Bent tube forming process prediction method based on truncated gaussian function virtual sample expansion

Abstract

The application discloses a kind of based on truncated gaussian function virtual sample expansion elbow forming process prediction method.It includes the following steps: constructing original sample dataset, establishing expansion domain range;Using original sample dataset to train and fuse multiple proxy models, obtain hybrid proxy prediction model;According to expansion domain range and hybrid proxy model, generate virtual sample dataset by multi-objective optimization algorithm;After virtual sample dataset and original sample dataset are combined, train hybrid proxy prediction model, obtain hybrid proxy model based on transmission;Using hybrid proxy model based on transmission, according to the process parameter to be predicted Prediction cross-section distortion rate.The method of the application effectively overcomes the defects of small sample size and low prediction accuracy in the field of mechanical engineering, by generating virtual sample dataset, increasing the sample size and prediction stability, and applying high-precision transmission hybrid proxy model, improving the prediction accuracy of the prediction model on cross-section distortion rate.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bend forming, in particular to a bend forming process prediction method based on truncated Gaussian function virtual sample expansion. BACKGROUND

[0002] In recent years, with the rapid development of advanced manufacturing technology and the increasing demand for personalization, the application of pipe design in engineering structures is becoming more and more widespread. However, in the actual pipe design process, due to the limitation of experimental conditions and the increase of manufacturing cost, the data samples available for design optimization are often very limited. Especially under complex mechanical environment, the mechanical performance analysis and optimization of pipe require a large amount of high-precision experimental data, and obtaining these data often requires a lot of time and resources.

[0003] At present, the prediction modeling under small sample condition mainly depends on surrogate models, such as response surface model, support vector machine and neural network, etc. These models extract the rules from limited experimental data to realize the prediction of pipe performance. However, when the sample size is insufficient, the prediction accuracy of surrogate model may be greatly affected. By generating virtual samples, the deficiency of original samples in feature space can be made up, so as to improve the prediction performance of surrogate model. However, the existing virtual sample expansion methods mostly rely on traditional interpolation or extrapolation technology, and the quality of generated samples is difficult to guarantee, and the real distribution of original data cannot be accurately reflected. SUMMARY

[0004] In view of the deficiencies of the prior art, the present application provides a bend forming process prediction method based on truncated Gaussian function virtual sample expansion. The method combines the virtual sample expansion method based on truncated Gaussian function with the transfer mixed surrogate model, and is used for small sample performance prediction in the production of bent pipe.

[0005] The technical scheme adopted by the present application is as follows:

[0006] The bend forming process prediction method comprises the following steps:

[0007] S1, obtaining different process parameter combinations and corresponding cross-section distortion data in the pipe bending forming process, and constructing an original sample data set.

[0008] The process of step S1 is specifically as follows:

[0009] S1.1, selecting at least one process parameter in the pipe bending forming process, pre-setting the value range of each process parameter, and generating a plurality of process parameter combination samples according to the value range of all process parameters by Latin hypercube sampling method, each process parameter combination sample being composed of specific values of all process parameters;

[0010] S1.2, respectively establish respective simulation models according to each process parameter combination sample, and obtain corresponding cross-section distortion data by simulating the simulation models corresponding to each process parameter combination sample;

[0011] Specifically, the cross-section distortion data includes cross-section distortion rates at multiple different positions of the bent part of the pipe;

[0012] S1.3, the specific values of all process parameters in each process parameter combination sample and the corresponding cross-section distortion data are combined into a sample pair, and all sample pairs constitute an original sample data set.

[0013] Preferably, the process parameter combination mainly consists of six process parameter combinations of core ball diameter, core ball width, core ball spacing, initial elongation, core ball number and bending radius.

[0014] Preferably, the cross-section distortion data includes cross-section distortion rates at 0, 1 / 4, 1 / 2, 3 / 4 and the end of the bent part of the pipe.

[0015] S2, obtain an extended domain range according to the original sample data set; simultaneously train multiple proxy models using the original sample data set, fuse all trained proxy models to obtain a hybrid proxy prediction model; then generate a virtual sample data set according to the extended domain range and the hybrid proxy model through a multi-objective optimization algorithm; and combine the virtual sample data set and the original sample data set into a hybrid sample data set.

