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Article

Prediction of Enthalpy of Mixing of Binary Alloys Based on Machine Learning and CALPHAD Assessments

by
Shuangying Huang
,
Guangyu Wang
and
Zhanmin Cao
*
School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing 100083, China
*
Author to whom correspondence should be addressed.
Metals 2025, 15(5), 480; https://doi.org/10.3390/met15050480
Submission received: 10 March 2025 / Revised: 17 April 2025 / Accepted: 18 April 2025 / Published: 24 April 2025
(This article belongs to the Special Issue Machine Learning in Metallic Materials Processing and Optimizing)
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Abstract

:
The enthalpy of mixing, a critical thermodynamic property in the liquid phase reflecting element interaction strength and pivotal for studying phase equilibria, can now be predicted efficiently using machine learning. This study proposes a model combining machine learning with the Calculation of Phase Diagram (CALPHAD) to predict the enthalpy of mixing. We obtained data for 583 binary alloy systems from the SGTE database, ensuring experimental constraints for accuracy. Using pure element properties and Miedema’s model parameters as descriptors, we trained and evaluated four machine learning algorithms, finding LightGBM to perform best (R2 = 92.2%, MAE = 3.5 kJ/mol). The model performance was further optimized through Recursive Feature Elimination (REF) and Maximal Information Coefficient (MIC) feature selection methods. Shapley Additive Explanations reveals that the primary factors affecting the mixing enthalpy, such as atomic radius and electronegativity, align with the key parameters of the Miedema model, thereby confirming the physical interpretability of our data-driven approach. This work offers an accelerated method for predicting complex multi-component system thermodynamics. Future research will focus on collecting more high-quality data to enhance model accuracy and generalization.

Graphical Abstract

1. Introduction

The enthalpy of mixing in a binary solution is the enthalpy change before and after the elements are mixed. This reflects the ability of the elements to interact with each other and is an important thermodynamic property for determining the stability of alloy phases and calculating phase diagrams. The enthalpy of mixing provides crucial information for understanding the glass-forming ability of metallic glasses [1] and serves as a key indicator for the stability of a given alloy [2]. Consequently, it plays a critical role in both metallurgical preparation and the research and development of new materials. In addition, machine learning techniques have been increasingly applied in materials science. Studies have shown that the thermodynamic properties of liquid phase mixing are essential descriptors for predicting phase equilibria, including the prediction of binary liquidus [3], ternary isothermal sections [4], and phase formation in high-entropy alloys [5,6,7,8].
At present, the methods for studying the enthalpy of mixing in alloy solutions can be broadly classified into three categories: experimental methods, theoretical modeling methods, and Calculation of Phase Diagrams (CALPHAD) methods. Calorimetry, as a classical experimental method, has helped researchers to obtain a large amount of the enthalpy of mixing data for alloy systems [9,10,11,12]. The Miedema semi-empirical model was originally proposed by Miedema et al. in the 1980s [13]. Since then, it has undergone numerous refinements and optimizations, including adjustments to the model parameters and empirical constants [14,15,16], significantly improving its predictive accuracy. The model has also been successfully applied to approximate mixing enthalpies in both liquid and solid metal solutions [17,18,19], particularly in alloy systems with limited experimental data. However, with the diversity and complexity of alloying systems, traditional methods are time- and resource-intensive in predicting multicomponent alloying systems.
The CALPHAD methodology is a thermodynamics-based computational framework. It enables robust prediction of phase equilibria and materials properties. This is achieved through critical assessment and self-consistent integration of multi-source experimental data. The approach effectively mitigates measurement uncertainties and enhances dataset reliability [20]. Nevertheless, its intrinsic dependency on pre-existing thermodynamic databases presents a critical limitation [21]. Although the most binary systems have been comprehensively parameterized, the current CALPHAD methodology faces increasing challenges in dealing with multicomponent alloy systems. A primary issue is the limited number of ternary thermodynamic descriptions added to established databases each year. Moreover, when addressing novel alloy compositions that lack sufficient experimental validation, the uncertainties in thermodynamic extrapolation become especially significant.
The rapid development of data-driven machine learning algorithms has significantly advanced materials science research. These computational tools provide novel methodologies to predict alloy mixing enthalpies by leveraging their robust capabilities in nonlinear fitting and predictive modeling. Recently, significant progress has been made in predicting mixing enthalpies for solid solution alloy systems. Mukhamedov et al. [22] developed two machine learning models for the high-throughput prediction of mixing enthalpies in multicomponent Fe-Cr alloys within the body-centered cubic phase. In addition, models based on neural networks have been developed [23,24], which significantly improve the accuracy of mixing enthalpy predictions for solid metal solution systems. However, there remains a notable gap in research regarding liquid-phase alloy systems. The pioneering work of Deffrennes et al. [25] utilized the Magpie universal feature set to develop a machine learning framework that incorporates error correction for the Miedema model. This approach successfully achieved the indirect prediction of liquid-alloy-mixing enthalpy with excellent performance (Mean Absolute Error (MAE) = 3.0 kJ/mol), significantly outperforming the Miedema model (MAE = 4.2 kJ/mol). However, it failed to effectively address cumulative errors that could arise during the “error prediction recalibration” process and overlooked the physical correlations between features.
This study introduces an innovative, cross-scale feature, fusion methodology that systematically combines elemental basic properties with alloy thermodynamic characteristics. The developed framework establishes a physically interpretable feature engineering system. The study leverages 583 rigorously validated, binary-system, thermodynamic data from the SGTE database, which expands the training sample size by 35% compared to Deffrennes et al. [25]. As a result, a comprehensive mapping relationship is established across composition, features, and performance. After systematically comparing multiple algorithms, the LightGBM model demonstrated optimal performance on the test set. Furthermore, the study investigates the impact of alloy composition and input features on model accuracy. Through feature importance analysis, the key physical factors influencing mixing enthalpy are identified, providing a quantitative basis for the selection of critical descriptors in alloy design.

