (<b>b</b>) MAE.</p> "> Figure 10
<p>The hardness of tungsten alloy varies with the grain size and reinforcement phase content of different carbon oxides: (<b>a</b>) actual values; (<b>b</b>) predicted values.</p> "> Figure 11
<p>A comparison of predicted hardness based on the current RF model with only RC and GZ features used and actual hardness, along with the percentage error.</p> ">
Versions Notes

Abstract

:
Hardness, as a typical mechanical property of dispersion-strengthened tungsten alloy, is influenced by various coupled factors. This paper aims to identify the key factors affecting the hardness of the dispersion-strengthened tungsten alloys with different carbides and oxides as the reinforcement phase in order to enable the high-throughput prediction of hardness. A dataset was established with alloy hardness as the target variable, and the features included the content of reinforcement phase, the Vickers hardness of reinforcement phase, the melting point of the reinforcement phase, the valence electron number of the reinforcement phase, the sintering temperature, the sintering time, pressure, relative density, and grain size. Seven regression models were trained, and we selected random forest, support vector regression, and XGBoost regression machine learning models with better performance to construct a hardness prediction model of the dispersion-strengthened tungsten alloy. SHAP analysis, based on random forests, shows that the content of reinforcement phase, grain size, and relative density have the most significant impact on the hardness. A random forest model is the most suitable machine learning method for predicting the hardness of dispersion-strengthened tungsten alloys in this work. The R2 values of the training and test sets are 0.93 and 0.80, and the MAE values of the training and test sets are 22.72 and 38.37. The influence of the most important features on the hardness was also discussed based on the random forest model. This study provides a data-driven approach for the accurate and efficient prediction of the hardness of dispersion-strengthened tungsten alloys, offering an important reference for the design and development of high-performance tungsten alloy materials.