[0016] The process of step S2 is specifically:

[0017] S2.1, normalize the original sample data set to obtain a normalized sample data set;

[0018] S2.2, use the truncated-experience Gaussian membership function as a domain range expansion tool to expand the normalized sample data set to obtain an expanded domain range;

[0019] S2.3, train multiple proxy models using the normalized sample data set, fuse all trained proxy models according to the prediction results of each trained proxy model to obtain a hybrid proxy prediction model;

[0020] The fusion of all trained proxy models is specifically: the error verification method is used to obtain the error verification results of each trained proxy model as weights, and all trained proxy models are weighted and summed;

[0021] The proxy model includes Kriging model, radial basis function and support vector machine;

[0022] Specifically, in step S2.3, all trained agent models are fused according to the following formula to obtain a hybrid agent prediction model:

[0023]

[0024] wherein F HSM represents the hybrid agent prediction model, f1(x) represents the trained Kriging model, R1 represents the mean square error of the prediction error of the trained Kriging model, f2(x) represents the trained radial basis function, R2 represents the mean square error of the prediction error of the trained radial basis function, f3(x) represents the trained support vector machine function, and R3 represents the mean square error of the prediction error of the trained support vector machine function;

[0025] S2.4, using a multi-objective optimization algorithm, taking the hybrid agent prediction model obtained in step S2.3 as a fitness function, taking the cross-section distortion rate at each position as the target to be minimized, taking the extended domain range obtained in step S2.2 as the new sampling space of each process parameter, generating a plurality of virtual process parameter combination samples through non-dominated sorting, and all virtual process parameter combination samples form a virtual sample dataset, and the virtual sample dataset and the normalized sample dataset are combined into a hybrid sample dataset.

[0026] S3, using the hybrid sample dataset to train the hybrid agent prediction model to obtain a transfer-based hybrid agent model.

[0027] The expression of the transfer-based hybrid agent model is as follows:

[0028]

[0029] wherein f1'(x) represents a Kriging model fitted using the hybrid sample dataset, f2'(x) represents a radial basis function model fitted using the hybrid sample dataset, f3'(x) represents a support vector regression machine model fitted using the hybrid sample dataset, R1 represents the mean square error of the prediction error of the trained Kriging model, R2 represents the mean square error of the prediction error of the trained radial basis function, and R3 represents the mean square error of the prediction error of the trained support vector machine function.

[0030] S4, using the transfer-based hybrid agent model to predict cross-section distortion data according to to-be-predicted process parameters.

[0031] The beneficial effects of the present application are as follows:

[0032] 1) Improved prediction accuracy under small sample data conditions: This invention generates virtual samples by using a truncated empirical Gaussian membership function, which compensates for the insufficient amount of original sample data, thereby improving the model prediction accuracy under small sample conditions. It is especially suitable for scenarios where experimental data is difficult to obtain or is costly, helping the model to better capture the distribution characteristics of the data.

[0033] 2) Performance optimization for complex structure design: This invention is particularly suitable for the design and optimization of complex structures (such as pipe fittings). In the design process of pipe fittings, such as bending and forming, and pressure bearing, virtual sample expansion can more accurately predict the performance after forming, optimize design parameters, and reduce design cycle and production costs.

[0034] 3) Adaptive modeling under multiple operating conditions: The hybrid proxy model of this invention can adapt to changes in different operating conditions and material parameters, and is particularly suitable for scenarios that require frequent switching of production operating conditions. In the early stage of operating condition switching, the virtual sample augmentation technology makes up for the lack of labeled data, improves the predictive ability of the model in the early stage, and ensures the stability of product quality. Attached Figure Description

[0035] Fig. 1 This is the domain extension based on the truncated-empirical Gaussian function and the distribution map of observation points and virtual sample points in this invention.

[0036] Fig. 2 This is a flowchart of the pipe bending forming process prediction method based on virtual sample expansion of truncated Gaussian function in this invention.

[0037] Fig. 3 This is a schematic diagram describing the mandrel parameters in this invention.

[0038] Fig. 4 These are cross-sectional profile diagrams and cross-sectional distortion diagrams of the bend in this invention at several different locations. Detailed Implementation

[0039] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0040] Specific embodiments of the present invention are as follows:

[0041] This invention uses a rotary traction bending forming process as an example. In rotary traction bending, cross-sectional distortion data is typically used to measure the bending quality of pipe fittings. However, the cross-sectional distortion rate cannot be measured using online sensors and can only be obtained through offline detection and experimental analysis. This results in significant time lag and leads to the production of defective products. Furthermore, the pipe fitting manufacturing process often requires frequent adjustments to forming parameters based on different bending radii and material properties. This leads to insufficient collected tagged data in the initial operation phase after process parameter switching, making it difficult to establish an accurate forming quality prediction model.

[0042] See Figs. 1-4 In this embodiment, the specific steps of the pipe bending forming process prediction method based on virtual sample expansion using truncated Gaussian functions are as follows:

[0043] S1. Obtain different combinations of process parameters and corresponding cross-sectional distortion data during the bending and forming of the pipe fitting, and construct the original sample dataset. In this embodiment, the cross-sectional distortion data comes from the simulation model of the pipe fitting's rotary traction bending.

