Domain knowledge-guided interpretive machine learning: formula discovery for the oxidation behavior of ferritic-martensitic steels in supercritical water

Dataset

A total of 184 oxidation data of FM steels in SCW are collected from the literature and given at the online Supplementary Information. Every datum in the FM steel oxidation (FMO) dataset includes oxidation caused weight gain in units of mg/dm$$ ^{2} $$ , testing conditions, exposure time and steel chemical compositions, with a total of four testing features and 11 alloying element features (the balance element Fe is not accounted for), as shown in Table 1.

Xgboost and SHAP values

Xgboost ^[40] is a powerful tree-based boosting ensemble algorithm. The present work employs the Xgboost algorithm to regress the oxidation data of FM steels in SCW, and the values of hyperparameters involved are optimized by cross-validation and/or a grid search in the hyperparameter space with the open python library scikit-learn ^[43]. Table S1 at Section 2.1 of the Supplementary Information lists all optimized values of the hyperparameters.

The SHAP algorithm is developed based on game theory ^[25]. A SHAP value, whether positive or negative, reflects the contribution of a feature to a predicted response in one datum, and the predicted response is given by an ML model. In the present work, SHAP values are calculated with Xgboost models. If there are $$ m $$ features of $$ (x_{1}, \cdots, x_{m}) $$ and one predicted response $$ \hat{y} = f(x_{1}, \cdots, x_{m}) $$ , the predicted response can be expressed by the SHAP values in an additive manner as

where $$ \phi_{0} $$ denotes the SHAP value of the tree root in the tree-based Xgboost algorithm and $$ \phi_{j} $$ is the SHAP value of feature $$ x_{j} (j = 1, \cdots, m) $$ . The SHAP value $$ \phi_{j} $$ can be decomposed into

where the SHAP value $$ \phi_{j, j} $$ denotes the pure contribution of feature $$ x_{j} $$ itself and $$ \phi_{j, k} $$ is known as the interaction between features $$ x_{j} $$ and $$ x_{k} $$ , and accordingly, $$ \phi_{j, j} $$ and $$ \phi_{j, k} $$ are deemed the pure and interaction SHAP values, respectively. Equation (1b) indicates that the contribution from feature $$ x_{j} $$ to the predicted response is the sum of its pure contribution plus its joint contributions with other features. Therefore, a comprehensive study of SHAP values will provide insights into the mechanisms of the oxidation of FM steels in SCW.

Integration of SHAP values with domain knowledge

The game theory-based SHAP value is an additive feature attribution method, where the output is a sum of contributions of each input feature ^[44]. If the contributions of variables to a function are not additive in the original variable space, but additive in a mapped space, the SHAP value will be calculated in the mapped space. For the oxidation of FM steels in SCW, there are 15 features and each feature contributes to the oxidation weight gain $$ \Delta w = f(x_{1}, \cdots, x_{15}) $$ , where $$ x_{j}(j = 1, \cdots, 15) $$ represent the 15 features.

It might be inaccurate to calculate the reasonable SHAP values in the original space. Based on the domain knowledge of oxidation, we take a dimensionless Arrhenius equation of

Feature ranking, selection and data screening

An Xgboost model is first developed with all features via ten-fold cross-validation (10-CV) and is used to calculate all SHAP values $$ \phi_{ij} $$ , where subscripts "$$ i $$ " and "$$ j $$ " denote sample $$ i (i = 1, \cdots, n) $$ and feature $$ x_{j} (j = 1, \cdots, m) $$ , respectively. The mean $$ \overline{| \phi_{j} |} $$ of the absolute SHAP values for each feature $$ x_{j} (j = 1, \cdots, m) $$ is then calculated from

where $$ n = 184 $$ and $$ m = 15 $$ are the numbers of data and features, respectively. The feature importance $$ \mathfrak{R}_{\phi_{j}} $$ indicates that the higher the mean of the absolute SHAP values of a feature, the greater the contribution of the feature to the response will be.