2. Materials and Methods

2.1. Datasets

The enthalpy of mixing data in this study were obtained from the SGTE alloy database in FactSage8.3 [26], and all data in the SGTE alloy database are based on available experimental information [27]. Temperature is a key parameter when calculating the enthalpy of mixing. However, the melting temperature is usually expressed as a temperature interval consisting of the liquidus and the solidus rather than a single value. In order to simplify the process, the method proposed by Chelikowsky and Anderson [28] is used in this study, i.e., the liquidus is defined as the melting temperature.
First, the liquidus of each binary alloy system was calculated using the Equilib module in FactSage software, and the highest temperature on the liquidus was selected. To ensure that the alloy system remained in the liquid phase, 100 K was added to this maximum temperature, providing the necessary temperature for calculating the enthalpy of mixing. Next, the Macro Processing of the Equilib module was used to batch calculate the enthalpy of mixing for each alloy system at the selected temperature. The composition of each binary alloy was sampled from 0 to 1 with a step size of 0.01, and after excluding the endmembers, 99 of the enthalpy of mixing data points were obtained per system [29].
Finally, a total of 583 binary alloy systems, optimized using the CALPHAD method, were collected, involving 66 metal elements, including aluminum alloys, copper alloys, silver alloys, nickel alloys, and magnesium alloys. The distribution of the dataset is shown in Figure 1, and a total of 57,717 of the enthalpy of mixing thermodynamic data points were generated, providing a rich database for subsequent research. We have examined the enthalpy of mixing data obtained from CALPHAD calculations for 583 systems, of which 45% are supported by direct calorimetric, experimental enthalpy of mixing data, and the others, although not experimentally available for the thermodynamic properties of liquids, are well constrained by measured data from solids and phase equilibrium experiments.

