Machine learning based quartz crucible high temperature deformation resistance prediction method

Abstract

The application belongs to the technical field of crucible deformation resistance prediction, and relates to a quartz crucible high-temperature deformation resistance prediction method based on machine learning; the application collects key historical data, covers aluminum alloy composition, crucible physical properties, use conditions and deformation resistance, etc.; data preprocessing is performed to ensure the quality and consistency of the data set; through statistical methods such as Pearson correlation coefficient, LASSO regression and principal component analysis, features that have a significant influence on deformation resistance are screened out, and feature engineering is performed to construct new interaction features, which can reflect the complex relationship between smelting conditions and crucible physical properties; the introduction of the deep learning model enables the prediction model to learn the deep patterns in the data, thereby significantly improving the prediction accuracy; finally, the trained model can accurately predict the high-temperature deformation resistance of the quartz crucible according to real-time data, providing a scientific basis for production decisions and helping to optimize the production process and reduce costs.

Description

Technical Field

[0001] This application belongs to the field of crucible deformation resistance prediction technology, and more specifically, relates to a machine learning-based method for predicting the high-temperature deformation resistance rate of quartz crucibles. Background Technology

[0002] Quartz crucibles, as high-temperature containers, play a crucial role in high-temperature processing such as metal smelting, glass manufacturing, and ceramic firing. Their performance directly affects product quality and production efficiency. Quartz crucibles need to possess excellent high-temperature deformation resistance to ensure the stability of their shape and dimensions under high-temperature environments, preventing container breakage or product contamination due to deformation.

[0003] The application of quartz crucibles is particularly crucial in the smelting of aluminum. Aluminum has a melting point of approximately 660 degrees Celsius, but in actual smelting processes, furnace temperatures often need to reach even higher levels to ensure complete melting of the aluminum and removal of impurities. Under such high-temperature conditions, the crucible material must be able to withstand the combined effects of thermal and mechanical stresses, and quartz crucibles, due to their high melting point, good thermal stability, and chemical stability, become an ideal choice.

[0004] The deformation resistance of quartz crucibles is a key indicator of their performance at high temperatures. A low deformation resistance means the crucible is prone to deformation at high temperatures, which can shorten its lifespan, increase production costs, and even affect product quality. Therefore, accurately predicting the deformation resistance is an effective way to achieve more precise material proportioning, improve efficiency, and enhance product quality. Consequently, with the development of aluminum alloy smelting processes, accurately predicting the deformation resistance of quartz crucibles, especially their deformation resistance under different aluminum contents, has become a core issue for improving production efficiency, reducing costs, and optimizing crucible material proportioning. Summary of the Invention

[0005] This invention provides a machine learning-based method for predicting the high-temperature deformation resistance of quartz crucibles, aiming to achieve accurate prediction of the deformation resistance of quartz crucibles.

[0006] A machine learning-based method for predicting the high-temperature deformation resistance of quartz crucibles, characterized by the following steps:

[0007] Step 1: Collect historical data related to the deformation resistance rate of quartz crucibles, including aluminum alloy composition data, crucible physical property data, crucible usage conditions, and deformation resistance rate data. The deformation resistance rate data is used as the true value label of the input data when training the corresponding prediction model.

[0008] Step 2: Preprocess the collected historical data, including missing value handling, outlier detection and handling, data normalization, and categorical data encoding, to obtain preprocessed data;

[0009] Step 3: Based on the preprocessed data, perform Pearson correlation coefficient, LASSO regression, and principal component analysis step by step to select important features;

[0010] Step 4: Construct the interaction features between smelting conditions and physical properties based on the preprocessed data to obtain the constructed new features;

[0011] Step 5: Train the prediction model based on the newly constructed features and the selected important features to obtain the trained prediction model;

[0012] Step 6: Based on the trained prediction model and real-time data, predict the high-temperature deformation resistance of the quartz crucible and obtain the prediction results.

[0013] This invention utilizes machine learning and deep learning technologies to achieve a highly efficient and accurate method for predicting the high-temperature deformation resistance of quartz crucibles. First, key historical data is collected from actual production, covering information such as aluminum alloy composition, crucible physical properties, usage conditions, and deformation resistance, providing a realistic and comprehensive training foundation for the model. Data preprocessing ensures the quality and consistency of the dataset. Statistical methods such as Pearson correlation coefficient, LASSO regression, and principal component analysis are used to screen features that significantly influence the deformation resistance, and feature engineering is performed to construct new interactive features that reflect the complex relationship between melting conditions and crucible physical properties. The introduction of a deep learning model enables the prediction model to learn deep patterns in the data, significantly improving prediction accuracy. Finally, the trained model can accurately predict the high-temperature deformation resistance of quartz crucibles based on real-time data, providing a scientific basis for production decisions and helping to optimize production processes and reduce costs.

[0014] Preferably, the steps for processing the preprocessed data using the Pearson correlation coefficient are as follows:

[0015] Calculate the Pearson correlation coefficient between each feature and its corresponding resistance to deformation:

[0016]

[0017] In the formula: This represents the correlation coefficient between characteristic X and resistance to deformation Y; X i and Y i This represents the feature value and the target variable value of the i-th sample; and This represents the mean of characteristic X and resistance to deformation Y; n represents the number of samples.

[0018] If the correlation coefficient r between a certain feature and the target variable is X,YIf the absolute value of a feature is lower than a set first threshold, the feature is removed, and the first dataset is obtained.

