Predicting stacking fault energy in austenitic stainless steels via physical metallurgy-based machine learning approaches
0 Abstract
Stacking fault energy (SFE) significantly influences plastic deformation, strength, and processing performance, making accurate assessment and prediction of SFE essential for material design and optimization. Traditional SFE calculations mainly rely on experimental measurements and thermodynamic theories, with the former usually being time-consuming and the latter limited in applicability at different compositions. To overcome these limitations, this study proposes a machine learning (ML) strategy introducing physical metallurgy (PM) parameters relevant to SFE, aiming to achieve robust predictions. Specifically, this study evaluates three methods for introducing PM information into ML (as an input, an intermediate parameter, and a transfer source), with transfer learning as the best strategy. Initially, various PM parameters were calculated based on alloy composition and temperature, and subsequently used as inputs to train a convolutional neural network (CNN). This source model was then transferred to the SFE prediction model. The results from the model transfer using different PM information show that incorporating phase-transformation driving force (DF) as a source model for SFE prediction provided the most accurate and reliable results. This approach of introducing PM parameters into ML significantly improves the predictive capability of SFE models, offering a new perspective and solution for the prediction of SFE. Furthermore, this method may also be applicable to the prediction of other material properties during material design and optimization.
Keywords
INTRODUCTION
Stacking fault energy (SFE) plays an important role in determining the susceptibility of a crystal to dislocation sliding and phase transformation during deformation. The outstanding mechanical properties of austenitic stainless steel, such as high plasticity and toughness, are partly attributed to its face-centered cubic (FCC) structure and relatively low SFE. Under specific conditions, a lower SFE facilitates deformation twinning and can induce a phase transformation from an FCC to a hexagonal close-packed (HCP) structure, enhancing the deformation mechanism, strength and processing performance of the alloy[1-4]. Therefore, accurate assessment and prediction of the SFE are essential for material design, as mastering SFE variations facilitates optimizing material microstructure, resulting in higher strength while maintaining plasticity and better addressing the need for high-performance materials in fields such as aerospace[5].
SFE calculation methods are categorized into experimental and theoretical approaches[6-10]. Experimental methods such as transmission electron microscopy (TEM), X-ray diffraction (XRD), and neutron diffraction (ND) directly measure stacking fault characteristics, which are then used to calculate the SFE. Whelan[11] established a theoretical foundation by examining extended dislocation nodes in steel grades; advancements in imaging techniques enabled researchers[12,13] to observe partial dislocation separations through weak-beam dark-field modes, a method widely applied in multi-component austenitic steels but limited to low SFE values. Reed and Schramm[14] determined the SFE using XRD line profile analysis, relating SFE to stacking fault probability, and rms microstrain. ND, similar to XRD but using thermal neutrons, also enables SFE determination. However, these experimental methods encounter a significant problem: the material constants in computational equations often derive from approximated values based on similar compositions, introducing uncertainties, and experimental measurement of the SFE is time-consuming and complex, hindering rapid material discovery. Given the dependence of SFE on alloy composition and temperature, empirical equations, thermodynamic models, and density functional theory (DFT) have emerged as alternative calculation methods. While several authors have developed empirical equations tailored to limited alloy compositions[15-17], their applicability is restricted. de Bellefon et al. collected 144 austenitic steel composition measurements and accurately predicted the SFE using linear regression[18]. As discussed in Supplementary Figure 1, Olson and Cohen[19] introduced a thermodynamic model conceptualizing stacking fault occurrence as an FCC-to-HCP transformation, calculating the change in energy per unit area, though its reliability depends heavily on the quality of the calculation of phase diagrams (CALPHAD) database[20-24]. The DFT suggested by Hohenberg and Kohn[25] calculates the SFE when the slip surface slides into a stable crystal structure by creating a crystal cell from an atomic perspective rather than depending on thermodynamic models or empirical equations. However, model building and optimization remain challenging and time-consuming.
