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Article

Machine Learning-Based Framework for Predicting Creep Rupture Life of Modified 9Cr-1Mo Steel

by
Mengyu Chai
1,*,
Yuhang He
1,
Yongquan Li
2,
Yan Song
1,*,
Zaoxiao Zhang
1 and
Quan Duan
1
1
School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China
2
China Special Equipment Inspection and Research Institute, Beijing 100029, China
*
Authors to whom correspondence should be addressed.
Appl. Sci. 2023, 13(8), 4972; https://doi.org/10.3390/app13084972
Submission received: 14 March 2023 / Revised: 9 April 2023 / Accepted: 12 April 2023 / Published: 15 April 2023
(This article belongs to the Section Materials Science and Engineering)
Browse Figures
Versions Notes

Abstract

:
Efficient and accurate predictions of creep rupture life are essential for ensuring the integrity of high-temperature components. In this work, a machine learning-based framework is developed for the quick screening of crucial features and accurate prediction of the creep rupture life of modified 9Cr-1Mo steels. A feature screening protocol based on correlation filtering and sequential feature selection techniques is established for identifying critical features that significantly affect the prediction performance from a set of numerous descriptors. Moreover, several machine learning algorithms are employed for model training to examine their ability to map the complex nonlinear interactions between multivariate features and creep life. The results show that the test stress, test temperature, tempering time, and the contents of S and Cr are identified as the crucial features that greatly influence the life prediction performance of modified 9Cr-1Mo steels. Moreover, the Gaussian process regression (GPR) model with these five selected crucial features exhibits the highest prediction accuracy among various machine learning strategies. Finally, an additional dataset out of model training and testing is used to further validate the efficacy of the constructed GPR model. The validated results demonstrate that most creep data are distributed inside the two-factor band lines. Results from this work show that the developed machine learning framework can offer high accuracy and excellent adaptability in predicting the creep life of modified 9Cr-1Mo steels under various environmental conditions.