1. Introduction

Tungsten has a high melting point, excellent high-temperature strength, and stable chemical properties. It is widely used, not only in the metallurgical, energy, and chemical industries, but also in aerospace, defense, and electronics [1,2]. However, tungsten alloys still face severe challenges related to low-temperature brittleness, recrystallization brittleness, and radiation-induced brittleness. Research has shown that adding reinforcement particles into the tungsten matrix can help overcome these brittleness defects by stabilizing the microstructure [3]. Dispersion-strengthened tungsten alloys include, but are not limited to, carbides (such as ZrC, TiC, HfC, TaC) and oxides (such as Y2O3, La2O3, Al2O3, ZrO2). These reinforcement phases, dispersed as fine particles in the tungsten matrix, can effectively hinder grain boundary and dislocation movement. Consequently, the grain size can be refined, and the mechanical properties can be improved [4,5,6,7,8,9,10]. Tungsten alloys are commonly prepared by traditional powder metallurgy methods, including hot isostatic pressing (HIP), spark plasma sintering (SPS), and microwave-assisted sintering (MAS), etc. [11,12]. Previous studies have shown that the mechanical properties of tungsten alloys are influenced by multiple coupled factors, including processing parameters, reinforcement phase content, grain size, porosity, and so on. Yar et al. [13] synthesized nano-sized W-1 wt%Y2O3 powder by a wet chemical method and consolidated it using SPS at 1100 °C and 1200 °C with identical holding times and pressures, achieving relative densities of 86% and 92%, grain sizes of 0.65 and 2.33 μm, and hardness values of 423HV and 518HV, respectively. W-(0.3–0.7) wt% TiC alloys with grain sizes of 0.06–0.2 μm and a relative density of 99% were fabricated by Kurishita et al. [14]. When the TiC content is close to 0.5%, the highest three-point bending fracture strength of the ultra-fine-grained W-TiC compacts achieved is about 1.6–2 GPa. Wang et al. [15] prepared W-0.2 wt% ZrC and W-0.8 wt% ZrC plates through ball milling, sintering, and multi-step hot-rolling processes. W-0.2 wt% ZrC alloy has a higher hardness and a larger tensile strength but a lower ductility than the W-0.8 wt% ZrC alloy.
Currently, the traditional “trial-and-error” approach of combining experience with experimentation to search for the appropriate microstructural combinations for achieving the desired mechanical properties in tungsten alloys offers poor quality control and is time-consuming and labor-intensive. There is an urgent need to find an effective and rapid strategy for predicting tungsten alloy performance, which can provide guidance for the development of tungsten alloys. With the rapid advancement of artificial intelligence, machine learning has gradually come into the spotlight, significantly accelerating the design and development of new materials [16,17]. Zhang et al. [18] proposed a new method that efficiently and accurately predicts the yield strength of heavy tungsten alloys using machine learning and hardness data. Among the eight machine learning models used, the results predicted based on the Gradient Boosting Decision Tree (GBDT) model were highly consistent, with an average error of 6.7%. Rajput et al. [19] employed a decision tree model to predict the ultimate tensile strength (UTS) of aluminum-based hybrid metal matrix composites, achieving a maximum accuracy of 92.029%. They also predicted the UTS of samples with Al7075 and Al6061 matrices and compared the results with actual experimental values, obtaining an error of less than 10%. Liu et al. [20] proposed a selection strategy, and the interpretable descriptors extracted using the Multilayer Perceptron (MLP) neural network demonstrated good interpretability and achieved a high prediction accuracy (R > 0.95). This indicates that the use of machine learning methods to predict the mechanical properties of dispersion-strengthened tungsten alloys is feasible.
Hardness is a key performance element of the dispersion-strengthened tungsten alloys, but it can be influenced by multiple factors. Few studies have explored the application of machine learning models to the performances of tungsten alloy matrix composites. This study aims to develop an accurate hardness prediction model for dispersion-strengthened tungsten alloys, incorporating different reinforcement phases. Spark plasma sintering (SPS) technology is a typical preparation method used to fabricate high-performance tungsten alloys. Experimental data about the dispersion-strengthened tungsten alloys with different reinforcement phases, prepared via the SPS method, were collected from previously published studies. Experimental data, taken from literature, underwent feature selection to identify key factors affecting hardness. Different machine learning regression models were evaluated, and 10-fold cross-validation and various evaluation metrics were used to identify the model with the least error. Parameter optimization was further performed to obtain the best machine learning model. The results provide insights into the relationship between key features and tungsten alloy hardness, aiding the development of advanced dispersion-strengthened tungsten alloys.

2. Methods

Machine learning method was utilized to predict the hardness of the dispersion-strengthened tungsten alloys featuring different carbides and oxides, and the illustration of the detailed process is presented in Figure 1.

2.1. Data Collection

Carbides and oxides are two typical reinforcements used in dispersion-strengthened tungsten alloys. Overall, 126 data entries were collected from experimental investigations in previously published studies on the hardness of tungsten alloys reinforced with different oxides and carbides as reinforcement phase particles. The alloys were prepared by the SPS method [1,2,3,4,5,6,7,8,9,10,12,13,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. The reinforcement phases include ZrC, HfC, TiC, TaC, Nd2O3, Pr2O3, Y2O3, Lu2O3, La2O3, HfO2, Al2O3, Sm2O3, and ZrO2. The data for pure tungsten were also collected. The data affecting the hardness of dispersion-strengthened tungsten alloys from studies were categorized into three types: alloy composition, processing parameters, and microstructure features. The target variable is hardness.