2.2. Feature Descriptor Construction

When constructing descriptors for machine learning models, it is crucial to select attributes that both uniquely define the material and reflect its fundamental physical and chemical characteristics. We have compiled 28 descriptors, the symbols and meanings of which are listed in Table 1. Of these, 19 descriptors are based on the general property set proposed by Ward et al. [30], known as the Magpie element feature set, which has recently been used to predict the liquidus temperature of binary alloys [3]. The Magpie element descriptors required for training the machine learning models were calculated using the ElementProperty module in the Matminer library [31]. These descriptors include statistical features of an element’s position in the periodic table, physical property statistics, and valence electron structure features [32]. For example, the valence electron structure features are information about the number of electrons in the s, p, d, and f orbitals of atoms [33,34].
Based on literature studies, we identified a set of key elemental features that significantly influence the alloy mixing enthalpy. These 9 descriptors are obtained from a database [35]. We first reviewed the Miedema model [36], in which Miedema et al. [37] developed ∆ n w s 1 / 3 (the cubic root difference of the electronic density at the Wigner–Seitz unit boundary) to predict the alloy mixing enthalpy. In binary-system mixing-enthalpy calculations, the molar volume, electronic density, and electronegativity are fundamental physical parameters of the Miedema model. Electronegativity provides information about the attraction of electrons by a given atom when forming ionic bonds, which is directly related to the work function [38]. For pure metals, the theoretical electronic density depends on the bulk modulus and molar volume. Another model that helps identify elemental features influencing the mixing enthalpy is the improved embedded atom method proposed by Ouyang et al. [39]. This method uses the cohesive energy, formation energy, and atomic volume of pure elements. These properties are applied to describe the work function in the Miedema model. Based on the above studies, we predicted that the following 5 features may be influential in predicting the mixing enthalpy: work function, electronegativity, electronic density, cohesive energy, and molar volume. Although these parameters are characteristics of solid alloys, they have been extended and refined for use in the calculation of liquid thermodynamic properties.
The pseudopotential radius Rs+p, first ionization energy, electrochemical equivalent, and bulk modulus, widely adopted in several studies, are considered important descriptors in the development of metal-formation enthalpy prediction models [40,41]. Therefore, these descriptors are also included in our consideration.
We calculated the averaging ( X m e a n ) and the average deviation ( X a v g _ d e v ) of elemental descriptors, as shown in Equations (1) and (2),
X m e a n = ∑ i = 1 n c i X i
X a v g _ d e v = ∑ i = 1 n c i X i − X m e a n
where c i and X i denote the mole content of the ith element and the value of the descriptor, respectively. The 56 descriptors for each binary alloy system are obtained by Equations (1) and (2).

2.3. Feature Selection Methods

Ideally, descriptors should be uncorrelated, as a large number of correlated features can hinder the efficiency and accuracy of the model. We first performed a correlation analysis to eliminate features with high correlation. The Maximal Information Coefficient (MIC) is used to evaluate the nonlinear correlation between each feature [42] and the nonlinear correlation between each feature and the target attribute. The MIC feature selection method ranks features by computing their MIC values with the target variable. Then, the one (feature1st) with the highest MIC was extracted out, and the MIC between feature1st and each of all remaining features were calculated. The features with a MIC with a feature1st higher than 0.8 were removed. For the left features, the above procedure repeats until only one feature left [43].

2.4. Model Explanation Methods

Shapley Additive Explanations (SHAP)

Shapley Additive Explanations (SHAP) [44] is a method used to explain machine learning model predictions by quantifying the importance of each feature. As a model-agnostic interpretability technique, SHAP provides both global and local explanations. Globally, it aggregates feature contributions across all samples to indicate their overall importance. Locally, it quantifies the contribution of each feature to an individual prediction, revealing how it influences the model’s output for a specific instance. The core idea of SHAP is rooted in cooperative game theory: it calculates the marginal contribution of each feature when added to all possible subsets of other features. The SHAP value is derived from the weighted average of these contributions, ensuring a fair allocation of predictive effects based on the feature’s impact across all permutations.