1. Introduction

The 9Cr-1Mo steel alloyed with niobium, vanadium, and nitrogen is known as modified 9Cr-1Mo steel or ASME Grade 91 steel, which is frequently used as an important structural material for high-temperature components in the fossil fuel industry due to its excellent high-temperature strength and adequate corrosion/erosion resistance [1]. The engineering components made of a heat-resistant alloy (e.g., modified 9Cr-1Mo steel) such as boiler tubes and reactor vessels generally suffer elevated temperature and low- to moderate-stress environments. Under these circumstances, creep rupture is one of the most critical failure mechanisms. The prediction of creep rupture life is thus indispensable for ensuring the integrity and reliability of high-temperature components.
Creep is a complicated material property that is strongly associated with a number of influencing factors, including the alloy composition, initial microstructures produced by various thermomechanical processes, and environmental conditions such as applied stress level and temperature [2,3,4,5,6]. Moreover, long-term creep testing following normal standards is generally expensive. These challenges make it difficult to determine the creep rupture life with high accuracy, which limits the structural integrity assessment and also impedes the creep-oriented design of new alloys. For the purpose of predicting the creep rupture life of alloys, various conventional models have been developed in the past few decades. The most widely accepted methods include the Larson–Miller parameter method [7], the Monkman–Grant method [8], the Theta projection method [9,10], the Omega method [11], etc. However, the time-temperature parameter (TTP) extrapolation models highly depend on the experimental data, and the extrapolation outcomes generally exhibit significant uncertainties [12]. On the other hand, the constitutive models are capable of describing the creep behavior and predicting creep rupture life with short/intermediate-term creep data compared to time-consuming experimental measurements. However, each of these models has a specific range of applications. Moreover, the parameters for these theoretical models are typically complex and alloy composition-sensitive [13]. Therefore, the practical application of traditional creep life prediction models is limited.
Recently, machine learning methods have demonstrated considerable promise for the life prediction of components [14,15,16,17]. As a data-driven approach, the significant advantage of machine learning is its exceptional ability to map the intricate nonlinear relationship between high-dimensional data, including material characteristics and environmental factors. Moreover, the application of machine learning techniques does not necessitate extensive specialist knowledge. Thus, machine learning can offer an alternative to complex traditional life prediction models and overcome their limitations. To date, some efforts have been made to improve the applications of machine learning methods in the prediction of the creep rupture life of alloys [13,18,19,20,21,22,23,24,25,26,27,28]. In the majority of these machine learning studies, the alloy composition, heat treatment parameters, and environmental factors such as testing temperature and stress are regarded as input variables, while the creep rupture life or the time-temperature parameters (e.g., Larson–Miller parameter and Manson–Haferd parameter) are employed as output variables [13,20,21,22,23,24,25,26,27,28]. Various machine learning strategies such as multilayer perceptron [13], linear regression [20], random forest regression [21], deep neural network [22], and ensemble learning methods [23,24,25] are developed to accurately describe the complex relationship between high-dimensional input data and output life. For instance, Zhang et al. [22] proposed a deep neural network for life prediction of 316 stainless steel items under creep, fatigue, and more complex creep-fatigue conditions. Tan et al. [23] evaluated the performances of several ensemble learning algorithms on creep life with a prediction of 9% Cr martensitic steel, and they found the extra trees (ETs) and extreme gradient boosting (XGB) methods exhibit higher accuracy than other models. Liu et al. [26] proposed a divide-and-conquer self-adaptive (DCSA) learning method to accurately predict the creep rupture life of Ni-based superalloys. Moreover, five microstructural factors such as the lattice parameter, stacking fault energy, and other characteristics were taken into consideration as input descriptors in addition to the basic input variables to reduce the prediction error.
Although extensive research has shown the capability of developing machine learning-based models for creep rupture life prediction, very few studies take into account the impact of material forms (such as tube, plate, and pipe) on prediction outcomes. The modified 9Cr-1Mo steel has been selected for the majority of the high-temperature components, such as boiler tubes, reactor vessels, and seamless pipe, in the fossil industry. However, the findings of a recent study by Cano and Stewart [29] suggest that despite the material (9Cr-1Mo-V-Nb) remaining the same for various product forms, namely tube, plate, and pipe, the parameters for the Wilshire life prediction model are different for each form. Therefore, there is a need to determine the degree to which the material form affects the machine learning-based creep life predictions. On the other hand, even though most earlier studies used an acceptable number of features, such as material characteristics and environmental factors as input variables for life prediction, many of them overlooked the feature screening procedure prior to the training and validation of machine learning models. An overly complex model may be created by using too many input variables, which will have a detrimental effect, such as the overfitting issue, on prediction performance. As a result, the machine learning model with a simple structure is required. Moreover, when predicting the creep life of new alloys supplied by various manufacturers, some input features might not be readily available. Therefore, feature screening must be carried out prior to the training of machine learning models in order to identify the crucial features that have a significant impact on the creep rupture life and to make the model as simple and applicable as possible.
The work begins with constructing a dataset that contains the creep rupture data of modified 9Cr-1Mo steel with different material forms, and focuses on developing a machine learning-based framework for the efficient identification of crucial features selected from a collection of numerous features, and the precise prediction of creep rupture life. Several machine learning algorithms are employed to train the model and examine the impact of different algorithms on prediction performance. These algorithms include least absolute shrinkage and selection operator (Lasso) [30], ridge regression (Ridge) [31], support vector regression (SVR) [32], Gaussian process regression (GPR) [33], decision tree (DT) [34] and random forest (RF) [35]. Finally, the developed creep rupture life prediction model that possesses the highest prediction accuracy is validated using a dataset out of model training and testing. Results from this study will provide valuable insights for identifying critical features contributing to improving the model performance, and also guide complementary studies on the life prediction of alloys using the machine learning strategy.

2. Materials and Methods

Figure 1 shows the machine learning-based framework for creep rupture life prediction. This study aims at establishing a high-accuracy machine learning-assisted creep life prediction model by screening critical features from a set of a large number of variables. The machine learning procedure that is applied in the current research contains five primary steps, namely, dataset construction, data preprocessing, feature screening, model training and evaluation, and final prediction. In the beginning, creep data for modified 9Cr-1Mo steel are collected from a variety of sources, and the input features associated with the target variable, namely creep rupture life, are determined. Specifically, the determination of input features should be based on both physical mechanisms and experimental observation. The constructed dataset is then preprocessed by normalization and logarithmic transformations, which helps to increase modeling accuracy. Before training the machine learning model, feature screening is performed with the primary goal of identifying the critical features that influence the model prediction via correlation filtering and sequential feature selection methods. After determining a critical number of important features, several machine learning algorithms are employed for model training with k-fold cross-validation. The most appropriate model is determined by evaluating some error indicators of various models, including the coefficient of determination (R2) and root mean squared error (RMSE). Finally, the developed model is applied for the prediction of creep rupture life using new data, and the predicted results are compared with the actual values.