2.2. Feature Selection

Feature selection in machine learning is an indispensable step during data preprocessing. Its purpose is to eliminate redundant and irrelevant features, as well as features that may have a negative impact on the hardness of dispersion-strengthened tungsten alloys, thereby improving the accuracy of the machine learning model. In this study, the Pearson correlation coefficient (PCC) and the SHAP (Shapley Additive Explanations) analysis method are employed to identify the key features that significantly influence the hardness of dispersion-strengthened tungsten alloys.
The Pearson correlation coefficient (PCC) can accurately identify features with high correlation and analyze them. The stronger the positive correlation, the closer value of the PCC to +1. Conversely, when the negative correlation is stronger, the value of the PCC is closer to −1. To ensure the stability and reliability of the model, highly correlated features should be removed.
There also exist features with nonlinear relationships, and for variables exhibiting nonlinear relationships, feature importance is typically calculated to perform selection. SHAP analysis, based on algorithmic models, is a method for analyzing feature importance. For a specific prediction result, the SHAP value represents the contribution of each feature to the prediction outcome, as shown in Equation (1):
Y i = Y base + f x i , 1 + f x i , 2 + … + f x i , j
where Y i and Y base represent the model’s prediction for the i-th sample and the baseline of the entire model, respectively; x i , j   represents the j-th feature of the i-th sample, respectively; f x i , j is the SHAP value for the feature x i , j .

2.3. Machine Learning Algorithm Model

Python 3.12 and the scikit-learn 1.0.2 library were applied for machine learning model building, data preprocessing, model evaluation, and hyperparameter tuning. Data processing was performed using pandas, numerical computations were handled with NumPy 1.26.4 and data visualization was achieved through matplotlib. Finally, the trained models were saved using joblib 1.4.2.
An appropriate algorithm is crucial in constructing a machine learning model to predict the hardness of dispersion-strengthened tungsten alloys. Since the data collected in this study consist of continuous variables that influence the hardness of dispersion-strengthened tungsten alloys, regression algorithms, which can predict continuous output values, are selected. Varied model performances can be obtained with the application of different machine learning models to similar problems. Therefore, the adoption of multiple machine learning models in the present work becomes crucial for developing a comprehensive understanding of their potential and obtaining the best model. Seven models are applied for the prediction of hardness, including random forest (RF), support vector regression (SVR), Extreme Gradient Boosting regression (XGB), Linear Regression (LR), K-Nearest Neighbors (KNN), Gamma Regression (GR), and Decision Tree Regression (DTR). In the present work, the collected experimental data are limited. The main limitation of a small dataset is the risk of overfitting. By adopting various strategies, including hyperparameter tuning and K-fold cross-validation, these models have demonstrated strong performances in terms of prediction accuracy and generalization ability in previous studies when used to predict the characteristic features of alloys, especially with the metal matrix composites [19,20,51,52]. The model performance is evaluated using the Mean Absolute Error (MAE) and the Coefficient of Determination (R2) to represent the accuracy of the model. Calculations are performed via Equations (2) and (3).
MAE = 1 N ∑ m = 1 N y m ′ − y m 2
R 2 = ∑ m = 0 N − 1 y m − y m ′ 2 ∑ m = 0 N − 1 y m − y m ′ ¯ 2  
where ym is the actual value of the m-th sample, y m ′ is the predicted value of the m-th sample, and y m ′ ¯ is the average of all actual values. The smaller MAE and the larger R2 obtained based on the machine learning prediction results indicate that the model is better suited for predicting the hardness of dispersion-strengthened tungsten alloys.
K-fold cross-validation is commonly used to evaluate model performance and select the best model or parameter configuration by dividing the dataset into K subsets. In each iteration, K − 1 subsets are used for training, and the remaining subset is used as the validation set. This process is repeated K times, ensuring that each subset has a chance to serve as the validation set. This method thus helps assess the stability and accuracy of the model, reducing errors introduced by data splitting and preventing both overfitting and underfitting issues. In this work, K is set to be 10 to balance evaluation effectiveness and computational efficiency, as shown in Figure 2.
A comparison of the MAE and R2 results was conducted for different models, and the models that performed better were selected for further optimization. GridSearchCV and RandomizedSearchCV was used to conduct automatic hyperparameter tuning. The model’s performance, validated through 10-fold cross-validation, is evaluated using MAE and R2. The model performed best was utilized to predict the hardness of dispersion-strengthened tungsten alloys.

