A method for predicting the oxidation rate constant of a nickel-based single crystal superalloy

Abstract

The application discloses a nickel-based single crystal superalloy oxidation rate constant prediction method, and the method comprises the following steps: collecting oxidation experiment data to construct a characteristic matrix and a K P data set; training a TabPFN agent model to predict Al activity a(Al) and Al diffusion coefficient D Ni (Al); constructing a thermodynamic coupling index Φ=ln(a(Al)×D Ni (Al)) and establishing an Arrhenius baseline, and extracting a composition-induced kinetic deviation amount Devi; clustering Devi based on a Gaussian mixture model, determining an optimal partition number K through BIC, and dividing a composition space into multiple zones; training a TabPFN regression model independently for each zone, and selecting corresponding prediction values according to the zone to which a new sample belongs. The application can effectively solve the prediction bottleneck of different contributions of other alloy elements under different Al supply states.

Description

Technical Field

[0001] This invention belongs to the field of high-temperature alloy design technology, and particularly relates to a method for predicting the oxidation rate constant of nickel-based single-crystal high-temperature alloys. Background Technology

[0002] Nickel-based single-crystal superalloys are key materials for hot-end components such as turbine blades in aero-engines, and their oxidation resistance depends on whether a continuous and dense Al2O3 protective film can be formed on the surface. The oxidation parabolic rate constant (K2) is... P K is the core parameter characterizing this performance, and accurate prediction of K P This is crucial for accelerating the design of novel alloys. However, existing K... P Prediction techniques have the following shortcomings:

[0003] 1) Traditional empirical models based on Arrhenius relationships are only suitable for temperature extrapolation in single-component systems and cannot describe the complex nonlinear coupling effects between more than ten elements such as Al, Cr, and Re, resulting in limited prediction accuracy; 2) Conventional machine learning methods are prone to overfitting and have poor generalization ability under typical high-dimensional, small-sample conditions; deep learning relies on massive training data, which is out of touch with the current situation of scarce experimental data in materials; moreover, the models exhibit "black box" characteristics, lacking physical interpretability and making it difficult to guide alloy design; 3) Existing methods have not effectively integrated key thermodynamic / kinetic parameters such as Al activity and diffusion coefficient, and have failed to establish their relationship with K P The mechanism-driven correlation between them; 4) Alloy samples with different Al supply states in the global model are forced to be included in the same feature manifold learning, resulting in the same alloy element affecting K P The contribution direction may reverse in different component intervals, making it impossible to capture the differences in nonlinear effects between partitions. Summary of the Invention

[0004] The main objective of this invention is to provide a method for predicting the oxidation rate constant of nickel-based single-crystal superalloys, aiming to achieve high-precision and interpretable prediction of the parabolic oxidation rate constant of nickel-based single-crystal superalloys, and to overcome the prediction accuracy bottleneck caused by the reversal of the contribution direction of alloying elements under different Al supply states.

[0005] Therefore, the present invention provides a method for predicting the oxidation rate constant of nickel-based single-crystal superalloys, comprising the following steps:

[0006] Step S1: Dataset construction and feature engineering. High-temperature oxidation experimental data of Al2O3-forming nickel-based single-crystal superalloys are collected from publicly published literature, and the parabolic rate constant K of the steady-state oxidation stage is extracted. PValues ​​were used to construct a feature matrix based on the oxidation temperature T, the inverse temperature 1 / T, and the nominal mass fractions of 13 key alloying elements (Ni, Al, Co, Cr, Mo, Re, Ru, Ti, Ta, W, Hf, Si, and Y). K was then used as the input feature. P Taking the natural logarithm of the value yields ln(K) P ) as the target variable for prediction;

[0007] Step S2: Thermodynamic descriptor surrogate model construction, using oxidation temperature T, reverse temperature 1 / T, and the content of the 13 elements as input features, and using the logarithmically transformed Al activity ln(a(Al)) and Al diffusion coefficient ln(D) as input features respectively. Ni Using (Al) as the target variable, two TabPFN surrogate models are trained to predict thermodynamic / kinetic parameters under arbitrary composition and temperature conditions.

[0008] Step S3: Construction of the thermodynamic coupling index Φ and extraction of the Arrhenius baseline. Based on Wagner's oxidation theory, Φ is defined as: Φ = ln(a(Al) × D Ni (Al) is used as a measure of effective Al supply capacity, and an Arrhenius-type baseline Φ is established. baseline = α × (1000 / T) + β, calculate the kinetic deviation Devi for each sample: Devi = Φ - Φ baseline To filter out the main contribution of temperature; based on the Gaussian mixture model, the composition space is partitioned, assuming that the dynamic deviation Devi follows a mixture distribution consisting of K Gaussian distributions, and the Devi is clustered by the Gaussian mixture model. The optimal number of clusters K is determined by the Bayesian information criterion BIC, and the composition space is divided into K regions.