The material genome initiative (MGI) has spurred data-driven, machine learning (ML)-based approaches[26,27] for SFE calculations. For example, Chaudhary et al. constructed a ML-based classifier for SFE prediction, facilitating the prediction of deformation mechanisms for unknown alloys[28]. Khan et al. proposed a framework combining DFT calculations, ML, and physical properties to predict SFE in entropy alloys[29]. Although these studies correlate alloy composition with SFE, predictive accuracy remains limited, and the ML model often lacks interpretability. Therefore, incorporating domain expertise [physical metallurgy (PM) parameters] into ML models to improve predictive accuracy and interpretability has become an important strategy, with promising results demonstrated[30-32]. For instance, Shen et al. incorporated thermodynamic methods on precipitated phases in ultra-high-strength stainless steels as ML model inputs for hardness prediction, achieving accurate, generalized predictions and optimizations in alloy design[33]. Basha et al. considered the role of fully connected layers in convolutional neural networks (CNNs) by analyzing layer-dataset feature relationships by changing their numbers, providing a novel approach for the incorporation of PM information into deep learning models[34]. Wei et al. showed that using source features related to target performance aids in predictive transfer, highlighting effective incorporation of PM mechanisms into ML models[35]. Despite these advancements, ML-based SFE prediction models introducing PM knowledge remain underexplored.
In this study, we systematically compare different strategies for introducing PM information into ML, examining PM variables affecting SFE to obtain a generic prediction model. Using experimentally collected datasets on alloy composition, temperature, and SFE, we derived relevant PM data through thermodynamic calculations. Three PM introduction methods were explored: (i) direct input as features; (ii) inclusion in CNN fully connected layers; and (iii) transfer learning, where the source model predicting PM data transfers to the SFE prediction model. We evaluated the performance of each method and identified transfer learning as the optimal approach, providing methodological insights for universal SFE prediction and contributing to alloy design and development.
MATERIALS AND METHODS
Dataset and data preprocessing
Multiple publicly available SFE databases for austenitic alloys[28] were referenced and employed, containing experimental SFE measurements for several steel grades at different temperatures. Since the majority of data were measured at 300 K and SFE measurements at other temperatures are scarce and highly variable, only the SFE data recorded at 300 K were retained for this study. In addition, the original database was further screened to remove data with large errors, and the same compositions were averaged. This process resulted in a high-quality dataset of 188 samples containing 11-dimensional compositional features and SFE values, which were used for model training. To further refine the quality of the data, the phase transformation driving force (DF), and the Gibbs free energies of the FCC (GFCC) and HCP (GHCP) phases of the 188 sample alloys were computed using Thermo-Calc® software and the TCFE9 database. The details of the dataset are listed in Table 1. The distribution of the main elements and related PM parameters of the alloys in the dataset is shown in Figure 1, which contains compositional data for austenitic stainless steel, high-manganese steel, etc.
Figure 1. Scatterplot of the distribution matrix of the main features in the dataset.
Input and output ranges of the various parameters in the SFE database
| Features | Minimum | Maximum | Mean | Standard deviation |
| Ferrum/wt.% | 48.623 | 85 | 67.208 | 6.537 |
| Cr/wt.% | 0 | 25.85 | 15.23 | 5.777 |
| Ni/wt.% | 0 | 26.4 | 11.93 | 6.473 |
| Mn/wt.% | 0 | 27.5 | 4.35 | 7.106 |
| Si/wt.% | 0 | 6.22 | 0.41 | 1.12 |
| Molybdenum/wt.% | 0 | 2.7 | 0.64 | 1.06 |
| Carbon/wt.% | 0 | 0.69 | 0.054 | 0.124 |
| Nitrogen/wt.% | 0 | 0.88 | 0.059 | 0.135 |
| Phosphorus/wt.% | 0 | 0.08 | 0.009 | 0.015 |
| Sulfur/wt.% | 0 | 0.043 | 0.002 | 0.007 |
| Aluminum/wt.% | 0 | 3.98 | 0.11 | 0.551 |
| DF/J·mol-1 | -1,663.99 | 1,892.676 | -513.779 | 834.899 |
| Gibbs free energy of the FCC phase/J·mol-1 | -18,546.5 | -3,087.44 | -6,174.99 | 3,186.666 |
| Gibbs free energy of the HCP phase/J·mol-1 | -17,033.4 | -3,823.53 | -6,688.77 | 2,550.788 |
| SFE/mJ·m-2 | 3.26 | 70.15 | 35.3 | 13.7 |
To ensure model stability and avoid the “lucky split” issue caused by random partitioning, this study rigorously divided the data into training and test sets at a ratio of 4:1, and repeated this random partitioning process 100 times to conduct a more in-depth analysis of the model’s robustness. Additionally, for a comprehensive evaluation of each model’s performance, we specifically selected 63 austenitic stainless steel SFE data[18] points as an independent validation set. This validation set is completely isolated from the training and test sets, with the aim of assessing the model’s generalization capability in handling novel, unseen data.