3. Results and Discussion

3.1. Feature Analysis

In order to distinguish the effect of different reinforcement phases on the hardness of the tungsten alloys, three physical properties of the reinforcement phase were introduced, including the hardness of the reinforcement phase (RV), the melting point of the reinforcement phase (RMP), and the valence electron number of the reinforcement phase (RVEN), with ranges of 0–25.005 (HV/GPa), 0–3920 (°C), and 0–10.8, respectively. The content of the reinforcement phase (RC) was used to represent the alloy’s composition, with ranges of 0–11.71 wt%. Processing parameters include sintering temperature (ST), sintering time (T), and sintering pressure (P), with ranges of 1100–2100 °C, 0–30 min, and 30–85 MPa, respectively. Relative density (RD) and grain size (GZ) were considered as microstructure features, with ranges of 83.2–99.9% and 0.36–22.2 μm, respectively. The target variable is hardness (HV), with a range of 200–1064.28 HV. Due to the limited data and incomplete feature data for some oxides (La2O3, Lu2O3, Sm2O3, Pr2O3), the corresponding data were removed. After data cleaning, 107 data entries remained and the main features used for machine learning on the dispersion-strengthened tungsten alloys are presented in Table 1.
Figure 3 shows the PCC heatmap, which reveals the inherent relationships among features influencing the hardness of dispersion-strengthened tungsten alloys. The color intensity in the PCC heatmap reflects the correlation strength between two various features. Deep red blocks represent a strong positive correlation between two features, meaning that an increase in one feature value is accompanied by a corresponding increase in another feature value. Conversely, deep blue blocks indicate a strong negative correlation. Lighter-colored blocks suggest a weaker correlation, implying that these features may be independent of each other. RV, RPM, and RVEN are all physical properties of the reinforcement phase and thus show relatively higher PCC values. Most PCC values in Figure 3 are less than −0.5, ensuring the generalization ability and stability of the model during the training process.

