A hybrid modeling method for performance prediction of iron-based gyromagnetic materials

Abstract

This invention discloses a hybrid modeling method for predicting the performance of iron-based gyromagnetic materials, belonging to the interdisciplinary field of magnetic materials and machine learning. Addressing the shortcomings of existing methods, such as insufficient feature processing targeting, low efficiency in high-dimensional feature processing, and limited system adaptability, this invention proposes a three-stage hybrid modeling approach: "symbolic nonlinear feature generation + principal component dimensionality reduction and screening + multi-model ensemble evaluation." First, it utilizes gene expression programming to automatically mine the nonlinear relationship between components and process parameters, generating interpretable analytical expressions. Then, it uses principal component analysis to reduce the dimensionality of high-dimensional features, retaining principal components with a variance contribution rate ≥95%. Finally, it inputs the reduced features and the original features into models such as linear regression and multilayer perceptron for parallel comparison and evaluation, outputting the optimal prediction scheme. This invention achieves high-precision prediction of core performance indicators of iron-based gyromagnetic materials in different systems, improving model interpretability and engineering practicality.

Description

Technical Field

[0001] This invention belongs to the field of magnetic material performance prediction and intelligent modeling technology. More specifically, this invention relates to a hybrid modeling method for predicting the performance of iron-based gyromagnetic materials. It is applicable to the modeling and prediction of complex nonlinear relationships between the composition parameters, process parameters and target performance indicators of iron-based gyromagnetic materials, and realizes the prediction of performance indicators such as remanent magnetization, coercivity and maximum energy product. Background Technology

[0002] Ferromagnetic materials, due to their excellent magnetic properties, have significant application value in fields such as high-performance motors, wind power generation, rail transportation, aerospace, and electronic devices. With the increasing demand for high-performance permanent magnet materials, material performance optimization has gradually shifted from traditional experience-driven approaches to data-driven and intelligent design.

[0003] Currently, the performance prediction of ferromagnetic materials mainly relies on experimental trial-and-error methods, empirical formula methods, and traditional statistical modeling methods. While experimental trial-and-error methods can obtain highly reliable results, they suffer from long development cycles, high experimental costs, and difficulty in revealing the coupling laws between variables. Empirical formula methods are usually established under specific material systems, limiting their applicability and making it difficult to handle high-dimensional nonlinear correlations under complex process conditions. Although traditional machine learning methods have made some progress in material performance prediction, they still have the following shortcomings.

[0004] Existing methods typically rely on manual experience to select input variables or simple linear combinations for feature construction, making it difficult to effectively characterize the potentially complex nonlinear mapping relationships between material composition, fabrication process, and target performance. Especially in multi-element doped iron-based gyromagnetic material systems, high-order couplings often exist between different factors, and traditional feature representation methods cannot fully reflect this mechanism.

[0005] Existing methods for high-dimensional feature processing are prone to dimensionality redundancy, model overfitting, and decreased generalization ability when dealing with multi-source input parameters. Conventional dimensionality reduction methods, if applied directly to the original variables, easily lose information that has a critical impact on performance prediction. Single prediction models often struggle to account for the differences in data distribution among different material systems, and their adaptability is often insufficient for different scenarios such as Ce-doped systems and pure NdFeB systems, resulting in limited prediction accuracy and robustness. There is an urgent need to propose a new method for predicting the performance of ferromagnetic materials that balances feature representation ability, dimensionality optimization ability, and model adaptability, in order to improve prediction accuracy and enhance model interpretability. Summary of the Invention

[0006] The purpose of this invention is to overcome the problems of insufficient feature processing, low efficiency of high-dimensional information utilization, and limited model adaptability in the existing technology. It proposes a hybrid modeling method for predicting the performance of iron-based gyromagnetic materials based on multi-gene expression programming and principal component analysis. By combining symbolic nonlinear feature generation, principal component dimensionality reduction screening, and multi-model integrated prediction, high-precision prediction of target performance indicators can be achieved.

[0007] A hybrid modeling method for predicting the properties of ferromagnetic materials includes the following steps: S1. Collect experimental data on iron-based gyromagnetic materials and perform preprocessing; S2. Generate symbolic nonlinear feature expressions representing the relationship between components, processes, and performance based on gene expression programming; S3. Principal component analysis is used to perform dimensionality reduction and screening on the nonlinear feature expression to form a dimensionality-reduced feature set; S4. Input the dimensionality-reduced feature set into the regression prediction model, and output the target performance prediction results and the optimal model combination; S5. Utilize the optimal model combination to predict the performance of ferromagnetic materials.