[0009] Step S4: Partitioning - Integrating TabPFN model training and prediction. For each region, a prediction pipeline is constructed that includes a region-specific standard scaler and a TabPFN regression model. New sample input features are simultaneously fed into all region pipelines, and the corresponding predicted value is selected as the final output based on the region to which the sample belongs.

[0010] Specifically, TabPFN is a table-based model that performs context learning on millions of synthetic prior datasets, performs Bayesian inference through a single forward propagation, and outputs a quantified result of prediction uncertainty.

[0011] Specifically, the test set determination coefficient R of the a(Al) proxy model described in step S2 2 The value is 0.9985, and the D is... Ni (A1) Test set determination coefficient R of the proxy model 2 It is 0.9367.

[0012] Specifically, in step S3, the slope α and intercept β of the Arrhenius baseline are obtained by linear regression on the complete dataset using ln(a(Al)×D). Ni (Al)) was fitted to 1000 / T, and α = -15.04, β = -28.21.

[0013] Specifically, the mixing coefficients of the Gaussian mixture model described in step S3 satisfy The mean of each Gaussian component is variance is The Bayesian information criterion BIC = k·ln(n) - 2·ln(L) is used, where k is the number of model parameters, n is the sample size, and L is the maximum likelihood function value. The value of K that minimizes BIC is selected as the optimal number of clusters.

[0014] Specifically, in step S3, the optimal cluster number K = 3, divides the component space into region 0 corresponding to the lowest effective supply level of Al, region 1 corresponding to the medium effective supply level of Al, and region 2 corresponding to the highest effective supply level of Al.

[0015] Specifically, in the partition-integrated TabPFN prediction module, the prediction pipeline corresponding to each partition includes a partition-specific standard scaler and a TabPFN regression model. The standard scaler scales the input features within each partition to zero mean and unit variance.

[0016] Specifically, in step S4, all partitioning models were evaluated using leave-one-out cross-validation. The global ensemble model achieved a leave-one-out cross-validation determination coefficient R0 on 60 sets of gold data. 2 The mean absolute error (MAE) reached 0.855, and the mean absolute error (MAE) was 0.880.

[0017] Specifically, the method can be applied to predict the parabolic rate constant of oxidation of Al2O3-forming nickel-based single-crystal superalloys in the temperature range of 700 °C to 1200 °C, and can be extended to other alloy systems with similar oxidation protection mechanisms.

[0018] Compared with the prior art, the present invention has the following beneficial effects:

[0019] 1. Prediction accuracy is significantly improved. Through the strategy of "physically guided decoupling + local partition regression", the global LOOCV determination coefficient R0 is significantly improved. 2 The mean absolute error (MAE) reached 0.855, and the mean absolute error (MAE) reached 0.880, verifying the effectiveness of the partitioning strategy in addressing the accuracy bottleneck caused by component heterogeneity.

[0020] 2. Strong physical interpretability. The partitioning is based on the classic Wagner oxidation theory, establishing a bridge from "black box prediction" to "white box understanding," providing materials designers with reliable predictive basis.

[0021] 3. Excellent adaptability to small samples. Leveraging TabPFN's context learning capabilities based on millions of synthetic prior datasets, high-precision predictions are achieved with only 60 experimental sets, avoiding the dependence of traditional deep learning methods on massive training data. Meanwhile, the LOOCV validation method provides a near-unbiased estimate of the generalization error on small datasets.

[0022] 4. Capable of handling nonlinear inversion of elemental effects. Physical clustering of the composition space is achieved through GMM partitioning, assigning alloy samples from different effective Al supply states to different partitions for independent modeling. This effectively resolves the nonlinear inversion of the effect of the same element on K under different Al supply states. P The challenges of different impacts. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0024] Figure 1 This is a flowchart of the method for predicting the oxidation rate constant of nickel-based single-crystal superalloys provided in the embodiments of the present invention. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0026] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0027] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0028] See Figure 1 A method for predicting the oxidation rate constant of nickel-based single-crystal superalloys includes the following steps:

[0029] Step S1: Dataset Construction and Feature Engineering

[0030] Step S1-1, Data Collection: Systematically collect high-temperature oxidation experimental data of Al2O3-forming nickel-based single-crystal superalloys from published literature, and extract the oxidation parabolic rate constant K. P Values ​​(only data from the steady-state oxidation stage controlled by Al2O3 are taken, excluding NiO-dominated data from the initial transient oxidation stage); finally, 60 sets of high-quality high-temperature oxidation experimental data and the corresponding Al activity and Al diffusion coefficient were obtained.