The data screening is carried out by evaluating the errors $$ \varepsilon_{i} $$ between the response $$ y_{i} $$ and the predicted response $$ \hat{y}_{i} $$ , viz., $$ \varepsilon_{i} = y_{i} - \hat{y}_{i} (i = 1, \cdots, n) $$ . The errors should follow a normal distribution $$ \varepsilon\sim N(0, \sigma) $$ . If the errors exhibit a nonzero mean, the prediction from an ML model deviates systematically from the real values. In this case, the ML model should be improved. If the errors show a normal distribution with a zero mean and few errors of data located far away from the mean, the $$ 3\sigma $$ criterion is usually taken as the threshold to gauge whether a datum is an outlier. Caution must be used in handling outliers. Repeated tests are strongly suggested to verify the reliability of the outliers and reliable outliers imply a new mechanism.

Feature selection and data screening

A total of 184 data on FM steel oxidation in SCW are collected from the literature and provided in the Supplementary Information. Fifteen features are employed here and categorized into two groups, namely, alloying elements and testing conditions. The feature analysis is carried out within each of the groups. The SHAP values $$ \phi_{ij} (i = 1, \cdots, 184) (j = 1, \cdots, 15) $$ of all features and errors $$ \varepsilon_{i} = y_{i} - \hat{y}_{i} (i = 1, \cdots, 184) $$ are calculated with the Xgboost model evaluated by 10-CV. Figure 2(A)-(D) shows the SHAP values of the four testing condition features, while Figure 2(E)-(I) illustrates the SHAP values of the five alloying elements of V, Si, Cr, Ni and Mn. The SHAP values of the other six alloying elements are shown in Figure S1 in Section 2.2 of the Supplementary Information, where the blue points are calculated SHAP values and the red dashed lines indicate the SHAP value of $$ \phi_{0} = 5.45 $$ , which serves as a reference baseline through the origin of the vertical SHAP value axis.

Figure 2. SHAP analysis of features. A-D: SHAP values of testing conditions, i.e., temperature, time, DOC and pressure. E-I: SHAP values of alloy compositions, i.e., V, Si, Cr, Ni and Mn. J: Feature importance ranking by $$ \mathfrak{R}_{\phi_{j}} $$ . K: Fitting effect of Xgboost model with selected eight features for 10-CV.

As expected, Figure 2A shows that the lower the value of $$ \frac{1}{RT} $$ , the higher the SHAP value of $$ \phi_{\frac{1}{RT}} $$ , meaning that in general, the oxidation of FM steels is severer at higher temperatures. In contrast, Figure 2A also shows that at a given temperature, the SHAP value of $$ \phi_{\frac{1}{RT}} $$ varies greatly, e.g., the SHAP value of $$ \phi_{\frac{1}{RT}} $$ varies from 0.31 to 1.22 at $$ \frac{1}{RT} = $$ 0.13029, from 0.93 to 1.22 at $$ \frac{1}{RT} = $$ 0.13775 and from $$ - $$ 2.04 to $$ - $$ 1.52 at $$ \frac{1}{RT} = $$ 0.17868. The variation in $$ \phi_{\frac{1}{RT}} $$ at a certain temperature is caused by other features, which will be analyzed later. The SHAP value of $$ \phi_{\ln \left(\frac{t}{t_{0}}\right)} $$ also exhibits the anticipated trend, as illustratedin Figure 2B, where the larger the value of $$ \ln \left(\frac{t}{t_{0}}\right) $$ , the higher the SHAP value of $$ \phi_{\ln \left(\frac{t}{t_{0}}\right)} $$ . This result is logical because a longer exposure time increases the oxidation weight gain. Furthermore, the variation in $$ \phi_{\ln \left(\frac{t}{t_{0}}\right)} $$ at a given value of $$ \ln \left(\frac{t}{t_{0}}\right) $$ is induced by other features and will be studied later.

The SHAP value of $$ \phi_{{\rm{DOC}}} $$ behaves interestingly, as shown in Figure 2C, where at the deaerated condition (DOC $$ < $$ 10 ppb), the SHAP value of $$ \phi_{{\rm{DOC}}} $$ changes greatly from $$ - $$ 0.84 to 0.17. Once DOC $$ > $$ 10 ppb, the higher the DOC, the larger the SHAP value of $$ \phi_{{\rm{DOC}}} $$ . The shape of the SHAP values of $$ \phi_{{\rm{DOC}}} $$ suggests a logarithm function of $$ g \left(\frac{({\rm{DOC}} + r)}{({\rm{DCO}})_{0}}\right) $$ . The result is rational in thermodynamics and kinetics because a higher DOC results in a stronger driving force for oxidation. Figure 2D indicates that the magnitude of the SHAP values of $$ \phi_{{\rm{pressure}}} $$ is one or two orders smaller in comparison with $$ \phi_{\frac{1}{RT}} $$ , $$ \phi_{\ln \left(\frac{t}{t_{0}}\right)} $$ and $$ \phi_{{\rm{DOC}}} $$ , ranging from $$ - $$ 0.013 to 0.009 except for one point at $$ - $$ 0.03 with a pressure of 30 MPa. This indicates that between the pressure range of 20.7–31 MPa, the pressure of SCW has a negligible contribution to the oxidation of FM steels in SCW.