3.2. Machine Learning

Based on the main features listed in Table 1 and the corresponding dataset, seven different regression models (RF, SVR, XGB, LR, KNN, GR, and DTR) were utilized and trained to ensure model diversity and comprehensiveness in evaluating the adaptability of different algorithms for hardness prediction. The use of multiple models helps to avoid model bias and provides the most robust and reliable prediction results. The model’s performance results are shown in Figure 4. The R2 values for the RF, SVR, XGB, LR, KNN, GR, and DTR models are 0.62, 0.58, 0.50, 0.49, 0.42, 0.38, and 0.34, respectively. The MAE values for the RF, SVR, XGB, LR, KNN, GR, and DTR models are 53.14, 68.93, 73.91, 75.28, 73.61, 77.05, and 73.83, respectively. The RF, SVR and XGB models exhibited higher accuracies and smaller errors, and were the thus focus of this work.
SHAP values were utilized to measure the importance of each feature and quantify their contribution to the target variable of hardness. The SHAP importance rankings of each feature, determined based on the average absolute values obtained via the random forest model for the hardness of the dispersion-strengthened tungsten alloys, are shown in Figure 5. It can be observed that RC has the highest importance in terms of influencing hardness, followed by the GZ and RD. These features were identified as key contributors based on their positive or negative SHAP values. Generally, the hardness of the carbides or oxides is much higher than that of tungsten, and the hard reinforcement phase in the tungsten matrix improves the hardness of the alloy. On the other hand, the addition of the reinforcement phase is designed to reduce the grain size and thereby improve the hardness of the dispersion-strengthened tungsten alloys. This agrees with the strengthening mechanism of the dispersion-strengthened tungsten alloys. As typical powder metallurgy materials prepared via the SPS method, the pores are detrimental to the hardness of the dispersion-strengthened tungsten alloys, and thus RD ranks third in terms of feature importance. All the three physical properties show limited importance, indicating that the type of reinforcement phase has less impact on the hardness than other factors. The higher melting point of the reinforcement phase showing higher feature importance is probably conducive to its sintering stability and a reduction in grain size. Since the RV and RVEN features are less important, they will be considered for exclusion during the model tuning of RF, SVR, and XGB.
Automatic hyperparameter tuning was conducted on the RF and SVR models using GridSearchCV. The XGB model was subjected to automatic hyperparameter tuning through RandomizedSearchCV. The training and test sets were randomly divided according to a certain proportion of data. The training set was used for model training and optimization, while the test set was used to evaluate the model’s performance. The method of 10-fold cross-validation was then adopted to determine the optimal proportions of data partition. The dataset was initially split into a training set and a test set in different ratios of 90:10, 80:20, 70:30, and 60:40. We used a ratio of 80:20 for the training set and the test set for the RF model due to this formulation demonstrating best model performance. When the ratio of training set to test set data is 70:30, the test set R2 of the SVR model is at its relative highest value. Therefore, the dataset was randomly divided into the training set and the test set according to a ratio of 80:20 for the RF model and a ratio of 70:30 for both the SVR and XGB models. We paid attention to ensure that a RV with a SHAP feature importance of less than 0.01 was removed during the hyperparameter tuning process of the RF model to optimize the model’s performance and reduce data dimensionality.
RF, SVR, and XGB machine learning algorithm models were used after parameter optimization to learn the training set and predict the test set. The results of the training and testing sets for the RF, SVR, and XGB models are shown in Figure 6, Figure 7 and Figure 8, respectively. The blue solid circles represent the actual values of the training set, the green solid circles represent the actual values of the test set, and the red dashed line represents the prediction baseline. The data on the baseline indicate that the true value of the hardness is exactly consistent with the predicted value. The closer the data points are to the dashed line, the smaller the absolute error between the predicted and actual values. Most data are closer to the baseline for the RF model than the SVR and XGB models, indicating that the RF model has a better performance. The hardness data higher than 700 HV are relatively far away from the baseline, probably as a result of the scarcity of data.
Different model metrics, R2 and MAE, were applied to evaluate the tuned models, and the results are shown in Figure 9. As shown in Figure 9a, the R2 values of the training and test sets for the RF model are 0.93 and 0.80, respectively. For the SVR model, they are 0.86 and 0.58, respectively. For the XGB model, they are 0.90 and 0.70, respectively. The discrepancy in the R2 values between the training and test sets may arise from the insufficient and scattered data of different carbides and oxides as the reinforcement phases obtained from previous studies. The RF model outperforms the SVR and XGB models in terms of R2 for both the training and test sets. As shown in Figure 9b, the MAE values of the training and test sets are 22.72 and 38.37 for the RF model, 43.96 and 43.78 for the SVR model, and 32.47 and 46.34 the XGB model. In the experimental measurement of the hardness of the dispersion-strengthened tungsten alloy, the error value of the hardness for several tests on the same sample is usually several tens of HV. Therefore, the errors for all models are within the expected range. In summary, the RF model is superior to the SVR and XGB model and is the optimal model.

3.3. Effect of the Most Important Features on Hardness

Figure 10 presents the variation in both the actual hardness values and predicted hardness values, along with the RC and GZ, for the dispersion-strengthened tungsten alloys. The hardness values for pure tungsten were also included in the figures. Both the actual hardness values shown in Figure 10a and predicted hardness values shown in Figure 10a show similar tendencies. The hardness increases as the content of reinforcement phase increases and the grain size decreases. Meanwhile, the grain size tends to decrease with the increase in the content of reinforcement, as indicated in Figure 10. Due to other factors including the powder information and processing conditions, the opposite also occurs. Several abnormal cone shapes can be observed, corresponding to data with low relative density. Due to complex potential interactions between these features, it is hard to simultaneously achieve small grain size, a high content of reinforcement, and a high relative density in experimental investigations. This leads to the limited data for high hardness, resulting in the lower prediction accuracy of the model in the high-hardness region, as indicated in the upper-right corner of Figure 10. By comparing the predicted values from the RF model with actual values, we found that the prediction error is relatively small (approximately 10 HV) when the reinforcement phase content is around 1 wt%. According to the rank of the important features shown in Figure 5, relative density is the third important feature and also has a great effect on the hardness of the tungsten alloys, especially in the case of pure tungsten.
As shown in Figure 5, RC and GZ are the most important features influencing the hardness of the dispersion-strengthened tungsten alloys based on the RF model. The hardness was also predicted based on the current RF model, with only RC and GZ features used to further determine the contribution of these two features. Figure 11 presents a comparison of predicted hardness and actual hardness, along with the percentage error. The yellow rectangles represent the actual values, while the blue triangles represent the predicted values. As can be seen in Figure 11, most predicted hardness values agree well with the actual values. The green circles represent the percentage error between the predicted and actual hardness values. Among all 107 data points, more than 87% (94 data points) show percentage errors below 15%. Even when only the two most important features are used, the current RF model exhibits a certain level of accuracy.
Reducing grain size and optimizing the content of the reinforcement phase are key factors for achieving high hardness of the dispersion-strengthened tungsten alloys. Meanwhile, the relative density should be paid attention to. The present RF model can be further improved by adding data entries in future work, especially for the data with high hardness values.