[0031] Step S1-2, Target Variable Processing: Given K P The values ​​span multiple orders of magnitude in the component space, for K P Taking the natural logarithm of the value yields ln(K) P This was used as the target variable for prediction to improve numerical stability and enhance the adaptability of the regression model to a wide dynamic range. For trace element content information missing in some literature, mean imputation was performed based on the global distribution of the training set.

[0032] Steps S1-3, Feature Engineering: Based on the above data, construct a feature matrix with oxidation temperature T, reverse temperature 1 / T, and the nominal mass fraction (wt.%) of 13 key alloying elements, including Ni, Al, Co, Cr, Mo, Re, Ru, Ti, Ta, W, Hf, Si, and Y, as input features.

[0033] Step S2: Construction of the thermodynamic descriptor surrogate model

[0034] Step S2-1, surrogate model training: Based on 60 sets of data, with oxidation temperature T, reverse temperature 1 / T, and the content (wt.%) of 13 elements as input features, the logarithmically transformed Al activity ln(a(Al)) and the logarithmically transformed Al diffusion coefficient ln(D) are used respectively. Ni Using (Al) as the target variable, two TabPFN surrogate models are trained. The dataset is randomly divided into a training set (48 groups) and a test set (12 groups) in an 8:2 ratio to achieve high-precision prediction of the thermodynamic / kinetic parameters under arbitrary alloy composition and temperature conditions.

[0035] Step S2-2, Proxy Model Performance: On a complete dataset containing 60 sets of data, LOOCV (one sample is kept as the validation set each time, and the remaining 59 samples are used to train the model, repeated 60 times) is used to evaluate the predictive performance of the machine learning algorithm.

[0036] Evaluation indicators include the coefficient of determination (R²). 2 The formulas for calculating the mean absolute error (MAE) and mean absolute error (MAE) are as follows:

[0037] ;

[0038] ;

[0039] in, It is the measurement of the i-th sample. , It is the prediction of the i-th sample. ,and It is the sample mean of the measured values, and n is the sample size.

[0040] The a(Al) proxy model on the test set R 2 = 0.9985, MAE = 1.29×10 -9 This confirms that the model can capture thermodynamic activity as a smooth, composition-driven function with high fidelity across the entire composition range. Ni (A1) The proxy model performs well on the test set in R. 2 = 0.9367, MAE = 2.18×10 -18 Given that subsequent partitioning depends on the overall trend of the thermodynamic coupling index Φ rather than its absolute accuracy, the accuracy of this surrogate model is sufficient to meet the requirements of this invention.

[0041] The ln(a(Al)) and ln(D) extracted in step S1 above Ni(Al)) are known values ​​obtained from various literatures through CALPHAD calculations, serving as the training target variables for the TabPFN surrogate model in S2. After the S2 surrogate model is trained, it can quickly predict ln(a(Al)) and ln(D) for any component without CALPHAD calculations, using only composition and temperature information. Ni (Al)) thus replaces the time-consuming and expensive CALPHAD calculation, and more importantly, effectively avoids the problem of different calculation results caused by different databases, which in turn leads to deviations in subsequent model predictions.

[0042] Step S3: Construction of the thermodynamic coupling index Φ and extraction of the Arrhenius baseline

[0043] Step S3-1: Based on Wagner's oxidation theory, the ability of Al atoms to diffuse into the oxide film / metal interface is proportional to the concentration gradient of All (characterized by the thermodynamic properties of Al activity a(Al)) and the diffusion coefficient D of Al in the Ni matrix. Ni The product of (Al) is defined by the following thermodynamic coupling index Φ: Φ = ln(a(Al) × D) Ni (Al)).

[0044] Step S3-2: Given D Ni (Al) dominates the temperature dependence of Φ, and the following Arrhenius-type baseline is established to extract the main effect of temperature: Φ baseline = α × (1000 / T) + β, where α and β are the slope and intercept of the baseline, respectively.

[0045] Using linear regression, ln(a(Al)×D) was applied to the complete 60 sets of data. Ni Fitting (Al) to 1000 / T yields α = -15.04 and β = -28.21. This baseline provides effective separation of the temperature effect.