2.1. Data Collection and Preprocessing

The creep rupture dataset of modified 9Cr-1Mo steel with different forms was obtained from the National Institute for Material Science (NIMS) creep database [36] and many published studies in the literature [37,38,39,40]. A total of 431 sets of data were collected, including 265 sets of tube data, 158 sets of plate data, and 38 sets of pipe data. Figure 2 plots an instance of the creep life distribution diagram for modified 9Cr-1Mo steel with different material forms at 600 ℃. It is obvious that creep data present remarkable dispersion because of different chemical compositions and manufacturing processes from various data sources. In addition, despite the same applied stress level and temperature, the material form also influences the creep rupture life. This shows that it is challenging to account for some factors, such as alloy compositions and other manufacturing features, when predicting creep rupture life using conventional approaches.
The collected dataset contains eighteen input features and one output feature (creep rupture life), and Table 1 provides a statistical summary of the information gathered. Both the material characteristics and environmental parameters were taken into consideration for the input features. The material characteristics contain eleven features of chemical element content, including C, Mn, P, S, Ni, Cr, Mo, N, Al, V, and Nb (in wt.%), and five processing parameters, namely the normalizing temperature (N Temp), normalizing time (N Time), tempering temperature (T Temp), tempering time (T Time), and material form. In particular, various material forms such as tube, plate, and pipe are represented by 1, 2, and 3 in the dataset, respectively. Each material form contains both short-term data (creep life smaller than 104 h) and long-term data (creep life larger than 104 h). In addition to material qualities, two significant environmental parameters, test stress and temperature, are also regarded as input features.
The input variables have different levels of change, as indicated in Table 1, and the creep life is distributed in a wide range with an extremely large standard deviation. To minimize issues caused by the significant range variation between various features, all input variables were normalized to a certain range between 0 to 1. Moreover, the logarithmic transformation is applied for the output feature. The applied preprocessing process makes it simpler to examine how the data are distributed and helps to improve the modeling accuracy. The standardized dataset is then divided into a training dataset (80% data) and a testing dataset (20% data) to evaluate the generalization ability of various machine learning models.

2.2. Feature Screening

There are two main objectives for feature screening. One is to identify the essential features that have a substantial impact on the target variable and explore each essential feature’s impact on the result. The other is to minimize the input dimensions, speed up the training process, and make the model interpretable and universal [41,42]. The selected features should be both easily accessible and responsive to the creep rupture life. In this study, Pearson correlation screening and sequential feature selection methods are used for feature screening.
The Pearson correlation coefficient is generally used to examine the strength of the linear relationship between any two features. In the feature screening process, the Pearson correlation analysis will provide a preliminary filter of features that exhibit strong linear correlation. The Pearson correlation coefficient is obtained according to the following formula:
r = ∑ i = 1 n ( X i − X ¯ ) ( Y i − Y ¯ ) ∑ i = 1 n ( X i − X ¯ ) 2 ∑ i = 1 n ( Y i − Y ¯ ) 2
where n is the total number of samples in the dataset, X i and Y i are the values of features X and Y of the ith sample, and X ¯ and Y ¯ represent the average values of features X and Y of all samples, respectively. Any two input features that have an absolute value of r greater than 0.95 are considered to be linearly dependent, meaning that they provide comparable information and thus have comparable impacts on the output. The feature that has the lower r with the output is thus deemed to be less significant and is then removed.
After correlation screening, the retained features are analyzed using the SHapley Additive exPlanation (SHAP) method [43], which is capable of reflecting the contribution of each feature to the prediction of the model. The global effect of the features is determined and the features are ordered from highest to lowest importance to the prediction. Subsequently, the sequential forward selection method is employed to choose the optimum set of features that can describe the data within a given machine learning model. This approach is a bottom-up search process in which features are sequentially added to an empty candidate set until the addition of additional features results in the highest model accuracy [44]. In this study, the input features are gradually added to the regression model following the order of feature importance provided by the SHAP approach. The critical number of features is determined when there is a negligible improvement of R2 or no decrease of RMSE. Based on the above selection process, the essential features that significantly affect the creep life are acquired and the importance of each feature is determined. Finally, these selected features are utilized to establish the machine learning models for creep rupture life prediction.