This study also explores PM parameters of existing alloy systems to establish an extended dataset. Specifically, as shown in Figure 2A, 6,000 sets of new alloy samples were randomly generated within specified compositional ranges [chromium (Cr) < 25.85 wt.%; nickel (Ni) < 26.4 wt.%; manganese (Mn) < 27.5 wt.%; silicon (Si) < 6.22 wt.%]. All PM parameters for these samples, including DF, GFCC and GHCP, were calculated using Thermo-Calc®.
Figure 2. SFE prediction framework. (A) Data processing and dataset expansion; (B) different frameworks for 3 PM parameter introduction methods. SFE: Stacking fault energy; PM: physical metallurgy.
For data normalization, the inputs and outputs were normalized using the z-score, which is a standard method for eliminating dimensional differences between feature ranges[36], as determined by:
where z denotes the normalized value, x is the original value from the dataset, and μ and σ represent the mean and standard deviation of the original values for a certain dimensional feature, respectively.
SFE prediction framework
As previous studies primarily established direct relationships between alloy composition, temperature, and SFE, this study further introduces PM parameters related to the SFE, such as the DF, GFCC, and GHCP. Three approaches for introducing PM parameters in increasing “depth” [Figure 2B] were considered: (1) Direct input: PM parameters, along with alloy composition, were used as input features to predict SFE; (2) Intermediate layer in CNN: PM parameters were introduced into the fully connected layer of the CNN; and (3) Transfer learning: PM parameters were used to train a source model that was then transferred to the SFE prediction model.
When the PM parameters were used as input features, five common ML algorithms were selected to construct the regression prediction models: random forest (RF), gradient boosting regression (GBR), multilayer perceptron (MLP), support vector regression with a radial basis function kernel (SVR), and extreme gradient boosting (XGB). After optimizing each model using grid search, RF was found to perform best for SFE prediction. Various PM-composition combinations were evaluated to assess the influence of PM parameters on SFE prediction accuracy.
In addition to the above approaches, this study explores the introduction of PM parameters into an intermediate layer of the deep learning model. A simple CNN structure was developed, comprising a sequential arrangement of a convolutional layer and a fully connected layer. As shown in Supplementary Figure 2, in this setup, the 11-dimensional constituent features of the alloy samples were finally reshaped into a 4 × 4 matrix (with the alloy compositional features filtered into the matrix by filling them sequentially, and the values of the remaining five elements set to zero) and then fed into a convolution module containing two convolutional layers (4 × 4 × 8 and 4 × 4 × 16). This convolutional module was followed by a fully connected layer with two architectures (64 and eight layers, respectively). At this fully connected layer, PM parameter information was incorporated, and the combined data was fed into the next fully connected layer to predict SFE.
For the transfer learning model, a CNN source model was initially constructed using the aforementioned PM parameter dataset to predict DF, GFCC and GHCP based on the alloy composition. This source model was then used to construct the SFE prediction model. Specifically, the convolutional and fully connected layers of the source model were replicated in the target transfer model, creating a new structure called the transferred feature layer. The parameters of this layer were frozen and did not participate in further training; only the remaining fully connected layers were randomly initialized for model training on SFE prediction.
All data preprocessing and model training were performed using the TensorFlow and Scikit-learn packages. For the CNN model, training was conducted over 500 iterations with mean square error (MSE) as the loss function, a learning rate of 0.1, and the Adam optimizer.
For model performance evaluation, to objectively evaluate the generalization ability of various models and identify the optimal model, the squared correlation coefficient (R2) and mean absolute error (MAE) were used as evaluation metrics[37,38]. These metrics are expressed as follows:
where n is the number of samples, and f(xi) and yi are the predicted and experimental values of the ith sample, respectively.