It is somewhat surprising that vanadium plays the most important role among the studied 11 alloying elements in the oxidation of FM steels in SCW, as shown in Figure 2E, where the SHAP value of $$ \phi_{\rm V} $$ statistically decreases from 0.22 to $$ - $$ 0.15 as the V content increases from 0 to 0.3 wt.%. When the V content is higher than 0.19 wt.%, the SHAP values of $$ \phi_{\rm V} $$ are all negative regardless of the great variation, thereby indicating that the V content should be higher than the critical value in anti-oxidation FM steels in SCW. The performance of silicon in the oxidation of FM steels in SCW is similar to that of vanadium. Figure 2F shows that the critical Si concentration is 0.38 wt.%, beyond which the SHAP values of $$ \phi_{{\rm{Si}}} $$ are all negative. However, for 0.5 wt.% Si, a few SHAP values of $$ \phi_{{\rm{Si}}} $$ appear as outliers, which will be analyzed later. Although chromium is the basic element of FM steel against oxidation, its anti-oxidation performance is unsatisfactory with a SHAP value of $$ \phi_{{\rm{Cr}}} > 0 $$ if its content is less than 9.5 wt.%, as shown in Figure 2G. Nickel has high corrosion resistance to acids and alkalis at higher temperatures ^[45] and thus alloying Ni with a content higher than 0.13 wt.% into FM steels enhances the oxidation resistance of FM steels in SCW, as shown in Figure 2H. Manganese might be detrimental to the oxidation of FM steels in SCW when its content is high. Figure 2I indicates that the SHAP value of $$ \phi_{{\rm{Mn}}} > 0 $$ if the Mn concentration is higher than 0.9 wt.%.

The feature importance defined in the SHAP method (see Methods), $$ \mathfrak{R}_{\phi_{j}} $$ , is calculated for all 15 features and plotted in Figure 2J. The three features of the testing conditions, $$ \frac{1}{RT} $$ , $$ \ln \left(\frac{t}{t_{0}}\right) $$ and DOC, rank as the top three in the feature importance, indicating that the oxidation environment and time play the most important roles in the oxidation of FM steels in SCW. In contrast, the influence of SCW pressure in the range of 20.7–31 MPa on the oxidation of FM steels is trivial. In general, the feature importance of alloying elements is lower than the three features of the testing conditions, as shown in Figure 2J.

The feature selection is then conducted by the sequential backward selector wrapped with Xgboost and 10-CV, which yields the three testing features of $$ \frac{1}{RT} $$ , $$ \ln \left(\frac{t}{t_{0}}\right) $$ and DOC, and five alloying elements of V, Si, Ni, Cr and Mn, i.e., the top eight in the feature importance order. It seems that the elements of V, Si and Ni play more important roles than Cr based on the feature importance ranking. As described later in the pure SHAP value analysis, Cr is the major element when its concentration is higher than 11 wt.% against FM steel oxidation in SCW. Furthermore, the errors $$ \varepsilon_{i} $$ between the response $$ y_{i} $$ and the predicted response $$ \hat{y}_{i} $$ , i.e., $$ \varepsilon_{i} = y_{i} - \hat{y}_{i} (i = 1, \cdots, n) $$ , are calculated for the original 184 data and follow a normal distribution with a zero mean and standard deviation of $$ \sigma = 0.35 $$ , as shown in Figure S2 in Section 2.3 of the Supplementary Information. However, there are six errors located outside of $$ \pm 3\sigma $$ of $$ \varepsilon_{i} $$ , meaning that the associated six data are outside the 99.7% confidence interval. These six data are removed from the following analysis and highlighted in the FMO dataset, and hereafter, only the surviving 178 data are used to explore analytic formulas for the oxidation of FM steels in SCW (the details regarding the outliers are given in Section 2.4 of the Supplementary Information). With the selected eight features, the Xgboost model predicts the values of $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ for the 178 data. The predicted values are plotted against the real values in Figure 2(K), showing an excellent prediction with a Pearson correlation value of $$ \rho = $$ 0.988, representing a slight enhancement from the original $$ \rho = $$ 0.986 with all features.