[0019] Based on the first dataset, calculate the Pearson correlation coefficient between each feature:

[0020]

[0021] In the formula: The correlation coefficient between feature X and feature Y′ is represented by X. i and Y′ i Let X and Y′ represent the values ​​of feature X and feature Y′ of the i-th sample, respectively. and Let X and Y′ represent the means of features X and Y′, respectively.

[0022] If the Pearson correlation coefficient r between any two feature pairs is X,Y′ If the absolute value of the feature pair is higher than the preset second threshold, then any one feature in the feature pair will be removed to obtain the second dataset.

[0023] Preferably, the feature selection steps based on the LASSO regression are as follows:

[0024] Cross-validation: The dataset obtained by filtering based on Pearson correlation coefficient is divided into several subsets. Each time, one subset is selected as the validation set and the other subsets are used as the training set. The mean squared error under each regularization parameter λ is calculated, and the regularization parameter λ with the smallest mean squared error is adopted.

[0025] Model Training: The LASSO regression model was trained using the dataset selected based on the Pearson correlation coefficient and the chosen regularization parameter λ. Features X and the deformation resistance rate Y from the feature set were used. The model was trained by minimizing the loss function of the LASSO regression model, and the gradient descent method was employed to solve the optimization problem.

[0026]

[0027] In the formula: Y i X represents the deformation resistance rate of the i-th sample; ij β represents the j-th feature value of the i-th sample; j λ represents the regression coefficient of the j-th feature; λ represents the regularization parameter. This represents the regression coefficient vector obtained through training, containing the weights of all features, and representing the contribution of each feature to the resistance deformation rate, where p is the total number of features; β0 represents the intercept term of the regression model; β j Let represent the regression coefficient of the j-th feature, where j∈(1,2,…,p);

[0028] Feature selection: If β j If β = 0, then the corresponding feature is removed;j If the value is not equal to 0, the corresponding feature is retained; a third dataset is obtained based on feature selection.

[0029] Preferably, the specific steps of the principal component analysis are as follows:

[0030] Calculate the covariance matrix: Let X be the data matrix in the dataset obtained after feature selection based on LASSO regression. norm The size is n×p, where n is the number of samples and p is the number of features. The covariance is calculated based on the data matrix.

[0031]

[0032] In the formula: Σ is the p×p covariance matrix, and each element σ ij This represents the covariance between the i-th feature and the j-th feature; Represents data matrix X norm Transpose of;

[0033] Calculate eigenvalues ​​and eigenvectors: Calculate the eigenvalues ​​and eigenvectors of the covariance matrix Σ.

[0034] Σv i =λ i v i ;

[0035] In the formula: λ i Let v represent the i-th eigenvalue of the covariance matrix, and v represent the variance along the eigendirection; i Let represent the i-th eigenvector of the covariance matrix, and let represent the eigendirection.

[0036] Constructing a dimensionality reduction space: for eigenvalues ​​λ1, λ2, ..., λ p Sort the eigenvalues ​​by size, select the eigenvectors corresponding to the first k eigenvalues, and form a matrix V based on the eigenvectors corresponding to the first k eigenvalues. k , with a size of p×k, where p represents the number of features in the dataset obtained after feature selection based on LASSO regression, and k represents the number of principal components selected;

[0037] Mapping the data to a reduced-dimensional space: Calculate the coordinates of each data sample in the reduced-dimensional space using matrix multiplication.

[0038] Z = X norm V k ;

[0039] In the formula: Z represents the score matrix after dimensionality reduction, with a size of n×k, and each row represents the coordinates of a sample in the dimensionality reduction space;

[0040] Principal component selection: Calculate the ratio of the eigenvalues ​​corresponding to each principal component to the sum of the total eigenvalues ​​to obtain the contribution ratio of each principal component to the data variance.

[0041]

[0042] Where: G i This represents the contribution of the i-th principal component;

[0043] Calculate the cumulative contribution: Calculate the cumulative contribution of the top k principal components;

[0044]

[0045] In the formula: L k This represents the cumulative contribution of the first k principal components;

[0046] The calculated cumulative contribution is compared with the contribution threshold. If the cumulative contribution is higher than the threshold, the fourth dataset is constructed using the currently selected top k principal components. If the cumulative contribution is lower than the threshold, the value of k is increased until the cumulative contribution is higher than the threshold. The selected principal components are then obtained, and the fourth dataset is constructed based on the selected principal components.

[0047] Preferably, the interaction characteristics between the melting conditions and physical properties include the interaction characteristics between aluminum content and crucible expansion coefficient and the load characteristics between melting time and temperature; wherein the interaction characteristics between aluminum content and crucible expansion coefficient are constructed as follows:

[0048] interaction feature =log(1+A1) content )×(expansion coefficient ) β ;

[0049] In the formula: β is an adjustable parameter, representing the degree of influence of the reaction expansion coefficient on the interaction; log(1+A1) content This indicates a logarithmic transformation of the aluminum content; expansion coefficient Indicates the coefficient of thermal expansion of the crucible; A1 content Indicates the aluminum content in an aluminum alloy; interaction feature This represents the interaction between aluminum content and crucible expansion coefficient, and is a continuous value.

[0050] The load characteristics of the melting time and temperature are constructed as follows:

[0051]

[0052] In the formula: melting timeλ represents the melting time; temperature represents the melting temperature; λ represents an adjustable hyperparameter that controls the rate at which the melting time decays with respect to the load effect; e represents the base of the natural logarithm; load feature It represents the load characteristics of melting time and temperature, and is a continuous value.