4. Conclusions

This study employed machine learning methods to establish a database, perform feature engineering, select and optimize models, and conduct the high-throughput prediction of the hardness of the dispersion-strengthened tungsten alloys. The main conclusions are as follows:
  • A dataset with 107 data entries for the dispersion-strengthened tungsten alloys was constructed with 9 kinds of features affecting the hardness. The reinforcement phase, grain size, and relative density were identified as the key features influencing the hardness according to the SHAP importance feature.
  • Seven regression models, including random forest, support vector regression, XGBoost regression, Linear Regression, K-Nearest Neighbors, Gamma Regression, and Decision Tree Regression, were trained. Only the random forest and support vector regression machine learning model exhibit better performance. The hard prediction model of the dispersion-strengthened tungsten alloy was established based on random forest, support vector regression, and XGBoost regression models.
  • The random forest model is the most suitable machine learning method for predicting the hardness of dispersion-strengthened tungsten alloys. The R2 values of the training and test sets are 0.93 and 0.80, and the MAE values of the training and test sets are 22.72 and 38.37.
  • The hardness was also predicted based on the random forest model, with only content of the reinforcement phase and grain size used. This was to determine their contribution. More than 87% of the data (94 pieces of data) show percentage errors below 15%, indicating the reliability of the current model, even when only the two most important features are used.
  • The present work provides the basis of data extraction and modeling for efficient alloy design in the future. Massive efforts should be made in future work to achieve real industry applications by considering other sintering methods, the way to introduce the powder information, other mechanical properties or functional performances, and the hot working process.

Author Contributions

Conceptualization, S.D. and C.C.; methodology, C.Z. and B.H.; software, C.Z. and B.H.; validation, K.P., H.Y. and L.X.; formal analysis, S.D., C.Z. and B.H.; investigation, S.D. and C.W.; resources, L.X.; data curation, S.W. and F.M.; writing—original draft preparation, S.D.; writing—review and editing, C.C., C.Z. and S.W.; visualization, F.M. and C.W.; supervision, K.P. and F.M.; project administration, K.P. and S.W.; funding acquisition, C.C., C.W. and H.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (No. 52201118), the Central Guidance on Local Science and Technology Development Fund of Henan Province (No. Z20241071031) and the Science and Technology Research and Development Plan Joint Fund of Henan Province (No. 235200810102).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