[0046] Step S3-3: Calculation of kinetic deviation. Subtract the baseline predicted value at the corresponding temperature from the actual Φ value of each alloy to obtain the kinetic deviation Devi: Devi = Φ - Φ baseline This deviation encodes information about the composition-induced changes in effective Al supply capacity, eliminating the dominant contribution of temperature. Therefore, the kinetic deviation Devi for the i-th sample is defined as: Devi = Φᵢ - Φ baseline ,ᵢ.

[0047] Step S3-4: GMM clustering, based on component space partitioning using a Gaussian mixture model (GMM). It is assumed that the kinetic deviation Devi follows a mixture distribution consisting of K Gaussian distributions, with each Gaussian component corresponding to a valid Al supply state: , (k = 1, ..., K), where Mixing coefficient ( ), and These are the mean and variance of the k-th Gaussian component, respectively.

[0048] Step S3-5: Perform Gaussian mixture model clustering on Devi, and determine the optimal number of clusters K using the Bayesian Information Criterion (BIC). Objectively divide the composition space into K spaces, where the alloys within each region have a relatively consistent effective Al supply level, and the same element has a positive effect on K. P The direction of influence tends to be constant within this regime. The minimum K value corresponds to the optimal model complexity. Calculation results show that K = 3 is the optimal number of clusters, dividing the component space into the following three regimes:

[0049] Zone 0: Alloy composition corresponding to the lowest effective Al supply level;

[0050] Zone 1: Alloy composition corresponding to a moderate effective Al supply level;

[0051] Zone 2: Alloy composition corresponding to the highest effective Al supply level.

[0052] This embodiment describes in detail the specific implementation process of step S4.

[0053] Step S4: Partitioning - Integrated TabPFN Model Training and Prediction

[0054] Step S4-1: Constructing Region-Specific Pipelines. Independent prediction pipelines are constructed for regions 0, 1, and 2. Each pipeline includes a region-specific standard scaler and a TabPFN regression model. The standard scaler scales the features within each region to zero mean and unit variance to eliminate the impact of dimensional differences on model training. When predicting new alloy compositions, the input features simultaneously correspond to the prediction pipelines for all regions, and each pipeline outputs ln(K) for the corresponding region. P The predicted value is then selected as the final output based on the region label to which the sample to be predicted belongs. This supply index-guided decoupling + local partitioned regression strategy avoids the dilution of accuracy in the global model caused by component heterogeneity.

[0055] Step S4-2: Integrated Prediction Mechanism

[0056] The oxidation temperature T (°C) and the nominal mass fraction (wt.%) of 13 key alloying elements (Ni, Al, Co, Cr, Mo, Re, Ru, Ti, Ta, W, Hf, Si, Y), the predicted activity of Al, and the predicted diffusion coefficient of Al are simultaneously fed into three prediction pipelines corresponding to three different regional regimes. Each pipeline performs scaling and prediction within its respective regional context, outputting ln(K) for the corresponding regional regime. P Predicted values. Finally, based on the region classification of the sample to be predicted in the feature space (determined by the principle of maximizing the posterior probability of the GMM), the predicted value of the corresponding pipeline is selected as the final output. Among them, the predicted values ​​of Al element activity and Al element diffusion coefficient are the predicted values ​​of the TabPFN surrogate model in step S2.

[0057] Step S4-3: Performance Evaluation

[0058] The predictive performance of the partition-ensemble model was evaluated using global LOOCV (one sample was kept as the validation set each time, and the remaining 59 samples were used to complete the partition-training-prediction process as described above, repeated 60 times). To verify the effectiveness of the physical guidance partitioning strategy proposed in this invention, a global single TabPFN model was designed for prediction. The global single TabPFN model uses the same input features, the same hyperparameter configuration, and the same LOOCV validation strategy as the partition-TabPFN model of this invention. The only difference is that the physical guidance decoupling and GMM partitioning of S2-S3 are not used, and a single TabPFN model is trained directly on the complete dataset. The results show that the global LOOCV R of the partition-ensemble TabPFN model is higher than that of the previous model. 2 The performance improved significantly from 0.713 to 0.855 for the global single TabPFN model, and the MAE decreased from 1.256 to 0.880, demonstrating the effectiveness of the partitioning strategy in addressing the accuracy bottleneck caused by component heterogeneity. (In ln(K)) P In the verification plot, the predicted data points converge tightly along the ideal 1:1 diagonal across the entire dynamic range spanning orders of magnitude, and the outliers observed in the global model have been significantly eliminated.