[0053] Preferably, the prediction model includes an input layer, a feature selection layer, a deep neural network layer, a regularization and optimization layer, and an output layer;

[0054] The input layer is used to input the newly constructed features and the selected important features, wherein the input is a complete feature vector formed by concatenating the newly constructed features and the selected important features;

[0055] The feature selection layer: The input feature vector passes through a self-attention mechanism to calculate the correlation between each feature and assign weights to each feature. The features are weighted based on the assigned weights to obtain weighted features. A Dropout layer is added after the self-attention mechanism, where the Dropout layer discards some features with a predetermined probability to obtain retained weighted features. The retained weighted features are then passed to the deep neural network layer.

[0056] The deep neural network layer includes multiple hidden layers. The output of each hidden layer is non-linearly transformed by the ReLU activation function. A Batch Normalization layer is added after each hidden layer for normalization. A Dropout layer is added after the Batch Normalization layer to discard some features with a predetermined probability, thus obtaining the retained deep features.

[0057] The regularization and optimization layers are used to apply L2 regularization constraints to the weights of the deep neural network layers to prevent the model from overfitting.

[0058] Output layer: The output of the deep neural network layer uses linear activation to output the predicted value of the deformation resistance rate.

[0059] Preferably, the loss function of the prediction model is as follows:

[0060]

[0061] In the formula: λ represents the mean squared error loss; l2 W represents the regularization hyperparameter of a deep neural network layer. attn λ represents the weight matrix in the self-attention mechanism. attn p represents the regularization hyperparameter in the self-attention mechanism. dropout,dnn p represents the dropout rate of the Dropout layer in a deep neural network. dropout,attnθ represents the dropout rate of the Dropout layer in the feature selection layer. j This represents the j-th parameter in the network; M represents the total number of parameters. This represents the total loss function.

[0062] The beneficial effects of this invention include:

[0063] This invention utilizes machine learning and deep learning technologies to achieve a highly efficient and accurate method for predicting the high-temperature deformation resistance of quartz crucibles. First, key historical data is collected from actual production, covering information such as aluminum alloy composition, crucible physical properties, usage conditions, and deformation resistance, providing a realistic and comprehensive training foundation for the model. Data preprocessing ensures the quality and consistency of the dataset. Statistical methods such as Pearson correlation coefficient, LASSO regression, and principal component analysis are used to screen features that significantly influence the deformation resistance, and feature engineering is performed to construct new interactive features that reflect the complex relationship between melting conditions and crucible physical properties. The introduction of a deep learning model enables the prediction model to learn deep patterns in the data, significantly improving prediction accuracy. Finally, the trained model can accurately predict the high-temperature deformation resistance of quartz crucibles based on real-time data, providing a scientific basis for production decisions and helping to optimize production processes and reduce costs. Attached Figure Description

[0064] To more clearly illustrate the technical solutions in the embodiments of this application, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0065] Figure 1 This is an overall step diagram provided for an embodiment of the present invention. Detailed Implementation

[0066] To make the technical problems, technical solutions, and beneficial effects to be solved by this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and are not intended to limit the scope of this application.

[0067] See Figure 1 As shown, the high-temperature deformation resistance prediction method for quartz crucibles based on machine learning is characterized by the following steps:

[0068] Step 1: Collect historical data related to the deformation resistance rate of quartz crucibles, including aluminum alloy composition data, crucible physical property data, crucible usage conditions, and deformation resistance rate data. The deformation resistance rate data is used as the true value label of the input data when training the corresponding prediction model.

[0069] The aluminum alloy composition data includes the content of aluminum, silicon, iron, and other elements (such as copper and magnesium); the aluminum alloy composition data is obtained based on chemical analysis techniques (such as X-ray fluorescence spectroscopy, laser-induced breakdown spectroscopy, etc.) or through formulation.

[0070] The physical property data of the crucible includes:

[0071] Density: The density of the crucible material affects its heat capacity and thermal conductivity;

[0072] Coefficient of thermal expansion: describes the expansion characteristics of crucible materials at high temperatures;

[0073] Compressive strength: The ability of a crucible material to withstand compressive forces.

[0074] Thermal conductivity: The thermal conductivity of crucible material affects its temperature distribution and thermal stability;

[0075] Specific heat capacity: The amount of heat absorbed per unit mass of crucible material, which affects the crucible's heat capacity;

[0076] Oxidation resistance: The oxidation resistance of crucible materials in high-temperature environments affects their service life;

[0077] Obtained through standard physical testing methods, such as using thermal expansion meters, thermal conductivity meters, and compressive strength testing equipment.

[0078] The crucible usage condition data includes:

[0079] Temperature: The temperature range that the crucible withstands during use, especially the operating temperature under high-temperature conditions.

[0080] Service life: The number of times or duration of use of the crucible affects the fatigue and deformation capacity of the material.

[0081] Type of smelting material: The type of metal being smelted (such as aluminum alloy, steel, copper, etc.) will affect the stress distribution in the crucible.

[0082] Melting speed: The speed at which metal melts is closely related to the crucible's resistance to deformation.

[0083] The environment in which the crucible is used: Environmental factors such as the oxygen concentration in the furnace and the atmosphere inside the furnace (oxidizing or reducing atmosphere) will also affect the performance of the crucible.

[0084] The deformation resistance data was obtained through experiments and existing records.

[0085] Step 2: Preprocess the collected historical data, including missing value handling, outlier detection and handling, data normalization, and categorical data encoding, to obtain preprocessed data;

[0086] For missing values, we use the median to fill them in. Outlier detection is performed by calculating the mean and standard deviation of the data, defining the normal range (e.g., data exceeding the mean ± 3 times the standard deviation is considered an outlier), and then removing or replacing outliers.