RESULTS AND DISCUSSION
Feature analysis of composition and PM parameters
In this study, SFE prediction was examined based on the alloy composition and PM parameters. To quantify the relationship and importance of these features in SFE prediction, the Pearson correlation coefficient, Shapley Additive exPlanations (SHAP) value and mean decrease accuracy (MDA) were used[39-41]. The Pearson correlation coefficient was defined as the covariance between two variables divided by the product of their standard deviations. It measures the degree of linear correlation between variables, ranging from -1 to 1, where absolute values closer to 1 indicate a stronger correlation. The SHAP value is a game-theory-based method for interpreting the predictive results of ML models. It helps us to understand how much each feature contributes to the prediction result of the model. MDA assesses the importance of specific features for SFE prediction by disrupting selected compositional or PM parameter features and calculating the resulting decrease in model prediction accuracy.
The Pearson correlation results [Figure 3A] revealed the relationship between alloying elements and SFE. Specifically, Cr, Ni, and Mo exhibited a positive correlation with the SFE values, whereas Mn and Si demonstrated a negative correlation. Notably, this finding partially contrasts with conventional understanding, particularly in austenitic stainless steels, where Cr is generally believed to lower the SFE. In this case, however, the Pearson analysis indicated a positive correlation, potentially influenced by the data on high-Mn steels, as Mn, a strong austenite-stabilizing element, can significantly increase the SFE values. Additionally, correlations among primary alloying elements, such as Cr, Ni, Mn, Si, and PM, are associated with one another, confirming the reasonableness of the selected parameters and underscoring the complexity of their influence on the SFE. The absolute values of the R2 between each input feature and the SFE were greater than 0 but less than 1, indicating the independent contribution of each feature on SFE and emphasizing the importance of considering the elemental interactions in alloy design.
Figure 3. Feature analysis of alloy compositions and PM parameters with respect to SFE. (A) Pearson’s correlation coefficient; (B) SHAP values; (C) MDA. PM: Physical metallurgy; SFE: stacking fault energy; SHAP: Shapley Additive exPlanations; MDA: mean decrease accuracy.
As shown in Figure 3B, the results of SHAP values demonstrate that both alloy composition and PM parameters contribute to SFE. Specifically, elements such as Ni and Si exhibit significant contributions, highlighting their core position in the process of predicting SFE using the model. Similarly, PM parameters also have an impact on SFE prediction. The SHAP values of DF, GFCC, and GHCP are sequentially ordered, with GFCC being higher than GHCP. This is similar to the results of MDA, as further analyzed in Figure 3C for the predictive significance of each feature: Ni contributed most significantly, with an MDA value exceeding 50%. Previous thermodynamic theoretical studies have shown SFE in FCC alloys to be closely linked with DF and other factors, indicating that PM parameters also contribute to SFE prediction. Among the PM parameters, the MDA values ranked as GFCC, DF, and GHCP, with means GFCC in austenitic steels has significantly higher importance than GHCP.
Reliability analysis of models with PM parameters
In the modeling strategy, where PM parameters are directly introduced as inputs, various common ML algorithms were considered, as outlined previously. In addition, modeling without the PM parameters was evaluated, meaning SFE prediction relied solely on the alloy composition. Figure 4 shows the R2 and MAE results of different algorithms before and after introducing different PM parameters. All models exhibited noticeable overfitting, with most achieving R2 values above 95% and MAE below 2 mJ/m2 for the training set, whereas the R2 for the testing set ranged from 65%-70% and MAE values were approximately 6 mJ/m2, indicating suboptimal prediction performance. Specifically, the simple introduction of PM information did not improve the predictive performance of the models, as the R2 and MAE values were similar between models with and without PM information. Furthermore, no significant difference was observed among different types or combinations of PM information.
Figure 4. Mean R2 and MAE results for different PM models using conventional ML algorithms. R2: The squared correlation coefficient; MAE: mean absolute error; PM: physical metallurgy; ML: machine learning.