2.5. Model Parameter Selection and Training

In order to make a comprehensive comparison, this study applies a variety of machine learning algorithms to train the data. Specifically, K-Nearest Neighbor (KNN) algorithm and Multilayer Perceptron Machine (MLP) based on distance metrics were chosen, as well as two integrated learning algorithms based on decision trees: the Random Forest (RF) algorithm and the LightGBM (LGBM) algorithm [45], which are both implemented in scikit-learn [46].
The selection of model parameters not only governs prediction accuracy but also determines computational efficiency, rendering hyperparameter optimization imperative during model training. To ensure methodological validity and prevent suboptimal parameter selection arising from manual adjustments, this investigation implemented grid search with 11-fold cross-validation for parameter determination. Each binary alloy system’s dataset was treated as an indivisible group, thereby preserving data integrity during training-test set partitioning.
During the cross-validation process, the MAE on the test set was adopted as the primary performance metric, where the final reported value represents the average over 11 iterations. This section elaborates on the optimal hyperparameter configurations for the four models, with all unspecified parameters maintained at their default settings.
For the Random Forest regression model, key hyperparameters include the number of base learners (n_estimators), the maximum depth of the decision trees (max_depth), and the maximum number of features to consider when splitting each tree (max_features). By adjusting these hyperparameters, the optimal configuration for the Random Forest regression model was determined. The best hyperparameter settings are shown in Table 2.
For the MLP regression model, common hyperparameters include the structure of the hidden layers (hidden_layer_sizes), the activation function (activation), and the initial learning rate (learning_rate_init), among others. For the dataset used in this study, the optimal configuration of the model was obtained by optimizing these hyperparameters. The specific parameter settings are shown in Table 3.
For the LightGBM model, key parameters that affect prediction performance include min_child_samples, max_depth, num_leaves, and n_estimators. Among these, min_child_samples controls the minimum number of samples required in each leaf node, with larger values helping to prevent overfitting; max_depth limits the maximum depth of the trees to prevent them from growing too deep and causing overfitting; num_leaves determines the maximum number of leaf nodes in each tree, and too many leaves may lead to overfitting; n_estimators represents the number of trees, and too many trees may cause overfitting, while too few trees may result in underfitting. By optimizing these parameters, the optimal hyperparameter configuration for the LightGBM model was determined. The specific parameter settings are shown in Table 4.
For the KNN regression model, n_neighbors is the most crucial hyperparameter. A smaller value of n_neighbors may make the model overly sensitive to noise and outliers, leading to overfitting, while a larger value may cause the model to underfit, failing to capture the local structure of the data effectively. Therefore, selecting an appropriate n_neighbors value is critical for improving model’s predictive ability. By optimizing this hyperparameter, the optimal configuration for the KNN regression model was determined. The specific parameter settings are shown in Table 5.

3. Results and Discussion

3.1. Prediction Results of Different Machine Learning Models

To accurately predict the enthalpy of mixing of binary alloy systems, we employed four different machine learning models: KNN, MLP, RF, and LightGBM. The performance of these models on different datasets was comprehensively evaluated, and their generalization ability was compared using quantitative metrics. Evaluation results of different machine learning models are as shown in Table 6.
In Figure 2, the scatter points are to the diagonal, the smaller the error between the predicted value and the actual values, indicating stronger predictive performance. As shown in the figure, the LightGBM model outperforms the other three models in terms of regression, achieving an R2 score of 92.2% and an MAE of 3.5 kJ/mol on the test set, demonstrating its excellent predictive performance. The RF algorithm follows as the second-best performer. This suggests that ensemble learning methods offer a clear advantage in fitting the prediction of the enthalpy of mixing, providing better training performance and predictive accuracy. Previous studies [29] have also shown that ensemble learning methods typically yield better results in the machine learning prediction of thermodynamic properties, and the current results are consistent with these findings.

3.2. LightGBM Model for Feature Selection in Binary Alloy Enthalpy of Mixing Prediction

To optimize the performance of the LightGBM model and enhance the interpretability of feature engineering, this study adopted a phased feature selection strategy. First, the initial 56 descriptors were ranked by importance using the Recursive Feature Elimination (RFE) method. By iteratively removing the features with the least contribution to predicting the target attribute, the feature space was reduced from 56 dimensions to 30. This process not only achieved initial dimensionality reduction but also improved model prediction accuracy. Building on this, further feature reduction was performed by identifying two highly correlated descriptor pairs (CohesiveEnergy_mean—MeltingT_mean and CovalentRadius_avg_dev—Rs+p_avg_dev) with a MIC greater than 0.8. Based on the feature importance ranking, one feature from each pair (MeltingT_mean and Rs+p_avg_dev) was retained due to their stronger associations with the target attribute, resulting in a final set of 28 statistically independent key descriptors. To systematically validate the independence of the selected features, the Pearson correlation coefficient matrix among the 28 features was calculated (as shown in Figure 3). The results indicated that the absolute values of the correlation coefficients for all feature pairs were below 0.8 [47], quantitatively confirming the low redundancy of the selected feature set. The red and blue color gradients in the figure represent positive and negative correlations between features, respectively.