2.3. Model Training and Evaluation

Several machine learning algorithms (i.e., Lasso, Ridge, SVR, GPR, DT, and RF) are used for model training. Lasso and Ridge regressions are types of linear regression, while SVR and GPR are capable of solving both linear and non-linear problems. DT is a non-parametric supervised learning method that is easy to interpret and understand, and requires less effort for data preprocessing. On the other hand, RF is an ensemble learning method that creates a number of decision trees at training. Compared to DT, RF has a more complex structure, but it prevents the overfitting issue and has a higher prediction accuracy. The selection of these different machine learning algorithms aims to explore and validate their effectiveness for the creep rupture life prediction of modified 9Cr-1Mo steels. More description about the selected machine learning models and hyperparameters optimization can be found in Appendix A.
The optimization of the hyperparameters of each regression model is conducted to acquire the parameters that exhibit the highest R2 and lowest RMSE. After the hyperparameters are optimized, the k-fold cross-validation (k = 5 in this study) is performed on the training dataset. That is to say, the training dataset is split into k equal folds. The (k-1) folds are chosen as the training set to train the model, and the remaining fold is used for validation. This process is repeated k times and the error is calculated. In this work, two error indicators such as R2 and RMSE are calculated to evaluate the trained model. These error terms can be obtained by the following equations:
R 2 = 1 − ∑ i = 1 n ( y i − y ^ i ) 2 ∑ i = 1 n ( y i − y ¯ ) 2
R M S E = ∑ i = 1 n ( y i − y ^ i ) 2 n
where n is the total number of data samples, y ^ i represents the predicted value, y i represents the actual value, and y ¯ is the mean of the observed data y . The higher R2 represents a better model fitting, while the lower RMSE indicates a higher model accuracy. By comparing the error indicators among the built models, the most appropriate machine learning algorithm that shows the lowest error is determined for the final prediction of creep rupture life. In addition, the accuracy of life prediction results is also assessed using the two-factor bands.

3. Results and Discussion

3.1. Feature Screening Results

Figure 3 displays the heat map of the Pearson correlation coefficient of all variables in the current dataset. Yellow indicates the positive correlation between features, while blue represents the negative correlation. It is clear that there is a significant nonlinear link between features because the correlation coefficient between any two features is always less than 0.95. This indicates that all the features in the current dataset can be considered independent variables. Consequently, it is reasonable and necessary to examine the effect and importance of each feature on creep rupture life.
Using a game-theoretic method known as the SHAP value, the individual contribution of each feature to the prediction of the model is revealed. The global effects of the features are shown in a bar plot, as can be seen in Figure 4a. The features are ordered from highest to lowest importance according to the absolute mean SHAP value. On the other hand, the local feature contribution for each sample is depicted by the beeswarm plot, as shown in Figure 4b. Each data point in this plot corresponds to a single observation in the dataset, and it is colored either in red or blue according to the feature value. Positive SHAP values indicate that the creep life is increased because of the feature value, while negative SHAP values suggest a negative influence. As expected, the test stress is shown to be the most crucial feature. The feature value of the test stress is high when the SHAP value is smaller than 0. This indicates that the stress with a high value corresponds to a lower predicted creep rupture life, whereas higher creep life can be obtained by applying lower stress. The second important feature is the test temperature, followed by the tempering time and S content. Higher values of these features also tend to negatively contribute to the creep rupture life. These key features also exhibit strong Pearson correlation coefficient values with creep life, as shown in Figure 3, indicating the compatibility of SHAP and correlation analysis. In addition, other features including the material form have a negligible contribution to the creep life because the SHAP values are distributed near 0, whether their values are high or low.
Figure 5 shows the results of sequential forward selection with the aid of GPR and SVR models. As the selection process proceeds, the GPR model achieves the highest R2 and lowest RMSE when there are five features. The error values also remain constant beyond a five-feature SVR model, as shown in Figure 5b. These facts show that selecting more than five features has a negligible impact on the predicted creep rupture life. Additionally, it can be seen in Figure 5b that there is a clear rise in error when the SVR model is trained using more than eleven features, demonstrating that the addition of redundant features has a detrimental effect on modeling accuracy. Therefore, feature screening must be conducted before a machine learning model is trained and validated in order to improve the robustness of the model.
The test stress, test temperature, tempering time, and the contents of S and Cr are determined as the five crucial features in order of the contribution of each feature to the model. According to Figure 4a, the importance of test stress and test temperature is about five times greater than that of tempering time, which is ranked third. This shows that environmental factors should be given far more attention in creep-oriented alloy design because they are substantially more significant than alloy constituents and processing conditions. The tempering time is the third most important feature, and a higher value has a negative impact on creep rupture life, as shown in Figure 4b. The long-term tempering of a martensitic 9Cr steel, according to Abe [45], caused a higher minimum creep rate and a shorter creep rupture life than the conventional quenching and tempering of the same steel, which is consistent with the present results. The composition features, namely the S and Cr content, are also two essential features that greatly influence the creep property. The role of S in capturing the inclusion information has been reported in earlier studies [46,47] which demonstrate that inclusion particles are the preferred sites for the nucleation of creep damage. This study also concludes that the inclusion is important for the accurate estimation of creep rupture life since a higher S content has a detrimental effect on creep property. The Cr content is shown to be another key element feature, and higher Cr content tends to negatively affect the creep rupture life. It is well known that Cr content is required to attain adequate oxidation resistance in high-temperature steam. However, it has been shown that a high content of Cr is not preferred for long-term stability in 9–12% Cr steels, because a large amount of M23C6 carbides would coarsen during service [20,48]. The current results of feature importance analysis are consistent with earlier theoretical studies, suggesting that the dataset employed in this work is extremely trustworthy and explicable. The above findings can also provide valuable insights and guidance for future creep-oriented alloy design.
It is also important to note that in addition to these five selected features, other features including alloy contents (e.g., Mo, V, Nb, and Ni) and processing parameters (e.g., normalizing temperature and tempering temperature) are insignificant for the model prediction due to their low variance in the current dataset. The variation in material form (e.g., tube, plate, and pipe) also has a negligible impact on creep rupture life according to both the results of the Pearson correlation coefficient and SHAP values. This is probably because the manufacturing process and microstructure vary from form to form, and consequently the differences in material form are characterized by alloy contents and heat treatment parameters. The use of insufficient data in this work is another possible cause. Most creep data are collected from the modified 9Cr-1Mo steel in the form of a tube, while limited data are obtained from the pipe. Building a larger, high-quality database is urgently required, because both the quantity and quality of data have a significant impact on the accuracy of machine learning models [49]. Although the material form and other features are demonstrated in this study to be insignificant for creep life prediction, more work still has to be done to make use of as many features as possible and investigate their effects on model performance. For example, it is possible to consider the material’s grain size as a crucial input feature in future machine learning-based predictions of creep rupture life. Previous works have highlighted that during high-temperature damage, inhomogeneous deformation tends to occur at the grain boundary and triple junctions where creep voids and cracks are most likely to be initiated [2,3,50]. Numerous studies have demonstrated that the grain size affects the creep strain rate differently in the high and low applied stress regimes [51,52]. Therefore, future research can take grain size into account together with alloy contents and other material characteristics for improving the model’s performance.