[0008] Preferably, in step S1, the experimental data for the iron-based gyromagnetic material includes material composition parameters, process parameters, and target performance indicators; wherein the composition parameters include rare earth element content, iron element ratio, and trace doping element content; the process parameters include sintering temperature, heat treatment regime, particle size parameters, and forming process parameters; and the target performance indicators include remanent magnetization. Br Coercivity Hcj and maximum magnetic energy product BH max ; The preprocessing includes missing value handling, outlier removal, standardization, and splitting the training and test sets.

[0009] Preferably, in S2, the gene expression programming uses a set of operators including addition, subtraction, multiplication, division, power functions, logarithmic functions, and absolute value functions to generate a symbolic nonlinear feature expression.

[0010] Preferably, in S2, the gene expression programming adopts a population evolution mechanism, and iteratively evolves through fitness evaluation, selection, crossover and mutation operations, with the goal of minimizing prediction error, and automatically generates multiple symbolic nonlinear feature expressions describing the relationship between material input and performance output.

[0011] Preferably, in step S3, the high-dimensional feature set generated in step S2 is input into the principal component analysis model to calculate the covariance matrix, eigenvalues, and eigenvectors. Principal components are selected based on the cumulative variance contribution rate, with a cumulative variance contribution rate of not less than 95% used as the principal component selection criterion to ensure that the main information is retained and redundant variables are eliminated, thus forming a dimensionality-reduced feature set.

[0012] Preferably, in S3, the filtered principal components constitute a dimensionality-reduced feature set, which is used in parallel with the original input features for performance prediction comparison.

[0013] Preferably, in S4, the regression prediction model includes one or more of the following: linear regression model, multilayer perceptron model, support vector regression model, random forest model, K-nearest neighbor model, and classification and regression tree model.

[0014] Preferably, in S4, the root mean square error, mean absolute error, mean absolute percentage error, and coefficient of determination are used as evaluation indicators for prediction performance to compare the prediction results of different regression prediction models and output the optimal model combination.

[0015] Preferably, in step S4, the optimal model combination consisting of the dimensionality reduction feature set and the target regression model is determined based on the evaluation index, and the obtained optimal model combination is used to predict the target performance index of the iron-based gyromagnetic material, thereby realizing the performance evaluation and auxiliary design of the iron-based gyromagnetic material.

[0016] Preferably, the hybrid modeling method for predicting the performance of iron-based gyromagnetic materials is applicable to the performance prediction of NdFeB-based iron-based gyromagnetic materials and Ce-doped iron-based gyromagnetic materials.

[0017] The present invention has at least the following beneficial effects: This invention automatically generates symbolic nonlinear features through gene expression programming (GEP), which, compared with traditional artificial feature construction methods, can more effectively reveal the complex coupling relationship between material composition, process parameters and performance, and improve feature expression capabilities.

[0018] By adopting a multi-model integrated evaluation mechanism, an appropriate prediction model can be automatically selected according to different material systems, thereby improving the stability and adaptability of prediction results.

[0019] The generated characteristic expressions have an explicit analytical form, which enhances the interpretability of the prediction process and is beneficial for the analysis of material design mechanisms and engineering applications.

[0020] The explicit analytical expressions output by GEP can directly reveal the performance-driven mechanism, solving the problem that "black box" models are difficult to guide engineering practice.

[0021] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0022] Figure 1 Comparison of MAPE feature processing predictions before and after for the Ce+NdFeB group-Hcj system; Figure 2 This is a comparison of MAPE feature processing predictions before and after processing of the Ce+NdFeB group-Br system. Detailed Implementation