Pure SHAP values and oxidation Cr equivalent

The SHAP value of each feature is decomposed into its pure SHAP value and the interaction SHAP values, as stated in Eq. (1b). Figure 3A-H shows the pure SHAP values of the selected eight features and there are 178 pure SHAP values in each figure. A comparison of Figure 3A-H to the corresponding Figure 2A-C and Figure E-I indicates that for a certain feature value, the pure SHAP value scattering is much smaller than the SHAP values. This is an expected result because a pure SHAP value eliminates all interaction SHAP values from its parent SHAP value. If pure SHAP values are calculated from a perfect model of a single function of variables, the pure SHAP value of a feature will correspond to the feature value one-to-one, i.e., for one feature value, there is only one pure SHAP value.

Figure 3. Pure SHAP value analysis of important features. A-C: Pure SHAP values of testing conditions, i.e., temperature, time and DOC. D-H: Pure SHAP values of alloy compositions, i.e., V, Ni, Cr, Si and Mn.

Figure 3A shows almost ideal pure SHAP values of feature $$ \frac{1}{RT} $$ . There are [0.90, 0.93], [0.83, 0.93], [1.06, 1.11], [0.12, 0.18], [$$ - $$ 0.12, $$ - $$ 0.09], [$$ - $$ 0.40, $$ - $$ 0.38] and [$$ - $$ 1.84, $$ - $$ 1.76] pure SHAP values at $$ \frac{1}{RT} = $$ 0.123598, 0.130292, 0.137753, 0.14612, 0.15557, 0.166327 and 0.178681, respectively, and the pure SHAP values at a given value of $$ \frac{1}{RT} $$ are almost the same such that the data are condensed to one point, as shown in Figure 3A. The near perfect result implies that the oxidation of FM steels in SCW is indeed a thermally active process and the pure SHAP value of feature $$ \frac{1}{RT} $$ catches the oxidation mechanism. Figure 3B illustrates the pure SHAP values of feature $$ \ln \left(\frac{t}{t_{0}}\right) $$ , where there are 36 values of $$ \ln \left(\frac{t}{t_{0}}\right) $$ and the maximum of the pure SHAP values is 0.41 at $$ \ln \left(\frac{t}{t_{0}}\right) = 7.600902 $$ .

There are a few reasons causing multiple pure SHAP values at a given feature. The first reason might be experimental errors, which measure the degree of the experimental scattering of repeated tests. The second reason might be attributed to the Xgboost model, which approximately estimates the response from the input feature data rather than a perfect function. The third reason might be the method used to calculate the SHAP values from a tree-based algorithm (Tree-Explainer model). Figure 3C shows that the pure SHAP values of the DOC feature can be expressed by a logarithm function of $$ 0.0682\ln \left(\frac{({\rm{DOC}} + 1)}{({\rm{DCO}})_{0}}\right) - 0.267 $$ , which is plotted by the dashed curve. The logarithm function of pure SHAP requires its argument to be larger than zero but beyond the restriction of a specific $$ r $$ value; hence $$ r = 1 $$ is arbitrarily assigned in the figure, and the real value of $$ r $$ for the oxidation in the deaerated SCW is determined in the following analysis. A similar trend is exhibited in the pure SHAP values of the element features. The outlier suspected points at 0.5 wt.% Si disappear in the pure SHAP values of the Si feature, thereby indicating they are caused by the interaction SHAP values. As expected, the critical values of 0.19 wt.% V, 0.38 wt.% Si, 9.5 wt.% Cr, 0.13 wt.% Ni and 0.9 wt.% Mn remain unchanged in the pure SHAP values. Figure 3D shows that the minimum pure SHAP value of V is $$ - $$ 0.122 at its highest concentration (0.3 wt.%) in the dataset, while Figure 3F indicates that the minimum pure SHAP value of Cr is $$ - $$ 0.186 at its highest concentration of 12 wt.%. The pure SHAP values based on the available data illustrate that Cr is still the basic alloying element for anti-oxidation FM steels in SCW.