3.2. Model Prediction Results

After the feature screening process, the five essential features are identified and used for training various machine learning models. Figure 6 shows the R2 and RMSE of different machine learning strategies. Among the six algorithms, the GPR model with a Matérn kernel function shows the best prediction accuracy because it displays the lowest RMSE and the highest R2, indicating that the GPR model is superior in capturing the complex nonlinear relationship between input features and creep rupture life. Although the SVR model with a polynomial kernel function shows a comparable R2 with GPR, its RMSE is slightly smaller than that of the GPR model. For other strategies, it is not unexpected that the Lasso and Ridge models perform worse than GPR and SVR models in terms of accuracy, because they are essentially linear regression models and cannot capture the inherent nonlinearity in the connections between creep life and input parameters. In addition, the DT model shows a relatively high R2 for the training set, while the value for the testing set remains the lowest. This indicates that the DT model is prone to overfitting, which has been also reported by previous studies [13,23]. The RF model, an ensemble learning technique that functions by building a collection of decision trees during training, can reduce this problem. Consequently, the accuracy of the RF model is much higher than that of the DT model and comparable with that of the GPR and SVR models, as shown in Figure 6. In summary, based on the above results, the GPR model is selected as the optimum approach for the creep rupture life prediction of modified 9Cr-1Mo steel in this investigation.
Figure 7 displays the parity plots comparing the predicted creep rupture life to the actual values of modified 9Cr-1Mo steel using the GPR model with five key input features. The left plot shows the regression values obtained from both training and testing datasets. The prediction is more accurate as the data points approach the 45° line, as shown in Figure 7a. The RMSE values for the training and testing datasets are exceptionally low at 0.102 and 0.127, respectively. The comparison between the predicted creep life and actual creep life is displayed in Figure 7b, in which all data points are colored by the test stress at which the creep rupture experiments were carried out. Moreover, the second important feature, namely the test temperature, is quantified by the symbol size. Over the whole applied stress and temperature ranges, it is demonstrated that the GPR model offers excellent quantitative agreement. Furthermore, almost all the predicted data are distributed in the two-factor band line. It is also worth noting that the prediction accuracy and stability for long-term creep data (>104 h), which is the primary focus in structural integrity assessment and creep-oriented new alloy design, are both extremely satisfactory. These findings demonstrate that the GPR model provides high accuracy and good adaptability in predicting creep life under various environmental conditions.
The prediction results of creep rupture life using the SVR model are shown in Figure 8 for comparison since the R2 of the SVR model is very close to that of the GPR model. Similarly, almost all creep data lie in the two-factor band line, as shown in Figure 8b, indicating adequate accuracy of the SVR model. However, compared to the predictions of the GPR model, an increasing proportion of data points are dispersed away from the 45° line as displayed in Figure 8a, leading to lower RMSE values for the SVR model. In summary, although almost all creep data points can be evaluated within the two-factor band lines by developing the GPR and SVR models with five crucial features, the constructed GPR model exhibits a better prediction performance and can be deemed applicable for estimating the creep rupture life of 9Cr-1Mo steels exposed to a variety of stress and temperature conditions.