This study further investigated the impact of the simple introduction of PM information on model prediction scalability by comparing models without PM information and those introducing all PM parameters (including GBR, MLP, RF, SVR, and XGB) for predicting SFE under single compositional changes. Figure 5 illustrates the trend of the model-predicted SFE variations with changes in the weight percentages (wt.%) of Cr, Ni, Mn, and Si alloying elements. Figure 5A and B shows the average values of SFE for 100 model predictions across five algorithms for Cr content variations (0-25 wt.%). Initially, SFE decreased with increasing Cr content, and then exhibited an upward trend once it decreased to a certain value. Compared to the non-PM control group, the model with the introduction of the PM parameter showed a significant reduction in the range of predicted values and an improvement in the error bars, indicating improved accuracy and stability in SFE predictions. Similarly, as Si increased, SFE decreased, and then increased again before declining [Figure 5G and H]. Guided by PM knowledge, the model with PM parameters effectively avoided negative SFE values. Although the introduction of PM information improved predictive results, its effect varied by element. For instance, the model prediction performance of Ni
Figure 5. Comparison of SFE trends for component prediction before and after the introduction of PM in different models. (A and B) Cr; (C and D) Ni; (E and F) Mn; (G and H) Si. SFE: Stacking fault energy; PM: physical metallurgy.
Impact of PM parameters on the model in different introduction methods
As discussed, introducing PM information as direct input yielded only limited improvement in SFE prediction, even when varying combinations of PM parameters were considered. This section discusses the effects of different PM parameter introduction approaches. Figure 6 shows the model prediction results with different PM information introduced under three frameworks, containing training, testing, and validation sets, including additional austenitic stainless steel samples. The prediction accuracies and errors of the ML models introduced across different PM information for both the training and testing sets were similar for all three frameworks. The main difference was in validation set predictions, highlighting significant differences in model scalability - important for practical applications and alloy design. Before introducing PM parameters, it is notable that, compared to RF algorithms, CNNs show higher flexibility in managing complex and nonlinear data. The convolutional layer of a CNN can automatically extract complex data features, effectively reducing prediction errors in SFE modeling and demonstrating the excellence of CNN modeling.
Figure 6. Comparison of model predictions under different introduction methods. (A) Input features by RF models; (B) fully connected layer by CNN models; (C) output of the source model by Transfer learning. RF: Random forest; CNN: convolutional neural network.
In the simple PM information introduction approach (e.g., Figure 6A), none of the PM parameter variations improved the generalization ability of the model, with the validation set MAEs consistently exceeding
This model is not limited to the original alloys in the dataset, but can deeply analyze PM information for both the alloys in the dataset and alloys of similar composition, enabling SFE predictions across a broad range of alloy systems and establishing a foundation for robust predictive scalability. However, this model constructs complex correlations between composition and PM information across diverse alloy composition-PM datasets, an approach absent in previous methods. As shown in Supplementary Figure 3, after expanding the database, the R2 of the source model increased from 98.5% to 99.8%. Next, we utilized these already-constructed models to further transfer and predict SFE, and verified the generalization ability of the models. Supplementary Figure 3C shows that the prediction accuracy has improved, with the R2 increasing from 72.3% to 78.5% while maintaining a low MAE value. This demonstrates that when the source model is trained with PM parameters as the output, generating sufficient virtual samples and fully training the model allows it to capture the underlying relationships between alloy composition and PM parameters. By transferring this complex association to predict SFE compositionally, introducing PM information as an intermediate step significantly enhances the predictive scalability of the model, enabling accurate prediction results even with small sample data, and truly achieving an enhancement in the interpretability of the ML model.
Prediction of SFE based on different PM introduction methods
This study considered three methods for introducing PM parameters into ML modes. As an example of the introduction of DF, Figure 7A shows the prediction results of these models, containing the training set, testing set, and additional sample data of austenitic stainless steel compositions as the validation set. For the training set, all models achieved high accuracy, with R2 > 85% and MAE below 4 mJ/m2, demonstrating the effectiveness of the RF algorithm and CNN in modeling SFE. Meanwhile, the slightly lower accuracy observed in the model incorporating PM parameters at the fully connected layer splicing may be due to the limitation of the small datasets used for deep learning models. For the testing set, the performances of the three models were similar, with prediction accuracies of approximately 70% and errors under 6 mJ/m2. The validation set, crucial for assessing the generalization ability of the model, revealed more significant differences between strategies. R2 values for the three models were above 70%, with the TR model having the greatest accuracy (close to 80%). For MAE, the RF model had a significantly higher prediction error (above 6.5 mJ/m2), whereas the CNN and TR models yielded lower errors (under 3 mJ/m2), indicating that the transfer-learn strategy exhibited the best prediction performance.