From the pure SHAP value of an individual feature, we defined the joint SHAP value of two features as

which measures the joint contribution of two features. In general, $$ \phi_{jk} $$ is a function of features $$ x_{j} $$ and $$ x_{k} $$ . As described above, the present industrial practice still employs Cr as the basic alloying element in manufacturing anti-oxidation FM steels in SCW. Therefore, Cr is taken as the basic feature $$ x_{j} $$ and V, Si, Ni and Mn are treated as the partner features $$ x_{k} $$ in the calculation of the joint SHAP value. A linear function is proposed to approximately estimate the interaction function, i.e., $$ \phi_{jk}(x_{j}, x_{k}) \approx \phi_{jk0} + a_{j}^{0}x_{j} + a_{k}^{0}x_{k} $$ . The two coefficients $$ a_{j}^{0} $$ and $$ a_{k}^{0} $$ represent the contribution weights of features $$ x_{j} $$ and $$ x_{k} $$ , respectively, to their interaction function. Then letting $$ a_{j}^{0} \equiv 1 $$ introduces an equivalent coefficient $$ a_{k} = a_{k}^{0}/a_{j}^{0} $$ for feature $$ x_{k} $$ and named it $$ x_{j} $$ equivalent coefficient of $$ x_{k} $$ . With the approach, we calculate the Cr equivalent coefficient for each of V, Si, Ni and Mn and then sum up the four Cr equivalent coefficients to yield the oxidation Cr equivalent concentration, $$ [\widetilde{{\rm{Cr}}}] $$ ,

where $$ [ \cdot] $$ represents the real concentration in wt.% of an element. Figure 4A-D shows the joint SHAP values of two features, $$ \phi_{{\rm{CrSi}}} $$ , $$ \phi_{{\rm{CrMn}}} $$ , $$ \phi_{{\rm{CrNi}}} $$ and $$ \phi_{{\rm{CrV}}} $$ , respectively, and each of the joint SHAP values of two features contains 178 data. Using the linear approximation and the approach mentioned above, we can then obtain the oxidation Cr equivalent concentration

Figure 4. Joint SHAP value of two features and the derived oxidation Cr equivalent concentration. A-D: Joint SHAP value of two features, i.e., Cr with Si, Mn, Ni and V, respectively. E: Predicted values of Xgboost model versus experimental values with the transferred four features. F: Pure SHAP analysis of oxidation Cr equivalent concentration feature.

Hereafter, we use one feature of the oxidation Cr equivalent concentration to replace the five element features. Thus, the total number of features is reduced to four, one chemical composition feature and three testing condition features. With the four features, the Xgboost model is retrained with 10-CV. The predictions on the 178 data are plotted in Figure 4E, showing a perfect fitting with $$ \rho = 0.985 $$ . With the Xgboost model, we calculate the pure SHAP value of the oxidation Cr equivalent concentration, $$ \phi_{[\tilde{Cr}], [\tilde{Cr}]} $$ and plot it in Figure 4F. The pure SHAP value $$ \phi_{[\tilde{Cr}], [\tilde{Cr}]} $$ is almost linearly proportional to the oxidation Cr equivalent concentration $$ [ \widetilde{{\rm{Cr}}}] $$ with a correlation of $$ \rho = - 0.97 $$ . Figure 4F shows that when $$ [ \widetilde{{\rm{Cr}}}] > 19.29 $$ wt.%, the pure SHAP values $$ \phi_{[\tilde{Cr}], [\tilde{Cr}]} $$ are all negative. It is important to consider that the oxidation behavior of FM steels in SCW depends on both the composition and microstructure of the steels. The oxidation chromium equivalent concentration is based only on the chemical composition and therefore $$ [ \widetilde{{\rm{Cr}}}] $$ is a purely compositional representative, which will provide guidance in the design of anti-oxidation FM steels.