4. Validation Testing

The validation testing was carried out utilizing additional data acquired from the published literature [53,54,55] to further demonstrate the efficacy of the constructed GPR model with five screened important features for dataset out of model training and testing. A total of 21 creep rupture life data for ASME Grade 91 steel are included in the validation testing dataset. The five most important features screened in this study, namely the test stress, test temperature, tempering time, and the contents of S and Cr, were collected as input variables for the created creep rupture life prediction model. The prediction results using the validation testing dataset are displayed in Figure 9. The majority of the data points are close to the 45° line and are distributed inside the two-factor band lines, indicating adequate prediction accuracy. However, it is important to note that the RMSE of the validation dataset is estimated to be 0.279, which is much higher than that of the testing dataset (0.127). This is due to the fact that five data points fall outside the two-factor band lines, as shown in Figure 9b. The increased error in the validation dataset could be due to the following reasons. First, due to the variations in the test equipment, test procedure, data collection and post-processing methods, and specimen geometry between different data sources [56], the experimental creep data of 9Cr-1Mo steels that are currently accessible vary greatly from source to source. Second, different creep specimens made of the same material may have different initial microstructures and initial internal stress states, which result in variances in the overall creep responses [57]. As a result, different rupture life data may be obtained from the creep tests carried out by the same testing apparatus with materials supplied by the same manufacturer. For instance, under the same 700 °C/60 MPa testing conditions, the NIMS database [36] shows that the creep rupture life of modified 9Cr-1Mo steel varies from 364.6 to 447.5 h, exhibiting approximately 20% inaccuracy. These possible reasons may cause a higher error in machine learning-based creep life prediction using the validation testing dataset. However, given that the majority of data points fall inside the two-factor band lines, it can be stated that the GPR model with five crucial features that are screened is effective in accurately predicting the creep rupture life of modified 9Cr-1Mo steel.

5. Conclusions

In summary, a machine learning-based computational framework has been developed for the quick screening of crucial features and accurate prediction of the creep rupture life of modified 9Cr-1Mo steels. First, a feature screening protocol based on correlation filtering and sequential feature selection techniques was developed for identifying critical features from a set of numerous descriptors, including material characteristics and environmental factors. The results show that the test stress, test temperature, tempering time, and the contents of S and Cr are identified as the five crucial features that greatly influence the creep life prediction of modified 9Cr-1Mo steels. The contributions of these features to creep rupture life are discussed and connected to the existing theoretical studies. Subsequently, various machine learning algorithms including Lasso regression, Ridge regression, support vector regression (SVR), Gaussian process regression (GPR), decision tree (DT), and random forest (RF) are employed for model training to explore their capability to map the intricate nonlinear interactions between multivariate variables and creep rupture life. Among the six algorithms, the GPR model with the five selected crucial features exhibits the highest prediction accuracy. Furthermore, the validation testing is performed using additional data out of model training and testing to further validate the efficacy of the constructed GPR model. The validated results show that most creep data of modified 9Cr-1Mo steels fall inside the two-factor band lines. Therefore, one can conclude that the developed GPR model with five crucial features can offer high accuracy and excellent adaptability in predicting creep life under various environmental conditions.
The work described here creates a foundation for addressing the enormous challenge of the efficient and precise estimation of the creep rupture life of components at high temperatures. Though the effect of some material characteristics (e.g., the material form and the contents of C, Mo, N) on prediction accuracy is negligible in this work, the generative feature screening method offers an efficient way to identify the critical features and minimize the model complexity, thus allowing a significant number of candidates that may influence the creep properties to be identified without having to experimentally analyze them in the lab. As more creep data are provided to the dataset and additional features such as the inclusions and hardness are integrated, there is a growing opportunity for identifying robust features and improving the life prediction performance of high-temperature components.