3.3. SHAP Analysis to Explore the Factors Influencing the Enthalpy of Mixing

The SHAP method is capable of interpreting the model output of individual predictions, and its main advantage lies in its ability to quantify the contribution of each feature to the prediction results. In this study, SHAP is used to analyze the importance of features and provide the specific impact of each feature on the results in individual predictions, thus helping us to reveal the predictive mechanism of the model and improve its transparency.
This paper performs feature analysis on the best-performing LightGBM model, and the SHAP feature importance of the model shown in Figure 4a represents the importance ranking of each input feature. Figure 4b shows the contribution of each input feature in individual predictions.
Feature importance analysis shows (Figure 4a) that the contribution of the column sequence average absolute deviation (Column_avg_dev) to the prediction of mixing enthalpy is the highest, followed by the pseudopotential radius average absolute deviation (Rs+p_avg_dev) and electronegativity average absolute deviation (Electronegativity_avg_dev). This finding aligns with traditional physical metallurgy theory in multiple dimensions. In the Miedema model, the electronegativity difference (ΔΦ) is a core parameter that directly affects the strength of chemical bonds. The Hume-Rothery rules [48,49] state that, when the atomic radius difference (Δr) exceeds 15%, the lattice distortion energy increases significantly. The larger the column sequence deviation, the more pronounced these property differences typically are, thus affecting the mixing enthalpy. The machine learning model successfully captures the synergistic effect of these multiple parameters, using the column sequence deviation as a high-order correlation feature, which helps explain the interaction between ΔΦ and Δr and their contribution to the mixing enthalpy. It is worth noting that, among the top 15 important features, 9 physical features constructed based on domain knowledge (accounting for 60%) show significant advantages, indicating that manually selected features based on prior physical knowledge can effectively capture relevant material characteristics while having lower computational costs.
The SHAP feature density scatter plot shows how the top features in the dataset influence the model output, as seen in Figure 4b. The sign polarity of the SHAP values reveals the direction of feature control over the mixing enthalpy. Positive values correspond to a positive gain in the predicted value, while negative values indicate a suppressive effect. Points accumulate along each feature axis to display the density. When the group number deviation is larger (e.g., combinations of elements from the IB group and VA group), the SHAP values become more negative, indicating that this feature has a strong negative impact on the prediction of the mixing enthalpy. The electronegativity feature has the widest color region, suggesting that it has the greatest influence on the prediction of the mixing enthalpy. This explains why the model’s prediction becomes more difficult when elements like B and C from the right-hand side of the periodic table are included, as their electronegativity changes significantly (e.g., from 2.04 for B to 2.55 for C), which can lead to the formation of highly polar bonds that the existing features may not fully capture.

3.4. LightGBM Model for Prediction of Enthalpy of Mixing of Binary Alloys

In order to avoid the bias caused by the division of the training and test sets, we adopted a cross-validation method. Specifically, the 583 binary systems were randomly divided into 11 subsets. In each dataset, one subset was selected as the test set, and the remaining 10 subsets were used as the training set. Figure 5 shows the training and testing errors for the 11 training-test dataset pairs. From the figure, it can be seen that the training errors remain almost the same, although there are moderate fluctuations in the testing errors between different dataset pairs. This suggests that the fitting ability of the model is more stable during the training process, while the variation of the testing error may be related to the specific composition of the dataset. Therefore, cross-validation effectively reduces the bias due to differences in data partitioning and enables a more comprehensive assessment of the model’s generalization performance.
To systematically investigate the influence of underlying chemical components on testing errors, this study conducted a comprehensive analysis of the mean absolute error across 583 alloy systems. Through error analysis, four systems (Ir-Ru, Ir-Zr, Bi-U, and Cu-Nb) exhibited notably high prediction deviations in the test set (Table 7). A literature review revealed that [50,51,52,53] these systems lack sufficient experimental thermodynamic property data in their liquid-phase regions, which likely contribute to the model’s prediction inaccuracies. Notably, when incorporating these high-error systems into the training set (e.g., Fold-02 where all four systems were included), the model accuracy improved substantially, with an MAE decreasing from 5.6 kJ/mol to 3.5 kJ/mol. This improvement underscores the critical influence of data coverage on model performance.
Figure 6 contrasts the prediction discrepancies of these four anomalous systems between the training and test sets. Their satisfactory performance during training versus significant deviations in independent testing highlights an overfitting phenomenon, further confirming that data scarcity limits model generalization capabilities. The study demonstrates that machine learning predictions based on the CALPHAD database inherently depend on the quality and reliability of foundational data. Systematic prediction biases may occur when test sets contain substantial unreliable data.