[0087] The normalization uses Z-score standardization: the data is transformed into a distribution with a mean of 0 and a standard deviation of 1 by subtracting the mean and dividing by the standard deviation;

[0088] Category data encoding: For data containing categories (such as classification information in aluminum alloy composition, type of smelting material in crucible usage conditions, etc.), category encoding is performed, for example, using unique thermal encoding: the category variables are converted into the form of a binary matrix, and each category becomes a binary feature. For example, if there are 3 categories for "aluminum alloy type", it can be converted into 3 binary columns.

[0089] The aforementioned preprocessing techniques are all conventional techniques in the field, and based on the above description, those skilled in the art can completely reconstruct the specific steps of the preprocessing based on the aforementioned techniques. Therefore, how to implement them will not be elaborated in this embodiment.

[0090] Step 3: Based on the preprocessed data, perform Pearson correlation coefficient, LASSO regression, and principal component analysis step by step to select important features;

[0091] As one possible implementation of this embodiment, the steps for processing the preprocessed data using the Pearson correlation coefficient are as follows:

[0092] Calculate the Pearson correlation coefficient between each feature and its corresponding resistance to deformation:

[0093]

[0094] In the formula: This represents the correlation coefficient between characteristic X and resistance to deformation Y; X i and Y i This represents the feature value and the target variable value of the i-th sample; and This represents the mean of characteristic X and resistance to deformation Y; n represents the number of samples.

[0095] If the correlation coefficient r between a certain feature and the target variable isX,Y If the absolute value of a feature is lower than a set first threshold, the feature is removed, and the first dataset is obtained.

[0096] For example, the first threshold is set to 0.2, and we have a total of five samples, each containing five aluminum contents, silicon contents and corresponding deformation resistance rates;

[0097] First, calculate the mean aluminum content, mean silicon content, and mean deformation resistance rate. Then, substitute the mean aluminum content and the corresponding mean deformation resistance rate into the formula for calculating the Pearson correlation coefficient between the above features and the corresponding deformation resistance rate. Next, substitute the aluminum content and the corresponding deformation resistance rate of each sample into the above formula to calculate the Pearson correlation coefficient between aluminum content and deformation resistance rate. The Pearson correlation coefficient between silicon content and the corresponding deformation resistance rate can be calculated by referring to the aluminum content calculation, i.e., the calculation logic is the same.

[0098] Finally, if the absolute value of the Pearson correlation coefficient between the obtained silicon content and the corresponding deformation resistance is less than 0.2, it is discarded; if it is greater, it is retained. This process continues until all features have been calculated in the above manner, and finally all retained features are obtained, forming the first dataset.

[0099] Based on the first dataset, calculate the Pearson correlation coefficient between each feature:

[0100]

[0101] In the formula: The correlation coefficient between feature X and feature Y′ is represented by X. i and Y′ i Let X and Y′ represent the values ​​of feature X and feature Y′ of the i-th sample, respectively. and Let X and Y′ represent the means of features X and Y′, respectively.

[0102] If the Pearson correlation coefficient r between any two feature pairs is X,Y′ If the absolute value of the feature pair is higher than the preset second threshold, then any one feature in the feature pair will be removed to obtain the second dataset.

[0103] The above calculation logic is the same as the Pearson correlation coefficient between the feature and the corresponding deformation resistance rate, only the data is different. The Pearson correlation coefficient is calculated between features, i.e., silicon content and aluminum content. Assuming we set the second threshold to 0.8; if the absolute value of the calculated Pearson correlation coefficient between silicon content and aluminum content is greater than 0.8, then any one of the features can be removed. By removing one of the two highly correlated features, the problem of multicollinearity is significantly reduced, improving the stability and generalization ability of the model. Moreover, the information provided by highly correlated features is redundant. Retaining features will increase the complexity of the model, resulting in long model training time and increased consumption of computing resources. By removing one of these highly correlated features, we reduce the number of parameters, reduce the training cost of the model, and after removal, the model will be more focused on the important features.

[0104] As one possible implementation of this embodiment, the steps for feature selection based on LASSO regression are as follows:

[0105] Cross-validation: The second dataset obtained based on the Pearson correlation coefficient is divided into several subsets. Each time, one subset is selected as the validation set and the other subsets are used as the training set. The mean squared error under each regularization parameter λ is calculated, and the regularization parameter λ with the smallest mean squared error is adopted.

[0106] Model Training: The LASSO regression model was trained using the second dataset obtained through Pearson correlation coefficient selection and the chosen regularization parameter λ. Features X and deformation resistance Y from the feature set were used. The model was trained by minimizing the loss function of the LASSO regression model, and the gradient descent method was employed to solve the optimization problem.

[0107]

[0108] In the formula: Y i X represents the deformation resistance rate of the i-th sample; ij β represents the j-th feature value of the i-th sample; j λ represents the regression coefficient of the j-th feature; λ represents the regularization parameter. This represents the regression coefficient vector obtained through training, containing the weights of all features, and representing the contribution of each feature to the resistance deformation rate, where p is the total number of features; β0 represents the intercept term of the regression model; β j Let represent the regression coefficient of the j-th feature, where j∈(1,2,…,p);

[0109] Feature selection: If β j If β = 0, then the corresponding feature is removed; j If the value is not equal to 0, the corresponding feature is retained; a third dataset is obtained based on feature selection.