Activation energy and time exponents

The oxidation mechanism of FM steels in SCW is embodied in the activation energy and time exponent ^{[29, 46]}, which are the coefficients of $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus features $$ \frac{1}{RT} $$ and $$ \ln \left(\frac{t}{t_{0}}\right) $$ , respectively. Figure S3 in Section 2.5 of the Supplementary Information shows the plots of $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \frac{1}{RT} $$ and $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \ln \left(\frac{t}{t_{0}}\right) $$ , respectively, indicating multiple values of $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ for a given value of $$ \frac{1}{RT} $$ or $$ \ln \left(\frac{t}{t_{0}}\right) $$ due to the contributions of other features. To investigate the effects of other features on the activation energy and time exponent, a novel "divide-and-conquer" approach is developed based on a tree model and hence named TCLR. Maximizing the information gain in conventional tree classifiers determines the splitting feature and the associated splitting value. TCLR is developed based on that the response $$ y = f(x_{1}, \cdots, x_{m} ) $$ is a linear function of features $$ x_{\hat{j}} $$ $$ (\hat{j} = 1, \cdots, \hat{m} ) \; \hat{m} \leq m $$ , where features $$ x_{\hat{j}} $$ $$ (\hat{j} = 1, \cdots, \hat{m} ) $$ are called the linear-features. The linear-feature set is a subset of the feature set $$ (x_{1}, \cdots, x_{m} ) $$ . A TCLR tree is a binary classification tree, where only the linear relationship between the response and one linear-feature is classified by using all features $$ (x_{1}, \cdots, x_{m} ) $$ in splitting. In the present work, there are two linear-features of $$ x_{\hat{j} = 1} = \frac{1}{RT} $$ and $$ x_{\hat{j} = 2} = \ln \left(\frac{t}{t_{0}} \right) $$ and one response $$ y = \ln\frac{w}{w_{00}} = f \left(\frac{1}{RT}, \ln \left(\frac{t}{t_{0}} \right), DOC, [ \tilde{{\rm{Cr}}} ] \right) $$ with four features. Thus, two TCLR trees are developed and in each of TCLR trees, the absolute change in Pearson correlation coefficient (Other metrics are embedded for nonlinear functions, as demonstrated in section 1 of SI), called the Linearity Goodness (LG), between function $$ y = f(x_{1}, \cdots, x_{m} ) $$ and a linear-feature $$ x_{\hat{j}} $$ $$ (\hat{j} = 1, \cdots, \hat{m} ) $$ on nodes is taken as a generalized loss function. The split feature and split value are determined, from n data with m features $$ \{x_{ij} \}(i = 1, \cdots, n )(j = 1, \cdots, m ) $$ , by maximizing the gain of average LG in the consequent nodes. For example, if the correlation coefficients in a parent node and two child nodes are denoted by $$ \rho^{p} $$ , $$ \rho^{c1} $$ and $$ \rho^{c2} $$ , respectively, the gain of average LG is defined by $$ \Delta | \rho | = \frac{1}{2}| \rho^{c1} + \rho^{c2} | - | \rho^{p} | $$ and the split feature and split value $$ x_{IJ} $$ are determined from

The TCLR tree must be pruned, otherwise, many leaves contain only two data per leaf, which destroys the model generalization considerably. A threshold of data number might also be introduced to prune the tree, i.e., the amount of data in a leaf should not be smaller than a pre-set threshold (default minsize $$ = $$ 3). Another threshold for the LG is introduced to pre-prune the tree. When the LG in a node is equal to or higher than a pre-set threshold, the node will become a leaf. In general, as the value of the LG threshold increases, the model complexity of our procedure tends to improve and the tree tends to overfit. The opposite behavior occurs as the LG threshold decreases. In the bias-variance trade-off, a lower value of the LG threshold corresponds to a larger value of the number threshold of data. Thus, TCLR has the function of data screening by discarding data, which cannot be described linearly (functionally) for some of the resulting leaves in such a framework. In the present work, the LG threshold is pre-set to be 0.90 for both $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \frac{1}{RT} $$ and $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \ln \left(\frac{t}{t_{0}}\right) $$ . The details of the TCLR algorithm are given in Supplementary Information.