4. Conclusions

In this work, we combined machine learning with the CALPHAD method to successfully predict the enthalpy of mixing of a binary alloy system by training and evaluating the model using the enthalpy of mixing data obtained from the SGTE alloy database. In terms of model selection, this paper compares the performance of four machine learning models, namely KNN, MLP, RF, and LightGBM. The results show that the LightGBM model performs the best, with an R2 of 92.2% and an MAE of 3.5 kJ/mol, which verifies the superiority of the ensemble learning approach in predicting thermodynamic properties. In addition, the adoption of RFE and MIC feature selection methods effectively improves the model efficiency and further enhances the prediction accuracy by reducing the number of input features. SHAP interpretability analysis identified the dominant factors influencing the enthalpy of mixing, particularly atomic radius and electronegativity. These factors align with the key parameters of the Miedema model and are consistent with knowledge in the field of metallurgical thermodynamics. Although the overall prediction accuracy is satisfactory, significant deviations are observed in specific alloy systems, especially in those with limited experimental datasets, highlighting the importance of data quality in machine learning models. Finally, this study demonstrates the feasibility and efficiency of data-driven models for predicting thermodynamic properties, while also emphasizing the challenges associated with accurate predictions in specific alloy systems. Future investigations will focus on collecting more high-quality experimental data to expand the training set, further reducing the dimensionality of descriptors, and incorporating thermodynamic constraints to reduce prediction uncertainty. Additionally, we will explore the applicability of this model to multicomponent systems to enable broader applications.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/met15050480/s1, File S1: dataset; File S2: experimental data; File S3: Predictions.