[0110] For example:

[0111] The LASSO regression model takes the following form:

[0112]

[0113] In the formula: Indicates the predicted resistance to deformation; X1, X2, ..., X p These represent the values ​​of different features; X p Represents the intercept term;

[0114] Suppose we have the following simplified dataset, which includes 3 features (X1, X2, X3) and target variable Y (deformation resistance rate), with specific values ​​as shown in Table 1 below;

[0115] Table 1

[0116] Sample number X1 X2 X3 Y 1 1 2 3 4 2 2 3 4 5 3 3 4 5 6

[0117] The steps for feature selection based on the LASSO regression are as follows:

[0118] Initialization: Select an initial regularization parameter λ;

[0119] Loss function calculation: For each regularization parameter λ, calculate the value of the loss function. For example, for λ = 0.1:

[0120] Loss(λ)=(4-(b0+b1·1+b2·2+b3·3)) 2 +(5-(b0+b1·2+b2·3+b3·4)) 2 +(5-(b0+b1·3+b2·4+b3·5)) 2 +0.1(|b1|+|b2|+|b3|);

[0121] Gradient descent: The gradient descent method is used to update the coefficients b0, b1, b2, b3 in order to minimize the loss function Loss(λ);

[0122] Cross-validation: Repeat the initialization to gradient descent process described above, each time choosing a different λ value, and calculate the mean squared error on the validation set;

[0123] Choosing the optimal λ: Find the λ value that minimizes the mean square error on the validation set;

[0124] Final Model: The final model is trained using the optimal λ value and the entire dataset, and based on the coefficient β. j To filter features.

[0125] As one possible implementation of this embodiment, the specific steps of the principal component analysis are as follows:

[0126] Calculate the covariance matrix: Let X be the data matrix in the third dataset obtained after feature selection based on LASSO regression. norm The size is n×p, where n is the number of samples and p is the number of features. The covariance is calculated based on the data matrix.

[0127]

[0128] In the formula: Σ is the p×p covariance matrix, and each element σ ij This represents the covariance between the i-th feature and the j-th feature; Represents data matrix X norm Transpose of;

[0129] Calculate eigenvalues ​​and eigenvectors: Calculate the eigenvalues ​​and eigenvectors of the covariance matrix Σ.

[0130] Σv i =λ i v i ;

[0131] In the formula: λ i Let v represent the i-th eigenvalue of the covariance matrix, and v represent the variance along the eigendirection; i Let represent the i-th eigenvector of the covariance matrix, and let represent the eigendirection. The eigenvalues ​​and eigenvectors are solved using numerical computation software, such as NumPy, MATLAB, and Mathematica. How to use these software programs is a conventional technique in this field, and therefore will not be elaborated further.

[0132] Constructing a dimensionality reduction space: for eigenvalues ​​λ1, λ2, ..., λ p Sort the eigenvalues ​​by size, select the eigenvectors corresponding to the first k eigenvalues, and form a matrix V based on the eigenvectors corresponding to the first k eigenvalues. k , with a size of p×k, where p represents the number of features in the dataset obtained after feature selection based on LASSO regression, and k represents the number of principal components selected;

[0133] Mapping the data to a reduced-dimensional space: Calculate the coordinates of each data sample in the reduced-dimensional space using matrix multiplication.

[0134] Z = X norm V k ;

[0135] In the formula: Z represents the score matrix after dimensionality reduction, with a size of n×k, and each row represents the coordinates of a sample in the dimensionality reduction space;

[0136] Principal component selection: Calculate the ratio of the eigenvalues ​​corresponding to each principal component to the sum of the total eigenvalues ​​to obtain the contribution ratio of each principal component to the data variance.

[0137]

[0138] In the formula: G i This represents the contribution of the i-th principal component;

[0139] Calculate the cumulative contribution: Calculate the cumulative contribution of the top k principal components;

[0140]

[0141] In the formula: L k This represents the cumulative contribution of the first k principal components;

[0142] The calculated cumulative contribution is compared with a contribution threshold. If the cumulative contribution is higher than the threshold, a fourth dataset is constructed using the currently selected top k principal components. If the cumulative contribution is lower than the threshold, the value of k is increased until the cumulative contribution is higher than the threshold, thus obtaining the selected principal components. The fourth dataset is then constructed based on these selected principal components. For example, if the contribution threshold is set to 0.95 and k is 8, and the cumulative contribution of the 8 principal components corresponding to the features is greater than 0.95, no additional features are needed. If the cumulative contribution is lower than 0.95, k is incremented by 1 sequentially until the contribution is greater than 0.95.

[0143] In actual production, quickly and accurately predicting the deformation resistance of quartz crucibles is crucial for optimizing production processes, reducing downtime, and improving production efficiency. In this embodiment, by calculating the correlation coefficient between features and the target variable (deformation resistance), features highly correlated with the target variable can be initially screened. This helps to remove irrelevant or weakly correlated features, reduce data dimensionality, thereby reducing the risk of model overfitting and improving prediction accuracy. LASSO regression, by introducing a regularization term, imposes constraints on feature weights, further screening out features that significantly affect the deformation resistance while compressing or excluding unimportant features. This not only simplifies the model but also improves the ability to identify key features, thus enhancing the model's performance. The system possesses the ability to capture and predict complex relationships. Furthermore, by calculating the Pearson correlation coefficient between feature pairs, highly correlated feature pairs can be identified and eliminated, preventing the model from learning redundant information and thus optimizing feature combinations and improving the model's generalization ability. Finally, by selecting the principal component with the largest explained variance, the number of features can be reduced while retaining important information, thereby lowering the model's complexity and computational cost. Therefore, this embodiment progressively implements feature selection techniques using Pearson correlation coefficient, LASSO regression, and principal component analysis. In predicting the high-temperature deformation resistance of quartz crucibles, by optimizing feature quality and model structure, the accuracy, efficiency, and interpretability of the prediction model are significantly improved, providing strong technical support for the production process.