Figure 7. Comparison of model predictions using different methods of introducing DF. (A) R2 and MAE for training set, testing set, and validation set; (B) distribution of predicted and measured values of SFE for austenitic stainless steels. DF: Driving force; R2: the squared correlation coefficient; MAE: mean absolute error; SFE: stacking fault energy.
Figure 7B shows the scatter plots of the predictions of the three models for the validation set. The RF model, in which DF was introduced directly as an input, showed the poorest performance, with predictions diverging from actual values. The prediction performance of the CNN model improved significantly, although performance in lower SFE ranges remained suboptimal. In contrast, most of the points predicted by the TR model aligned with the actual values along a line with a slope of 1, achieving the best prediction ability in lower SFE ranges, thereby reducing error and improving accuracy. In addition to DF, similar results were obtained for other PM parameters, indicating that the transfer learning strategy is the most effective approach. The above results indicate that as the “depth” of the PM parameter introduction increases - corresponding to enhanced guidance from PM mechanisms in ML - model generalization ability also improves. The improvement in the generalization ability of the model is still limited under the simple introduction mode of directly taking PM parameter inputs, but improves significantly when PM information is the intermediate (fully connected layer) input to the model. The most substantial improvement in generalization arises from the transfer learning approach, where a source model with deeply mined PM information is constructed and subsequently transferred to the SFE prediction model. These results provide a reliable strategy for building ML models for SFE prediction, leveraging PM knowledge effectively in small data samples.
CONCLUSIONS
To obtain a generic and accurate prediction of the SFE, this study explored different strategies for introducing PM information into ML models, successfully establishing a framework for efficient prediction using PM information-guided ML. The specific conclusions are as follows:
(1) After comparing various strategies, we found that transfer learning with PM information as an intermediate step yielded the best predictive performance. Specifically, when the RF algorithm is directly applied, the model’s MAE reaches as high as 7.1 mJ/m2. On the other hand, when PM information is simply introduced into the fully connected layer of CNN, the best-achieved R2 value is only 69.8%. However, when PM information is used as an intermediate step in transfer learning, the model not only achieves a higher R2 value (up to 78.5%) but also significantly reduces the MAE to 2.9 mJ/m2, resulting in a substantial improvement in prediction accuracy.
(2) By establishing a component-PM information association as an intermediate section for the composition-SFE relationship, the transfer learning model demonstrated significant improvements in generalization ability. This transfer learning approach allowed the model to identify the key roles of different PM information in SFE prediction, with DF as the source attribute providing the most accurate SFE transfer model, outperforming models that incorporated other PM parameters or their combinations.
DECLARATIONS
Authors’ contributions
Made substantial contributions to conception and design of this review, writing and editing: Song L, Wei X, Wang C, Li Y
Made substantial contributions to collation of literature, figure preparation, and writing: Song L, Wei X
Performed data analysis, discussion and review writing: Song L, Wei X
Provided administrative, technical, and material support: Wei X, Li Y
Availability of data and materials
The data supporting the findings of this study are available within the article and its Supplementary Materials. Further data can be obtained from the corresponding authors upon request.
Financial support and sponsorship
The research was financially supported by the National Key Research and Development Program (Grant No. 2022YFB3707501), the Applied Basic Research Program of Liaoning Province (Grant No. 2023JH2/101300147) and the Xingliao Talent Program of Liaoning Province (Grant No. XLYC2203027).
Conflicts of interest
All authors declared that there are no conflicts of interest.
Ethical approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Copyright
© The Author(s) 2025.
Supplementary Materials
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Song, L.; Wang, C.; Li, Y.; Wei, X. Predicting stacking fault energy in austenitic stainless steels via physical metallurgy-based machine learning approaches. J. Mater. Inf. 2025, 5, 2. http://dx.doi.org/10.20517/jmi.2024.70
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