Author Contributions

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

Funding

This study was funded by the National Natural Science Foundation of China (No. 22208258), China Postdoctoral Science Foundation (No. 2021M692532), the Fundamental Research Funds for the Central Universities (No. xzy012021044), and the SAMR project (No. 2020MK182).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Machine Learning Models and Hyperparameters Optimization

Appendix A.1. Lasso and Ridge Regression

As a fundamental machine learning algorithm, linear regression tries to find the following function to make predictions by linearly integrating various variables.
y ^ i = X β
where y ^ i represents the predicted value, X =   ( x 1 , x 2 ,   ⋯ ,   x m ) are selected feature values from the dataset, β =   ( β 0 , β 1 ,   ⋯ ,   β m ) are fitted parameters, and m is the number of features. The loss function can be expressed as follows:
L L i n e a r ( β ) = ∑ i = 1 n ( y i − ∑ j = 0 p β j x i j ) 2
A Lasso regression is a type of regularization, which adds the absolute value sum of fitted parameters to the loss function, as shown in Equation (A3):
L L a s s o ( β ) = ∑ i = 1 n ( y i − ∑ j = 0 p β j x i j ) 2 + λ 1 ∑ j = 1 p | β j |
Similarly, Ridge also adds a regularization term (e.g., the sum of squares of parameters) to the loss function, which can be expressed as:
L R i d g e ( β ) = ∑ i = 1 n ( y i − ∑ j = 0 p β j x i j ) 2 + λ 2 ∑ j = 1 p β j 2
where n is the total number of data samples, and λ 1 and λ 2 are the regularization parameters in Lasso and Ridge regressions, respectively, which are two important hyperparameters that need to be optimized during modeling. In this study, the optimized λ 1 and λ 2 are 0.0043 and 0.24, respectively.

Appendix A.2. Support Vector Regression

Support vector regression (SVR) is a widely used regression algorithm that works on the principle of the support vector machine (SVM). While Lasso and Ridge regressions only support linear regressions, SVR allows for both linear and non-linear regressions. The main concept of SVR is to create a hyperplane with the smallest possible distance between it and the sample points.
The main hyperparameter of SVR is the kernel, which converts the observed data into a higher-dimensional dataset so that it can linearly separate the high-dimensional space. There are some widely used standard kernel functions including the linear kernel, the polynomial kernel, and the Gaussian kernel. The selection of the kernel function has a great impact on the performance of the regression algorithm. In this work, the polynomial kernel with a d of three is determined to be the most optimal kernel function through tuning, which can be expressed as:
k ( x i , x j ) = ( x i T x j ) d
where x i is the feature in the original dataset and x j is the feature after its transformation into a higher dimensional space, d is the integer parameter and d ≥ 1.

Appendix A.3. Gaussian Process Regression

The Gaussian process regression (GPR) model is a nonparametric kernel-based probabilistic machine learning method. It is capable of making predictions that take into account prior knowledge (kernels) and offering uncertainty measures for predictions. The selection of the kernel function has a significant impact on how well the Gaussian process performs. There are several standard kernel functions such as the squared exponential kernel, rational quadratic kernel, Matérn kernel, and exponential kernel. In the present research, the most optimal kernel function is found to be the Matérn kernel function (as shown in Equation (A6)) with a ν of 5/2 after hyperparameter tuning.
k ( x , x ′ ) = 2 1 − ν Γ ( ν ) ( 2 ν ‖ x − x ′ ‖ l ) ν K ν ( 2 ν ‖ x − x ′ ‖ l )
where l is the length scale parameter, ν is the smooth factor, Γ is the Gamma function, and Γ ( ν ) is the Bessel function.

Appendix A.4. Decision Tree and Random Forest

A decision tree (DT) is a non-parametric supervised learning method that constructs the regression or classification models in the shape of a tree-like structure. A decision tree consists of three types of nodes (e.g., the root node, internal nodes, and leaf nodes) and branches. The minimum leaf size is an important hyperparameter that needs fine-tuning to obtain the optimal DT model. In this study, the optimal minimum leaf size is determined to be one.
On the other hand, a random forest (RF) is made up of multiple decision trees (DTs). It is an ensemble learning algorithm that allows for both regression and classification problems. The fundamental principle of a RF is that it creates multiple DTs during training, and the final predicted result is obtained as the average prediction value of the individual DTs. The major advantage of an RF over a DT is that it shows improved resistance to overfitting because there is a robust number of DTs in an RF. In this work, the RF algorithm has two important hyperparameters (e.g., the number of trees and the maximum depth of the tree) which need to be tuned before training. After hyperparameter optimization, the optimal number of trees and the optimal maximum depth of the tree are obtained as 300 and 15, respectively.