[0144] Step 4: Construct the interaction features between smelting conditions and physical properties based on the preprocessed data to obtain the constructed new features;

[0145] As one possible implementation of this embodiment, the interaction characteristics between the melting conditions and physical properties include the interaction characteristics between aluminum content and crucible expansion coefficient, and the load characteristics between melting time and temperature; wherein the interaction characteristics between aluminum content and crucible expansion coefficient are constructed as follows:

[0146] interaction feature =log(1+A1) content )×(expansion coefficient ) β ;

[0147] In the formula: β is an adjustable parameter, representing the degree of influence of the reaction expansion coefficient on the interaction; log(1+A1) content This indicates a logarithmic transformation of the aluminum content; expansion coefficient Indicates the coefficient of thermal expansion of the crucible; A1 content Indicates the aluminum content in an aluminum alloy; interaction feature This represents the interaction between aluminum content and crucible expansion coefficient, and is a continuous value.

[0148] The load characteristics of the melting time and temperature are constructed as follows:

[0149]

[0150] In the formula: melting time λ represents the melting time; temperature represents the melting temperature; λ represents an adjustable hyperparameter that controls the rate at which the melting time decays with respect to the load effect; e represents the base of the natural logarithm; load feature It represents the load characteristics of melting time and temperature, and is a continuous value.

[0151] Step 5: Train the prediction model based on the newly constructed features and the selected important features to obtain the trained prediction model;

[0152] As one possible implementation of this embodiment, the prediction model includes an input layer, a feature selection layer, a deep neural network layer, a regularization and optimization layer, and an output layer;

[0153] The input layer is used to input the newly constructed features and the selected important features, wherein the input is a complete feature vector formed by concatenating the newly constructed features and the selected important features;

[0154] The feature selection layer: The input feature vector passes through a self-attention mechanism to calculate the correlation between each feature and assign weights to each feature. The features are weighted based on the assigned weights to obtain weighted features. A Dropout layer is added after the self-attention mechanism, where the Dropout layer discards some features with a predetermined probability (0.2 or 0.3) to obtain retained weighted features. The retained weighted features are then passed to the deep neural network layer.

[0155] The deep neural network layer includes multiple hidden layers. The output of each hidden layer is non-linearly transformed by the ReLU activation function. A Batch Normalization layer is added after each hidden layer for normalization. A Dropout layer is added after the Batch Normalization layer to discard some features with a predetermined probability, thus obtaining the retained deep features.

[0156] For example, a three-layer hidden layer can be used. The first hidden layer contains 128 neurons and uses the ReLU activation function; the second hidden layer contains 64 neurons and uses the ReLU activation function; and the third hidden layer contains 32 neurons and uses the ReLU activation function.

[0157] After the output of each hidden layer, a Batch Normalization layer is added to normalize the output of each hidden layer, ensuring that the mean of the output is 0 and the variance is 1. Batch Normalization helps to speed up training and reduces the network's sensitivity to initialization.

[0158]

[0159] Where: h BN represents the output vector of the Batch Normalization layer; h represents the output feature vector of the hidden layer; μ represents the mean of the current batch features; σ represents the standard deviation of the current batch features; γ and β represent the learnable parameters;

[0160] Each hidden layer is followed by a Dropout layer to prevent overfitting. For example, a Dropout probability of 0.3 means that 30% of neurons will be randomly dropped during training.

[0161] The regularization and optimization layers are used to apply L2 regularization constraints to the weights of the deep neural network layers to prevent the model from overfitting.

[0162] The regularization is achieved by adding the sum of squares of the weights to the loss function:

[0163]

[0164] In the formula: L2loss represents the loss due to regularization; λ represents the regularization coefficient; W i This represents the i-th weight of the model; please refer to the loss function for details.

[0165] The Adam optimizer is used during model training, combining gradient descent and adaptive learning rate adjustment to optimize the model;

[0166] Output layer: The output of the deep neural network layer uses linear activation to output the predicted value of the deformation resistance rate.

[0167] The loss function of the prediction model is as follows:

[0168]

[0169] In the formula: λ represents the mean squared error loss; l2 W represents the regularization hyperparameter of a deep neural network layer. attn λ represents the weight matrix in the self-attention mechanism. attn p represents the regularization hyperparameter in the self-attention mechanism. dropout,dnn p represents the dropout rate of the Dropout layer in a deep neural network.dropout,attn θ represents the dropout rate of the Dropout layer in the feature selection layer. j This represents the j-th parameter in the network; M represents the total number of parameters. This represents the total loss function. This represents the regularization loss in the attention mechanism; This represents the regularization term in a deep neural network, which penalizes the squares of all parameters in the network. Specifically, each hidden layer includes a weight and a bias, and the squares of the weights and biases of each hidden layer are penalized.

[0170] In this embodiment, through the self-attention mechanism, the model can automatically learn and identify the correlation between input features, assign different weights to each feature, thereby highlighting important features and suppressing unimportant features, improving the efficiency and quality of feature representation; a Dropout layer is introduced after each hidden layer of the feature selection layer and the deep neural network layer, which reduces the model's over-reliance on training data and reduces the risk of overfitting by randomly dropping a certain proportion of neurons.