The support of the Provincial and Ministerial Collaborative Innovation Center for New Non-ferrous Metal Materials and Advanced Processing Technology is acknowledged.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. An illustration of the machine learning process for dispersion-strengthened tungsten alloys.
Figure 1. An illustration of the machine learning process for dispersion-strengthened tungsten alloys.
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Figure 2. Diagram illustrating the 10-fold cross-validation used in this work. The blue rectangle represents the training set, and the green rectangle represents the test set.
Figure 2. Diagram illustrating the 10-fold cross-validation used in this work. The blue rectangle represents the training set, and the green rectangle represents the test set.
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Figure 3. A Pearson correlation coefficient diagram of the features affecting the hardness of the dispersion-strengthened tungsten alloy.
Figure 3. A Pearson correlation coefficient diagram of the features affecting the hardness of the dispersion-strengthened tungsten alloy.
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Figure 4. A comparison of R2 and MAE for the seven regression models.
Figure 4. A comparison of R2 and MAE for the seven regression models.
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Figure 5. SHAP feature importance rank for target variable hardness.
Figure 5. SHAP feature importance rank for target variable hardness.
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Figure 6. Predicted hardness vs. actual hardness for RF algorithm model: (a) training set; (b) test set.
Figure 6. Predicted hardness vs. actual hardness for RF algorithm model: (a) training set; (b) test set.
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Figure 7. Predicted hardness vs. actual hardness for SVR algorithm model: (a) training set; (b) test set.
Figure 7. Predicted hardness vs. actual hardness for SVR algorithm model: (a) training set; (b) test set.
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Figure 8. Predicted hardness vs. actual hardness for XGB model: (a) training set; (b) test set.
Figure 8. Predicted hardness vs. actual hardness for XGB model: (a) training set; (b) test set.
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Figure 9. Comparison of model metrics of the training sets and test sets for the RF, SVR and XGBoost models: (a) R2; (b) MAE.
Figure 9. Comparison of model metrics of the training sets and test sets for the RF, SVR and XGBoost models: (a) R2; (b) MAE.
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Figure 10. The hardness of tungsten alloy varies with the grain size and reinforcement phase content of different carbon oxides: (a) actual values; (b) predicted values.
Figure 10. The hardness of tungsten alloy varies with the grain size and reinforcement phase content of different carbon oxides: (a) actual values; (b) predicted values.
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Figure 11. A comparison of predicted hardness based on the current RF model with only RC and GZ features used and actual hardness, along with the percentage error.
Figure 11. A comparison of predicted hardness based on the current RF model with only RC and GZ features used and actual hardness, along with the percentage error.
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Table 1. Main features used for machine learning on the dispersion-strengthened tungsten alloys after data cleaning.
Table 1. Main features used for machine learning on the dispersion-strengthened tungsten alloys after data cleaning.
Feature NameFeature AnnotationFeature Range
RC (wt%)Content of reinforcement phase0–11.71
RV (GPa)Vickers hardness of reinforcement phase0–25.005
RMP (°C)Melting point of reinforcement phase0–3920
RVENValence electron number of reinforcement phase0–10.8
ST (°C)Sintering temperature1100–2100
T (min)Sintering time0–30
P (MPa)Pressure30–85
RD (%)Relative density83.2–99.9
GZ (μm)Grain size0.36–22.2
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MDPI and ACS Style

Dai, S.; Chen, C.; Zhang, C.; Wei, S.; Han, B.; Wang, C.; Pan, K.; Xu, L.; Mao, F.; Yu, H. Machine Learning-Assisted Hardness Prediction of Dispersion-Strengthened Tungsten Alloy. Metals 2025, 15, 294. https://doi.org/10.3390/met15030294

AMA Style

Dai S, Chen C, Zhang C, Wei S, Han B, Wang C, Pan K, Xu L, Mao F, Yu H. Machine Learning-Assisted Hardness Prediction of Dispersion-Strengthened Tungsten Alloy. Metals. 2025; 15(3):294. https://doi.org/10.3390/met15030294

Chicago/Turabian Style

Dai, Shaowu, Chong Chen, Cong Zhang, Shizhong Wei, Beibei Han, Changji Wang, Kunming Pan, Liujie Xu, Feng Mao, and Hua Yu. 2025. "Machine Learning-Assisted Hardness Prediction of Dispersion-Strengthened Tungsten Alloy" Metals 15, no. 3: 294. https://doi.org/10.3390/met15030294

APA Style

Dai, S., Chen, C., Zhang, C., Wei, S., Han, B., Wang, C., Pan, K., Xu, L., Mao, F., & Yu, H. (2025). Machine Learning-Assisted Hardness Prediction of Dispersion-Strengthened Tungsten Alloy. Metals, 15(3), 294. https://doi.org/10.3390/met15030294

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