Author Contributions

Conceptualization, S.H. and Z.C.; methodology, S.H. and G.W.; software, S.H.; validation, S.H., G.W. and Z.C.; formal analysis, S.H.; investigation, S.H. and G.W.; resources, Z.C.; data curation, S.H.; writing—original draft preparation, S.H.; writing—review and editing, S.H. and Z.C.; visualization, S.H.; supervision, Z.C.; project administration, S.H., G.W. and Z.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the Supplementary Materials. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Number of occurrences of each element in the dataset.
Figure 1. Number of occurrences of each element in the dataset.
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Figure 2. The regression effect of the four models on the training set and test set.
Figure 2. The regression effect of the four models on the training set and test set.
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Figure 3. The Pearson correlation matrix of the 28 descriptors after feature selection.
Figure 3. The Pearson correlation matrix of the 28 descriptors after feature selection.
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Figure 4. The importance ranking of features used to achieve the best prediction performance in the LightGBM model and the feature density scatter plot.
Figure 4. The importance ranking of features used to achieve the best prediction performance in the LightGBM model and the feature density scatter plot.
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Figure 5. MAE of the 11-fold cross-validation model on the evidence training set and test set.
Figure 5. MAE of the 11-fold cross-validation model on the evidence training set and test set.
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Figure 6. (a–h) show enthalpy of mixing prediction results for the four high-error systems on the test set (left) and train set (right). ● The blue solid circles represent CALPHAD optimized values, and red X represents the predicted values of the LGBM model. (a) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Ir-Ru systems. (b) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Ir-Ru systems. (c) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Bi-U systems. (d) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Bi-U systems. (e) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Ir-Zr systems. (f) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Ir-Zr systems. (g) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Cu-Nb systems. (h) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Cu-Nb systems.
Figure 6. (a–h) show enthalpy of mixing prediction results for the four high-error systems on the test set (left) and train set (right). ● The blue solid circles represent CALPHAD optimized values, and red X represents the predicted values of the LGBM model. (a) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Ir-Ru systems. (b) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Ir-Ru systems. (c) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Bi-U systems. (d) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Bi-U systems. (e) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Ir-Zr systems. (f) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Ir-Zr systems. (g) Enthalpy of mixing prediction results on the test set and the CALPHAD calculation results for Cu-Nb systems. (h) Enthalpy of mixing prediction results on the train set and the CALPHAD calculation results for Cu-Nb systems.
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Table 1. List of descriptors used in the text.
Table 1. List of descriptors used in the text.
DescriptorMeanings
NumberAtomic number
MendelevNumberMendeleev number
AtomicWeightRelative atomic mass
MeltingTMelting point
ColumnWhat column in the periodic table of elements
RowWhat row in the periodic table of elements
CovalentRadiusCovalent radius
ElectronegativityElectronegativity
NsValenceNumber of filled s orbitals electrons
NpValenceNumber of filled p orbitals electrons
NdValenceNumber of filled d orbitals electrons
NfValenceNumber of filled f orbitals electrons
NValenceNumber of valence electrons
NsUnfilleds orbital not filled with electrons
NpUnfilledp orbital not filled with electrons
NdUnfilledd orbital not filled with electrons
NfUnfilledf orbital not filled with electrons
NUnfilledPeriphery not filled with electrons
SpaceGroupNumberSpace group number
Electronegativity_MBElectronegativity (Martynov and Batsanov)
Rs+pZunger’s pseudopotential radius
Work functionWork function
1st ionization energyEnergy of ionization first
n w s 1 / 3 The cube roots of the electron densities at the Wigner–Seitz cell boundary
V 2 / 3 Molar volume to the 2/3 power
Electrochemical equivalentElectrochemical equivalent
Bulk modulusValue of bulk modulus
Cohesive energyCohesive energy
Table 2. Optimal parameters of Random Forest regression model.
Table 2. Optimal parameters of Random Forest regression model.
ParameterN_EstimatorsMax_DepthMin_Samples_SplitMin_Samples_LeafMax_Features
Value800None206Log2
Table 3. Optimal parameters of Multilayer Perceptron Machine regression model.
Table 3. Optimal parameters of Multilayer Perceptron Machine regression model.
ParameterLearning_Rate_InitHidden_Layer_SizesActivationMax_Iter
Value0.01(50, 50)relu800
Table 4. Optimal parameters of LightGBM regression model.
Table 4. Optimal parameters of LightGBM regression model.
ParameterN_EstimatorsMin_Child_SamplesMax_DepthNum_LeavesReg_Lambda
Value10004015310.1
Table 5. Optimal parameters of K-Nearest Neighbor regression model.
Table 5. Optimal parameters of K-Nearest Neighbor regression model.
ParameterN_Neighbors
Value10
Table 6. Evaluation results of different machine learning models.
Table 6. Evaluation results of different machine learning models.
ModelDatasetR2MAE (kJ/mol)RMSE (kJ/mol)
LGBMTrain0.990.71.0
Test0.923.55.0
KNNTrain0.932.64.8
Test0.487.712.8
RFTrain0.990.81.4
Test0.903.85.6
MLPTrain0.923.55.1
Test0.854.96.8
Table 7. Comparison of MAE for the four high-error systems acting as the test set and training set.
Table 7. Comparison of MAE for the four high-error systems acting as the test set and training set.
Cross-Validation FoldHigh-Error SystemsMAEtest (kJ/mol)MAEtrain (kJ/mol)
Fold-06Ir-Ru62.551.19
Fold-10Bi-U39.80.88
Fold-05Ir-Zr36.032.73
Fold-11Cu-Nb31.693.86
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Huang, S.; Wang, G.; Cao, Z. Prediction of Enthalpy of Mixing of Binary Alloys Based on Machine Learning and CALPHAD Assessments. Metals 2025, 15, 480. https://doi.org/10.3390/met15050480

AMA Style

Huang S, Wang G, Cao Z. Prediction of Enthalpy of Mixing of Binary Alloys Based on Machine Learning and CALPHAD Assessments. Metals. 2025; 15(5):480. https://doi.org/10.3390/met15050480

Chicago/Turabian Style

Huang, Shuangying, Guangyu Wang, and Zhanmin Cao. 2025. "Prediction of Enthalpy of Mixing of Binary Alloys Based on Machine Learning and CALPHAD Assessments" Metals 15, no. 5: 480. https://doi.org/10.3390/met15050480

APA Style

Huang, S., Wang, G., & Cao, Z. (2025). Prediction of Enthalpy of Mixing of Binary Alloys Based on Machine Learning and CALPHAD Assessments. Metals, 15(5), 480. https://doi.org/10.3390/met15050480

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