[0171] The specific steps for optimizing the model using the Adam optimizer, combined with gradient descent and adaptive learning rate adjustment, are as follows:

[0172] Initialization: For each parameter θ of the prediction model (such as the weight matrix in the self-attention mechanism, the weight matrix and bias term in the deep neural network layer, and the learnable parameters γ and β in the Batch Normalization layer), initialize its value to a small random number from a normal or uniform distribution; set the learning rate η, representing the step size of parameter updates during model training; set two decay rates β1 and β2, used to calculate the exponential decay rate of the first and second moment estimates of the gradient, respectively; set a small constant ∈, which is 10. -8 ;

[0173] Training: For each iteration t:

[0174] Forward propagation: The input data is propagated forward through the model to obtain the predicted value;

[0175] Calculate the loss: Calculate the loss function

[0176] Backpropagation: Perform backpropagation to calculate the loss function with respect to each parameter θ and the gradient g. t ;

[0177] Update first-order moment estimate m t Update the first-moment estimate (i.e., the exponentially weighted moving average of the gradient) based on the gradient and the decay rate β1:

[0178] mt =β1·m t-1 +(1-β1)·g t ;

[0179] Where: m t m represents the order moment estimate of the t-th iteration; t-1 G represents the order moment estimate of the (t-1)th iteration; t Let represent the gradient at the t-th iteration;

[0180] Update the second moment estimate v t Update the second-order moment estimate (i.e., the exponentially weighted moving average of the squared gradient) based on the gradient and the decay rate β2:

[0181]

[0182] In the formula: v t v represents the second-order moment estimate of the t-th iteration; t-1 Let represent the second moment estimate for the (t-1)th iteration;

[0183] Bias correction: Bias correction is performed on the first and second moment estimates.

[0184]

[0185] In the formula: This represents the first-order moment estimate after bias correction; This represents the second-order moment estimate after bias correction;

[0186] Update parameters: Update parameter θ using the bias-corrected estimate.

[0187]

[0188] In the formula: θ t Indicates the parameters before the update; θ t+1 This indicates the updated parameters;

[0189] Repeated iteration: Repeat the forward propagation to parameter update steps until the stopping condition is met, such as reaching the predetermined number of iterations or the value of the loss function no longer decreasing significantly.

[0190] Step 6: Based on the trained prediction model and real-time data, predict the high-temperature deformation resistance of the quartz crucible and obtain the prediction results.

[0191] This invention utilizes machine learning and deep learning technologies to achieve a highly efficient and accurate method for predicting the high-temperature deformation resistance of quartz crucibles. First, key historical data is collected from actual production, covering information such as aluminum alloy composition, crucible physical properties, usage conditions, and deformation resistance, providing a realistic and comprehensive training foundation for the model. Data preprocessing ensures the quality and consistency of the dataset. Statistical methods such as Pearson correlation coefficient, LASSO regression, and principal component analysis are used to screen features that significantly influence the deformation resistance, and feature engineering is performed to construct new interactive features that reflect the complex relationship between melting conditions and crucible physical properties. The introduction of a deep learning model enables the prediction model to learn deep patterns in the data, significantly improving prediction accuracy. Finally, the trained model can accurately predict the high-temperature deformation resistance of quartz crucibles based on real-time data, providing a scientific basis for production decisions and helping to optimize production processes and reduce costs.

[0192] The above are merely preferred embodiments of this application and are not intended to limit this application. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A method for predicting the high-temperature deformation resistance of quartz crucibles based on machine learning, characterized in that, Includes the following steps: Step 1: Collect historical data related to the deformation resistance rate of quartz crucibles, including aluminum alloy composition data, crucible physical property data, crucible usage conditions, and deformation resistance rate data. The deformation resistance rate data is used as the true value label of the input data when training the corresponding prediction model. Step 2: Preprocess the collected historical data, including missing value handling, outlier detection and handling, data normalization, and categorical data encoding, to obtain preprocessed data; Step 3: Based on the preprocessed data, perform Pearson correlation coefficient, LASSO regression, and principal component analysis step by step to select important features; Step 4: Construct the interaction features between smelting conditions and physical properties based on the preprocessed data to obtain the constructed new features; The interaction characteristics between the melting conditions and physical properties include the interaction characteristics between aluminum content and crucible expansion coefficient, and the load characteristics between melting time and temperature. The interaction characteristics between aluminum content and crucible expansion coefficient are constructed as follows: ; In the formula: The degree of influence of the reaction expansion coefficient on the interaction is an adjustable parameter. This indicates a logarithmic transformation of the aluminum content; Indicates the coefficient of thermal expansion of the crucible; This indicates the aluminum content in the aluminum alloy; This represents the interaction between aluminum content and crucible expansion coefficient, and is a continuous value. The load characteristics of the melting time and temperature are constructed as follows: ; In the formula: Indicates the smelting time; Indicates the melting temperature; This represents an adjustable hyperparameter that controls the rate at which the effect of melting time on the load decays. The base of the natural logarithm; It represents the load characteristics of melting time and temperature, and is a continuous value; Step 5: Train the prediction model based on the newly constructed features and the selected important features to obtain the trained prediction model; Step 6: Based on the trained prediction model and real-time data, predict the high-temperature deformation resistance of the quartz crucible and obtain the prediction results; The steps for processing the preprocessed data using the Pearson correlation coefficient are as follows: Calculate the Pearson correlation coefficient between each feature and its corresponding resistance to deformation: In the formula: Representation of features With resistance to deformation The correlation coefficient between them; and This represents the feature value and the target variable value of the i-th sample; and Representation of features and resistance to deformation The mean; n represents the number of samples; If the correlation coefficient between a certain feature and the target variable is If the absolute value of a feature is lower than a set first threshold, the feature is removed, and the first dataset is obtained. Based on the first dataset, calculate the Pearson correlation coefficient between each feature: ; In the formula: Representation of features and characteristics The correlation coefficient between them and Represent the features of the i-th sample respectively. and characteristics The value; and Representing features respectively and characteristics The mean; If the Pearson correlation coefficient between any two feature pairs is If the absolute value of the feature pair is higher than the preset second threshold, then any one feature in the feature pair will be removed to obtain the second dataset. The steps for feature selection based on the LASSO regression are as follows: Cross-validation: The dataset selected based on Pearson correlation coefficient is divided into several subsets. Each time, one subset is selected as the validation set, and the other subsets are used as the training set. Each regularization parameter is calculated. The mean squared error is used to determine the regularization parameter that minimizes the mean squared error. ; Model training: Dataset selected based on Pearson correlation coefficient and chosen regularization parameters Training the LASSO regression model using features from the feature set and resistance to deformation The model is trained by minimizing the loss function of the LASSO regression model, and the optimization problem is solved using gradient descent. ; In the formula: This represents the deformation resistance rate of the i-th sample; This represents the j-th feature value of the i-th sample; Represents the regression coefficient of the j-th feature; Represents the regularization parameter; This represents the regression coefficient vector obtained through training, containing the weights of all features, where represents the contribution of each feature to the resistance deformation rate. The total number of features; This represents the intercept term of the regression model; Indicates the first The regression coefficients of each feature, where ; Feature filtering: If If so, then remove the corresponding feature; Then the corresponding features are retained; The third dataset was obtained based on feature selection.