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Figure 1. Machine learning-based framework for creep rupture life prediction.
Figure 1. Machine learning-based framework for creep rupture life prediction.
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Figure 2. Instance of creep life distribution of modified 9Cr-1Mo steel with different forms at 600 °C.
Figure 2. Instance of creep life distribution of modified 9Cr-1Mo steel with different forms at 600 °C.
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Figure 3. Heat map of the Pearson correlation coefficient.
Figure 3. Heat map of the Pearson correlation coefficient.
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Figure 4. Results of SHAP value: (a) Importance of input features, and (b) summary plot of SHAP value.
Figure 4. Results of SHAP value: (a) Importance of input features, and (b) summary plot of SHAP value.
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Figure 5. Variations in R2 and RMSE as a function of the number of input features. (a) GPR model and (b) SVR model.
Figure 5. Variations in R2 and RMSE as a function of the number of input features. (a) GPR model and (b) SVR model.
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Figure 6. Comparison of accuracy for different machine learning models: (a) RMSE and (b) R2.
Figure 6. Comparison of accuracy for different machine learning models: (a) RMSE and (b) R2.
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Figure 7. Prediction results of creep rupture life by GPR model: (a) Training and prediction results of creep life in log scale; (b) Predicted creep life vs. measured creep life.
Figure 7. Prediction results of creep rupture life by GPR model: (a) Training and prediction results of creep life in log scale; (b) Predicted creep life vs. measured creep life.
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Figure 8. Prediction results of creep rupture life by SVR model: (a) Training and prediction results of creep life in log scale; (b) Predicted creep life vs. measured creep life.
Figure 8. Prediction results of creep rupture life by SVR model: (a) Training and prediction results of creep life in log scale; (b) Predicted creep life vs. measured creep life.
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Figure 9. Prediction results of creep rupture life by GPR model with the validation dataset: (a) Prediction results of creep life from various literature in log scale; (b) Predicted creep life vs. measured creep life.
Figure 9. Prediction results of creep rupture life by GPR model with the validation dataset: (a) Prediction results of creep life from various literature in log scale; (b) Predicted creep life vs. measured creep life.
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Table
Table 1. Summary of the statistical description of collected information.
Table 1. Summary of the statistical description of collected information.
DataAbbreviationDescriptionMinimumMaximumMeanStandard Deviation
InputsCCarbon/wt.%0.0800.1100.0960.010
MnManganese/wt.%0.3500.5600.4320.049
PPhosphorus/wt.%0.0050.0210.0110.005
SSulfur/wt.%0.0010.0090.0020.002
NiNickel/wt.%0.0400.4600.1410.092
CrChromium/wt.%8.3109.0008.5300.175
MoMolybdenum/wt.%0.8600.9900.9180.032
NNitrogen/wt.%0.0300.0840.0510.010
AlAluminum/wt.%0.0010.0230.0100.007
VVanadium/wt.%0.1800.2300.2070.015
NbNiobium/wt.%0.0600.0900.0750.006
N TempNormalizing Temperature/°C104010651051.6245.859
N TimeNormalizing Time/min109033.27126.986
T TempTempering Temperature/°C750790771.9499.984
T TimeTempering Time/min3012055.91617.941
FormMaterial Form131.4730.653
Test StressTest Stress/MPa30450162.01891.278
Test TempTest Temperature/°C450725589.25861.697
Outputslg(life)Logarithm of creep life−1.7065.0913.5360.954
Creep lifeCreep life/h0.02123,442.114,613.521,862.8
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Chai, M.; He, Y.; Li, Y.; Song, Y.; Zhang, Z.; Duan, Q. Machine Learning-Based Framework for Predicting Creep Rupture Life of Modified 9Cr-1Mo Steel. Appl. Sci. 2023, 13, 4972. https://doi.org/10.3390/app13084972

AMA Style

Chai M, He Y, Li Y, Song Y, Zhang Z, Duan Q. Machine Learning-Based Framework for Predicting Creep Rupture Life of Modified 9Cr-1Mo Steel. Applied Sciences. 2023; 13(8):4972. https://doi.org/10.3390/app13084972

Chicago/Turabian Style

Chai, Mengyu, Yuhang He, Yongquan Li, Yan Song, Zaoxiao Zhang, and Quan Duan. 2023. "Machine Learning-Based Framework for Predicting Creep Rupture Life of Modified 9Cr-1Mo Steel" Applied Sciences 13, no. 8: 4972. https://doi.org/10.3390/app13084972

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