[0023] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0024] It should be understood that terms such as “having,” “comprising,” and “including” as used herein do not exclude the presence or addition of one or more other elements or combinations thereof. Example 1 A hybrid modeling method for predicting the properties of iron-based gyromagnetic materials, using the Ce+NdFeB group-Hcj system, includes the following steps: S1. Collect experimental data on iron-based gyromagnetic materials. This data includes material composition parameters, process parameters, and target performance indicators. Composition parameters include rare earth element content, iron ratio, and trace doping element content. Process parameters include sintering temperature, heat treatment regime, particle size parameters, and forming process parameters (obtained from academic journals and patent publications). Target performance indicators include remanent magnetization. Br Coercivity Hcj and maximum magnetic energy product BH max The target performance indicators were obtained using magnetic testing equipment (such as a vibrating sample magnetometer (VSM) or a hysteresis loop measurement system). Preprocessing of the experimental data for ferromagnetic materials involved missing value handling, outlier removal, standardization, and splitting the training and testing sets. Data of different dimensions (such as elemental proportions, temperature, rate, and performance indicators) were standardized or normalized to eliminate the impact of numerical range differences on model training. For data with a low missing proportion, mean imputation, neighborhood interpolation, or other interpolation methods were used to supplement the missing data. If the missing data was severe and might introduce bias, the sample was removed. Statistical analysis and visualization were used to detect and remove outliers and duplicate data to ensure the authenticity and consistency of the dataset. The cleaned dataset was then randomly divided into training and testing sets in an 8:2 ratio using a random sampling strategy. In this embodiment, the dataset used for the Ce+NdFeB group-Hcj system contains 44 samples, 4 feature variables, and 1 target variable. The feature variables are the contents of Ce, Nd, Fe, and B, and the target variable is coercivity. Hcj The dataset used for the Ce+NdFeB-Br system contains 45 samples, 4 feature variables, and 1 target variable. The feature variables are the contents of Ce, Nd, Fe, and B, and the target variable is the remanent magnetization. Br ; S2. Gene Expression Programming (GEP) generates symbolic nonlinear characteristic expressions representing the component-process-performance relationship. The set of operators used in GEP includes addition, subtraction, multiplication, division, power functions, logarithmic functions, and absolute value functions. The operator set is defined as {F=+,−,*, / ,√,} 4 √, log, abs}, where (+, −, *, / ) are arithmetic operations, (√, log, abs) are arithmetic operations, and (√, log, abs) are arithmetic operations. 4 √) represents exponentiation, log represents logarithmic operation, and abs represents absolute value operation, covering the core nonlinear transformations related to the properties of magnetic materials. Variable definition: The pre-processed composition parameters and process parameters (such as sintering temperature and particle size) are used as input variables (denoted as...). X 1,X2,..., X n Gene expression programming employs a population evolution mechanism, iteratively evolving through fitness evaluation, selection, crossover, and mutation operations, with the goal of minimizing prediction error. It automatically generates multiple symbolic nonlinear feature expressions describing the relationship between material input and performance output. The population size is set to 500, the number of iterations to 100 generations, the crossover probability to 0.8, and the mutation probability to 0.1. Candidate feature expressions are generated through gene encoding, crossover, and mutation, with the goal of minimizing prediction error through iterative optimization. Finally, five optimal symbolic nonlinear characteristic expressions are generated, for example: the output expression for the Ce+NdFeB group-Hcj system: GEP The feature construction results show that the final nonlinear expression is mainly composed of Fe( X 2 represents Fe content), B ( X 3 represents B content) and Ce ( X 4 represents the Ce content) variable, Nd ( XThe fact that Nd content (1) was not included in the final expression indicates that, under the current data distribution conditions, its statistical contribution to the target performance is low. Ce and Nd are substitutes for each other; Nd is correlated with Fe, B, and Ce to some extent, and some of its information can be indirectly characterized by other variables. GEP i ( i =1,2,3,4,5) represents the combination relationship formed under the combined action of Fe, B and Ce elements, used to characterize the synergistic effect among the three and their comprehensive impact on changes in material properties; And by using a fixed random seed, we ensure that the results are reproducible; S3. Principal component analysis is used to reduce the dimensionality of the nonlinear feature expression. The high-dimensional feature set generated in S2 is input into the principal component analysis model to calculate the covariance matrix, eigenvalues ​​and eigenvectors. Principal components are selected based on the cumulative variance contribution rate. The cumulative variance contribution rate of not less than 95% is used as the selection criterion for principal components. The selected principal components form a dimensionality-reduced feature set and are used in parallel with the original input features for performance prediction comparison. S4. Input the dimensionality-reduced feature set into the regression prediction model. The regression prediction models include linear regression model (Linear), multilayer perceptron model (MLP), support vector regression model (SVR), random forest model (RF), K-nearest neighbor model (KNN) and classification and regression tree model (CART). The specific calculation methods for each regression prediction model are as follows: (1) The linear regression model assumes a linear relationship between the input features and the target variable, and its expression is: y = β 0+ β 1 x 1+ β 2 x 2+…+ β n x n + e in, x 1, x 2,..., x n Indicates input features, β 0, β 1, β 2,..., β n Represents the regression coefficient, where, e This is the error term.