2. The method for predicting the high-temperature deformation resistance of quartz crucibles based on machine learning according to claim 1, characterized in that, The specific steps of the principal component analysis are as follows: Calculate the covariance matrix: Let the data matrix in the dataset obtained after feature selection based on LASSO regression be... Size is ,in For the sample size, To determine the number of features, calculate the covariance based on the data matrix: ; In the formula: for The covariance matrix, each element This represents the covariance between the i-th feature and the j-th feature; Representing a data matrix transpose; Calculate eigenvalues ​​and eigenvectors: Calculate the covariance matrix Eigenvalues ​​and eigenvectors: In the formula: Let represent the i-th eigenvalue of the covariance matrix, and let represent the variance along the eigendirection. Let represent the i-th eigenvector of the covariance matrix, and let represent the eigendirection. Constructing a dimensionality reduction space: for eigenvalues Sort by size, select the eigenvectors corresponding to the first k eigenvalues, and construct a matrix based on the eigenvectors corresponding to the first k eigenvalues. Size is ,in represents the number of features in the dataset obtained after feature selection based on LASSO regression, and k represents the number of principal components selected. Mapping the data to a reduced-dimensional space: Calculate the coordinates of each data sample in the reduced-dimensional space using matrix multiplication. ; In the formula: This represents the score matrix after dimensionality reduction, with a size of . Each row represents the coordinates of a sample in the reduced-dimensional space; Principal component selection: Calculate the ratio of the eigenvalues ​​corresponding to each principal component to the sum of the total eigenvalues ​​to obtain the contribution ratio of each principal component to the data variance. ; In the formula: This represents the contribution of the i-th principal component; Calculate the cumulative contribution: Calculate the cumulative contribution of the top k principal components; ; In the formula: This represents the cumulative contribution of the first k principal components; The calculated cumulative contribution is compared with the contribution threshold. If the cumulative contribution is higher than the threshold, the fourth dataset is constructed using the currently selected top k principal components. If the cumulative contribution is lower than the threshold, the value of k is increased until the cumulative contribution is higher than the threshold. The selected principal components are then obtained, and the fourth dataset is constructed based on the selected principal components.

3. The method for predicting the high-temperature deformation resistance of quartz crucibles based on machine learning according to claim 1, characterized in that, The prediction model includes an input layer, a feature selection layer, a deep neural network layer, a regularization and optimization layer, and an output layer. The input layer is used to input the newly constructed features and the selected important features, wherein the input is a complete feature vector formed by concatenating the newly constructed features and the selected important features; The feature selection layer: The input feature vector passes through a self-attention mechanism to calculate the correlation between each feature and assign weights to each feature. The features are weighted based on the assigned weights to obtain weighted features. A Dropout layer is added after the self-attention mechanism, where the Dropout layer discards some features with a predetermined probability to obtain retained weighted features. The retained weighted features are then passed to the deep neural network layer. The deep neural network layer includes multiple hidden layers. The output of each hidden layer is non-linearly transformed by the ReLU activation function. A Batch Normalization layer is added after each hidden layer for normalization. A Dropout layer is added after the Batch Normalization layer to discard some features with a predetermined probability, thus obtaining the retained deep features. The regularization and optimization layers are used to apply L2 regularization constraints to the weights of the deep neural network layers to prevent the model from overfitting. Output layer: The output of the deep neural network layer uses linear activation to output the predicted value of the deformation resistance rate.

4. The method for predicting the high-temperature deformation resistance of quartz crucibles based on machine learning according to claim 3, characterized in that, The loss function of the prediction model is as follows: ; In the formula: Indicates the mean square error loss; This represents the regularization hyperparameter of a deep neural network layer; This represents the weight matrix in the self-attention mechanism; This represents the regularization hyperparameter in the self-attention mechanism; This represents the dropout rate of the Dropout layer in a deep neural network. This represents the dropout rate of the Dropout layer in the feature selection layer; This represents the j-th parameter in the network; M represents the total number of parameters. This represents the total loss function.

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