The entire feature space is estimated from the regions of the four features, i.e., $$ [ \widetilde{{\rm{Cr}}}] \in [ 10.38, 30.319] $$ wt.%, $$ T \in [ 673.15,973.15] $$ K, $$ t \in [ 30, 2000] $$ h and $$ {\rm{DOC}} \in [ 0 , 8000] $$ ppb. The TCLR separates the entire feature space into many subdomains and each subdomain corresponds to one leaf, as shown in Figure S4, Table S2 and Table S3 in Supplementary Information. For $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ 1/RT $$ , the results indicate that 17 leaves pass the LG threshold and two fail, while for $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \ln \left(\frac{t}{t_{0}}\right) $$ , the leaves that pass and fail are 28 and 7, respectively. The amount of data in each passed leaf ranges from 3 to 30 for $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ 1/RT $$ and from 3 to 10 for $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ vs.$$ \ln \left(\frac{t}{t_{0}}\right) $$ , indicating that reliable values of activation energy Q and time exponent $$ m $$ will be statistically obtained by TCLR. The ideal case for the measurement of activation energy Q (or time exponent $$ m $$ ) is that the tests are carried out at various temperatures (or times) for a certain FM steel under other fixed testing conditions. The TCLR realizes such ideal cases from the complex data by partitioning the whole feature space of FM steels into many small rectangle domains, with one domain for a passed leaf. The samples on each passed leaf may have the same feature or varied feature values, but the variation must be too small to influence the measured activation energy Q (or time exponent m). On each of the failed leaves, the feature values of samples vary greatly so that they will not follow the same mechanism of oxidation, or it is too narrow to cover enough data subject to the capacity of the FMO dataset. One passed leaf and one failed leaf are taken as examples to illustrate the typical cases, as shown in Figure 5A-D.

Figure 5. Data located on the leaf of TCLR. (A), (B) One passed leaf and one failed leaf on TCLR of activation energy. (C), (D) One passed leaf and one failed leaf on TCLR of time exponents.

Figure 5A shows one passed leaf for $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ 1/RT $$ and the feature values, where four data all have the same $$ {\rm{DOC}} = $$ 8 ppb, three data have $$ [ \widetilde{{\rm{Cr}}}] = 17.27 $$ wt.% and one datum with $$ [ \widetilde{{\rm{Cr}}}] = 18.27 $$ wt.%, two data with $$ t = 40 $$ h and two data with $$ t = 50 $$ h. The linear fitting yields the statistically reasonable activation energy $$ Q = $$ 47.38 kJ/mol. Figure 5C shows one completely ideal passed leaf for the linear fitting of $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus $$ \ln \left(\frac{t}{t_{0}}\right) $$ , yielding time exponent $$ m = $$ 0.412, where all five data have the same values of $$ [ \widetilde{{\rm{Cr}}}] = 12.95 $$ wt.%, $$ {\rm{DOC}} = 200 $$ ppb and $$ T = 823.15 $$ K. There are four data on the failed leaf shown in Figure 5B, where the datum with $$ t = $$ 2000 h makes the LG not qualified. Although there are two data in Figure 5B that share the same feature values except for temperature, the two data do not allow to stay in a leaf because the threshold of data number is three. Figure 5D shows a failed leaf, where except for time, two data share the same feature values and another shares the same feature values. For the same reason that two data do not allow to stay in one leaf, the four data are restricted on a leaf with a low LG value.

The TCLR yields the values of activation energy Q and time exponent $$ m $$ as functions of the four features. The activation energy Q varies from 43.38 to 139.18 kJ/mol in the region $$ R_{\rm Q} = \{ [ \widetilde{{\rm{Cr}}}] \in [ 10.38, 30.319] $$ wt.%, $$ {\rm{DOC}} \in [ 0, 8000] $$ ppb, $$ T \in [ 673.15,973.15] $$ K and $$ t \in [ 30, 2000]\,{\rm{h}} \} $$ , where region $$ R_{\rm Q} $$ is the union set of all leaf domains in the TCLR on $$ \ln\frac{\Delta w}{\Delta w_{00}} $$ versus 1/RT withholds the inner interval it covers, containing all 178 data, including seven failed data. Similarly, the time exponent $$ m $$ varies from 0.112 to 0.71 in the region $$ R_{m} = \{ [ \widetilde{{\rm{Cr}}}] \in [ 10.38, 30.319] $$ wt.%, $$ {\rm{DOC}} \in [ 0, 8000] $$ ppb, $$ T \in [ 673.15,923.15] $$ K, and $$ t \in [ 30, 2000]\,{\rm{h}}\} $$ , containing 176 data including 27 failed data after removing an outlier highlighted on the red leaf of three data in Figure S4 in the Supplementary Information. The outlier gives $$ m = $$ 6.49, which is too larger than other $$ m $$ values. Two data on the red leaf share the same feature values except for time, which yields an unreasonable $$ m $$ of 15.398. This outlier should be carefully examined by experiments. The datasets contain data with four determinate features, and their corresponding activation energies and time exponents are extracted from the original FMO dataset by TCLR.