[0059] This invention leverages the contextual learning capabilities of TabPFN, based on millions of synthetic prior datasets, to achieve high-precision predictions with only 60 experimental data sets, avoiding the dependence of traditional deep learning methods on massive training data. Simultaneously, the LOOCV validation method provides a near-unbiased estimate of the generalization error for small datasets. These methods are particularly suitable for predicting the oxidation parabolic rate constant of Al2O3-forming nickel-based single-crystal superalloys in the temperature range of 700℃ to 1200℃.

[0060] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for predicting the oxidation rate constant of nickel-based single-crystal superalloys, characterized in that, Includes the following steps: Step S1: Dataset construction and feature engineering. Collect high-temperature oxidation experimental data of Al2O3-forming nickel-based single-crystal superalloys and extract the parabolic rate constant K of the steady-state oxidation stage. P Values ​​were used to construct a feature matrix based on the oxidation temperature T, the inverse temperature 1 / T, and the nominal mass fractions of 13 key alloying elements (Ni, Al, Co, Cr, Mo, Re, Ru, Ti, Ta, W, Hf, Si, and Y). K was then used as the input feature. P Taking the natural logarithm of the value yields ln(K) P ) as the target variable for prediction; Step S2: Thermodynamic descriptor surrogate model construction. Using oxidation temperature T, reverse temperature 1 / T and the content of the 13 elements as input features, and using the logarithmically transformed Al activity and Al diffusion coefficient as target variables, two TabPFN surrogate models are trained to achieve the prediction of thermodynamic / kinetic parameters under arbitrary composition and temperature conditions. Step S3: Construction of the thermodynamic coupling index Φ and extraction of the Arrhenius baseline. Based on Wagner's oxidation theory, Φ is defined as: Φ = ln(a(Al) × D Ni (Al) is used as a measure of effective Al supply capacity, and an Arrhenius-type baseline Φ is established. baseline = α× (1000 / T) + β, where a(Al) is the Al activity, D Ni (Al) represents the diffusion coefficient of Al in the Ni matrix, and α and β are the slope and intercept of the baseline, respectively. The kinetic deviation (Devi) for each sample is calculated as: Devi = Φ - Φ baseline To filter out the main contribution of temperature; Based on the Gaussian mixture model, the component space is partitioned by assuming that the dynamic deviation Devi follows a mixture distribution consisting of K Gaussian distributions. The Devi is clustered by the Gaussian mixture model, and the optimal number of clusters K is determined by the Bayesian information criterion (BIC) to divide the component space into K regions. Step S4: Partitioning - Integrating TabPFN model training and prediction. For each region, a prediction pipeline is constructed that includes a region-specific standard scaler and a TabPFN regression model. New sample input features are simultaneously fed into all region pipelines, and the corresponding predicted value is selected as the final output based on the region to which the sample belongs.

2. The method according to claim 1, characterized in that: The TabPFN is a table-based model that performs context learning on millions of synthetic prior datasets, performs Bayesian inference through a single forward propagation, and outputs a quantified result of prediction uncertainty.

3. The method according to claim 1, characterized in that, The test set determination coefficient R of the a(Al) proxy model described in step S2 2 The value is 0.9985, and the D is... Ni (A1) Test set determination coefficient R of the proxy model 2 It is 0.9367.

4. The method according to claim 1, characterized in that, In step S3, the slope α and intercept β of the Arrhenius baseline are obtained by linear regression on the complete dataset using ln(a(Al)×D). Ni (Al)) was fitted to 1000 / T, and α = -15.04, β = -28.

21.

5. The method according to claim 1, characterized in that, The mixing coefficients of the Gaussian mixture model described in step S3 satisfy The mean of each Gaussian component is variance is The Bayesian information criterion BIC = k·ln(n) - 2·ln(L) is used, where k is the number of model parameters, n is the sample size, and L is the maximum likelihood function value. The value of K that minimizes BIC is selected as the optimal number of clusters.

6. The method according to claim 1, characterized in that, In step S3, the optimal number of clusters K = 3, which divides the component space into region 0, corresponding to the lowest effective supply level of Al, region 1, corresponding to the medium effective supply level of Al, and region 2, corresponding to the highest effective supply level of Al.

7. The method according to claim 1, characterized in that, In step S4, all partitioned models were evaluated using leave-one-out cross-validation. The global ensemble model achieved a leave-one-out cross-validation determination coefficient R0 on 60 sets of gold data. 2 The value is 0.855, and the mean absolute error (MAE) is 0.

880.

8. The method according to claim 1, characterized in that, The method is applicable to the prediction of the oxidation parabolic rate constant of Al2O3-forming nickel-based single-crystal superalloys in the temperature range of 700℃ to 1200℃.

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