[0025] (2) The multilayer perceptron (MLP) model establishes a nonlinear mapping relationship between input and output through a multilayer neural network. Its calculation process is as follows: in h j For the next level j The output of each neuron f (·) is the activation function. w ij For weight parameters, x i For the next level i The input value of each neuron. b j This is a bias term.

[0026] (3) The Support Vector Regression (SVR) model achieves prediction by constructing the optimal hyperplane, and its function is expressed as: f ( x )= w T f ( x ) +b in f (x) represents the kernel function mapping. w T This is the weight coefficient vector. b This is a deviation.

[0027] (4) Random Forest Regression (RF) achieves prediction by integrating multiple decision trees, and its output is: Where represents the output of the i-th decision tree, i =1,2,…, N .

[0028] (5) K-Nearest Neighbors (KNN) model finds the nearest neighbor sample based on the distance between samples: The prediction result is based on the mean of the K nearest neighbor samples. d ( x i , x j ) represents a sample x i , x j The Euclidean distance between them k =1,2,…, n Index variables representing features, x ik Indicates the firsti The first sample k 1 eigenvalue, x jk Indicates the first j The first sample k Each feature value.

[0029] (6) The Classification and Regression Tree (CART) model uses feature partitioning to reduce node error. Its objective function is: in, i =1,2,…, N Indicates the number of training samples. y i Indicates the first i The true label value of each sample For the first i The predicted value for each sample.

[0030] The regression prediction models described above establish the mapping relationship between feature variables and target variables in different ways, and utilize RMSE, MAE, MAPE, and the coefficient of determination R. 2 The model performance is evaluated.

[0031] The root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R²) are used. 2 The prediction performance is evaluated by comparing the prediction results of different regression prediction models and outputting the optimal model combination; the optimal model combination consisting of the dimensionality reduction feature set and the target regression model is determined based on the evaluation index. Dimensionality reduction of the nonlinear features generated by GEP is performed, specifically including: Feature input: The five nonlinear features output by GEP are combined into a high-dimensional feature matrix, which is used as the input for PCA; Principal component calculation: The covariance matrix, eigenvalues, and eigenvectors of the high-dimensional feature matrix are calculated using the PCA algorithm; The original feature matrix is ​​standardized to construct a standardized data matrix. X Then the covariance matrix is ​​calculated. C : For covariance matrix C Perform eigenvalue decomposition: Cw = λw in, w For feature vectors, l For eigenvalues; Sort the eigenvalues ​​by size and calculate the cumulative variance contribution rate.P k : in, l i ( i =1,…, k ) is the former k The eigenvalues ​​of the principal components l i ( i =1,…, n The eigenvalues ​​are the eigenvalues ​​of all principal components. When the cumulative variance contribution rate reaches 95% or more, the top eigenvalues ​​are selected. k The principal components form a dimensionality reduction matrix, thus achieving dimensionality reduction.

[0032] Screening criteria: Sort by eigenvalue from largest to smallest, accumulate the variance contribution rate, and select the top eigenvalues ​​with a cumulative variance contribution rate ≥ 95%. k One principal component, for example: results of the Ce+NdFeB group-Hcj system. PC 1. As follows: Output result: the filtered result k The principal components form a dimensionality-reduced feature set, which is used for subsequent model training. S5. Utilize the obtained optimal model combination to predict the target performance indicators of iron-based gyromagnetic materials, thereby realizing the performance evaluation and auxiliary design of iron-based gyromagnetic materials; Model training and prediction: The “GEP+PCA” feature set and the original feature set are respectively input into six models: Linear, SVR, RF, MLP, KNN, and CART. After training is completed based on the training set, performance prediction is carried out on the test set.