The SISSO algorithm ^[41], with minimization of mean absolute percentage error (MAPE, see Supplementary Information) is carried out to find analytic expressions of activation energy Q and time exponent $$ m $$ as functions of the four features of $$ T $$ , $$ t $$ , DOC and $$ [ \widetilde{{\rm{Cr}}}] $$ , which yields

Note that $$ [ \widetilde{{\rm{Cr}}}] $$ , $$ T $$ and DOC are all dimensionless as pre-divided by the reference values of 1 wt.%, 1 K and 1 ppb, respectively, and the MAPE $$ = $$ 12.97% for $$ Q^{{\rm{SISSO}}} $$ and MAPE $$ = $$ 39.13% for $$ m^{{\rm{SISSO}}} $$ . It is interesting that feature $$ t $$ does not show up in both $$ Q^{{\rm{SISSO}}} $$ and $$ m^{{\rm{SISSO}}} $$ , meaning that time does not statistically play an important role in activation energy Q and time exponent $$ m $$ . As expected, the higher the oxidation Cr equivalent concentration $$ [ \widetilde{{\rm{Cr}}}] $$ , the larger the activation energy and the smaller the time exponent. The effects of DOC and $$ T $$ on Q and $$ m $$ are not completely clear in Eqs. (5a) and (5b) and deserve more systematic investigations.

Oxidation kinetic equations

As mentioned above, the pure SHAP values of the DOC feature suggest that the oxidation kinetic in logarithmic space yield in the form of

To have an analytic expression of $$ \ln \left(\frac{\Delta w_{0}}{\Delta w_{00}}\right) $$ , the SISSO algorithm is conducted again, which yields

Putting all the analytic expressions together gives

The analytic formula of Eq. (5d) has strong predictive power, as shown in Figure 6, with a fitting performance of $$ \rho = $$ 0.91 and MAPE $$ = $$ 6.17%. There are three data that cannot be accurately predicted by the formula, which fall outside of the $$ 3\sigma $$ error range, and the fitting performance will be improved to $$ \rho = $$ 0.94 and MAPE $$ = $$ 5.65% if these three data are excluded. The three steels share the same $$ [ \widetilde{{\rm{Cr}}}] = $$ 24.255 wt.%, DOC $$ = $$ 10 ppb, testing to 600, 1000 and 1500 h and coincidentally arrive at the upbound of testing temperature (923.15 K) on domain $$ R_{Qm} $$ . This may lead to the slightly underfitting of Eq. (5d) on them. In addition, the graphical user interfaces of the Xgboost prediction model were developed and are available at the end of this manuscript.

Figure 6. Predicted values of Eq. 5(d) versus experimentally measured values. Each dot represents an FM sample, and the dots on the dotted line indicate the equation predicted values are consistent with experimental observations. The light blue region covers $$ 3\sigma $$ error range, where Eq. (5d) fitted extremely well on.

Authors' contributions

Performed the research, analysed data, wrote the programmers, and drafted the manuscript: Cao B

Collected the data. Tong-Yi Zhang and Ziqiang Dong supervised the project: Yang S, Sun A

Studied and revised the manuscript more on the oxidation mechanism: Dong Z

Designed the study, performed the research, analysed data, revised, and finalized the manuscript: Zhang TY

Discussed the results: Cao B, Yang S, Sun A, Dong Z, Zhang TY

Availability of data and materials

All experimental data collected in the study are contained in Supplementary Information (FM steel oxidation dataset) and are also available at: https://github.com/Bin-Cao/TCLRmodel. The ML methodology described in the present work was implemented in Python. Source codes of the programmers and algorithms are available at: https://github.com/Bin-Cao/TCLRmodel.

Financial support and sponsorship

This work was sponsored by the National Key Research and Development Program of China (No. 2018YFB0704400), Key Program of Science and Technology of Yunnan Province (No. 202002AB080001-2), Key Research Project of Zhejiang Laboratory (No. 2021PE0AC02), and Shanghai Pujiang Program (Grant No. 20PJ1403700).

Conflicts of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2022.

Supplementary Materials