[0033] Indicator calculation and comparison: such as Figure 1 As shown in Figure 2, the mean absolute percentage error (MAPE), mean absolute error (MAE), and root mean square error (RMSE) of the two sets of features were calculated. The results show that the MLP model has the most significant optimization effect, with MAPE decreasing from 57% to 22%. The errors of SVR, KNN, and RF models all decreased to varying degrees, while the errors of Linear and CART models increased.

[0034] Optimal solution output: Based on the comparison of indicators, for the target performance indicators ( Br / Hcj / BH max For nonlinear error control scenarios, the preferred option is "GEP+PCA feature set+MLP model". For scenarios seeking stability and low error, the option is "GEP+PCA feature set+SVR model". The above optimal combination scheme can be used for performance prediction of practical iron-based ferromagnetic materials.

[0035] The number of devices and processing scale described herein are for the purpose of simplifying the description of the invention. Applications, modifications, and variations of the invention will be readily apparent to those skilled in the art.

[0036] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A hybrid modeling method for predicting the properties of iron-based gyromagnetic materials, characterized in that, Includes the following steps: S1. Collect experimental data on iron-based gyromagnetic materials and perform preprocessing; S2. Generate symbolic nonlinear feature expressions representing the relationship between components, processes, and performance based on gene expression programming; S3. Principal component analysis is used to perform dimensionality reduction and screening on the nonlinear feature expression to form a dimensionality-reduced feature set; S4. Input the dimensionality-reduced feature set into the regression prediction model, and output the target performance prediction results and the optimal model combination; S5. Utilize the optimal model combination to predict the performance of ferromagnetic materials.

2. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, In S1, the experimental data for the iron-based gyromagnetic material includes material composition parameters, process parameters, and target performance indicators; wherein, the composition parameters include rare earth element content, iron element ratio, and trace doping element content; the process parameters include sintering temperature, heat treatment regime, particle size parameters, and forming process parameters; and the target performance indicators include remanent magnetization. Br Coercivity Hcj and maximum magnetic energy product BH max ; The preprocessing includes missing value handling, outlier removal, standardization, and splitting the training and test sets.

3. The hybrid modeling method for predicting the properties of iron-based gyromagnetic materials according to claim 1, characterized in that, In S2, the gene expression programming uses a set of operators including addition, subtraction, multiplication, division, power functions, logarithmic functions, and absolute value functions to generate symbolic nonlinear feature expressions.

4. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, In S2, the gene expression programming adopts a population evolution mechanism, and iteratively evolves through fitness evaluation, selection, crossover and mutation operations, with the goal of minimizing prediction error, and automatically generates multiple symbolic nonlinear feature expressions describing the relationship between material input and performance output.

5. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, In S3, the high-dimensional feature set generated in S2 is input into the principal component analysis model to calculate the covariance matrix, eigenvalues ​​and eigenvectors. Principal components are selected based on the cumulative variance contribution rate, with a cumulative variance contribution rate of not less than 95% as the selection criterion for principal components.

6. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, In S3, the filtered principal components constitute a dimensionality-reduced feature set, which is used in parallel with the original input features for performance prediction comparison.

7. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, In S4, the regression prediction model includes one or more of the following: linear regression model, multilayer perceptron model, support vector regression model, random forest model, K-nearest neighbor model, and classification and regression tree model.

8. The hybrid modeling method for predicting the properties of iron-based gyromagnetic materials according to claim 1, characterized in that, In S4, the root mean square error, mean absolute error, mean absolute percentage error, and coefficient of determination are used as evaluation indicators for prediction performance. The prediction results of different regression prediction models are compared, and the optimal model combination is output.

9. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 8, characterized in that, In S4, the optimal model combination consisting of the dimensionality reduction feature set and the target regression model is determined based on the evaluation index. The obtained optimal model combination is used to predict the target performance index of the iron-based gyromagnetic material, thereby realizing the performance evaluation and auxiliary design of the iron-based gyromagnetic material.

10. The hybrid modeling method for predicting the properties of ferromagnetic materials according to claim 1, characterized in that, The hybrid modeling method for predicting the performance of iron-based gyromagnetic materials is applicable to the performance prediction of NdFeB-based iron-based gyromagnetic materials and Ce-doped iron-based gyromagnetic materials.

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