[0044] The process of step S1 is as follows:

[0045] S1.1 Select at least one key process parameter in the bending and forming process of the pipe fitting, pre-set the value range of each process parameter, and generate a preset number (pre-set to 120 in this embodiment) of process parameter combination samples based on the value range of all process parameters using the Latin hypercube sampling method. Combine the 120 process parameter combination samples into a sample set. Each process parameter combination sample is a combination of all process parameters, and these process parameters are presented in the form of specific numerical values.

[0046] In this embodiment, six process parameters are selected to form a process parameter combination. These six parameters are: core ball diameter, core ball width, core ball spacing, initial elongation, number of core balls, and bending radius. The value ranges of each process parameter are set according to the empirical formulas in the table below:

[0047]

[0048] S1.2. Using the simulation software ABAQUS (Engineering Simulation Finite Element Software), establish corresponding simulation models for each combination of process parameters. Obtain the corresponding cross-sectional distortion data by simulating the simulation models corresponding to each combination of process parameters.

[0049] The cross-sectional distortion data includes the cross-sectional distortion rate at five locations: 0, 1 / 4, 1 / 2, 3 / 4, and the end of the bend.

[0050] The cross-sectional distortion rate, namely the short-axis and long-axis distortion rates of the cross-section, is mainly obtained by processing the short-axis distortion rate and the long-axis distortion rate using the following formula:

[0051] Cross-sectional distortion rate = minor axis distortion rate × major axis distortion rate

[0052]

[0053] Wherein, the minor axis is the minor axis of the approximately elliptical cross-section of the pipe fitting after distortion, and the major axis is the major axis of the approximately elliptical cross-section of the pipe fitting after distortion.

[0054] S1.3. Combine the specific values ​​of all process parameters and the corresponding cross-sectional distortion data in each process parameter combination sample into a sample pair. All sample pairs constitute the original sample dataset.

[0055] The original sample dataset is represented as: {x train ,y train},in, These are input variables, consisting of the specific values ​​of six process parameters. The output variable is the cross-sectional distortion data, which consists of the cross-sectional distortion rate at five locations. N1 is the number of sample pairs. In this embodiment, N1 = 120.

[0056] S2. Obtain the extended domain range based on the original sample dataset; train multiple surrogate models using the original sample dataset, and fuse all trained surrogate models based on the prediction results of each trained surrogate model to obtain a hybrid surrogate prediction model; generate a virtual sample dataset based on the extended domain range and the hybrid surrogate model using a multi-objective optimization algorithm; merge the virtual sample dataset and the normalized sample dataset into a hybrid sample dataset.

[0057] The process of step S2 is as follows:

[0058] S2.1. Divide the original sample dataset into training and test sets in an 8:2 ratio, and normalize the original sample dataset to obtain a normalized sample dataset. Normalization accelerates model convergence and reduces training time. Normalization is performed according to the following formula:

[0059]

[0060] In the formula, x represents the normalized process parameter, a represents the original process parameter data, and a min a represents the minimum value in the original data of the process parameters. max This represents the maximum value in the original data of the process parameters.

[0061] S2.2. The truncated-empirical Gaussian membership function is used as a domain range expansion tool to expand the normalized sample dataset, resulting in an expanded domain range. The truncated Gaussian membership function is composed of a Gaussian membership function and a cutoff line. The cutoff line restricts the range of the Gaussian function, thus forming an expanded domain range. The truncated-empirical Gaussian membership function, on the other hand, further optimizes the applicability of the function by intersecting with the empirical data range based on this expanded domain range.

[0062] Step S2.2 specifically includes:

[0063] Set up sampling sets for each of the six process parameters. Where X1, X2, X3, X4, X5, and X6 represent the sampling sets corresponding to the core ball diameter, core ball width, core ball spacing, initial elongation, number of core balls, and bending radius, respectively, and N1 represents the number of virtual sample points in each sampling set. In this embodiment, N1 = 120.

[0064] The centroid of the sampling set corresponding to each process parameter is the cutoff line CL. i CL cut-off line i Set it according to the following formula:

[0065]

[0066] In the formula, CL i Represents the sample set X i The cutoff line, N1 represents the number of virtual sample points X in each sampling set. ij Represents the sample set X i The j-th sample point.

[0067] The left and right skewness of the sampling set corresponding to each process parameter are set according to the following formulas:

[0068]

[0069] In the formula, sk iL Represents the sample set X i The left skewness, sk iU Represents the sample set X i The right skewness, N iL Represents the sample set X i Medium smaller than the cutoff line CL i The number of sample points, N iU Represents the sample set X i Medium greater than the cutoff line CL i The number of sample points, s p This represents the adaptive parameter.

[0070] For each process parameter corresponding to the sampling set, the cutoff line CL is calculated based on the initial 120 sets of sampling point data. i Left skewness sk iL and right skewness sk iU The values ​​of the cutoff line, left skewness, and right skewness vary depending on the process parameters. Subsequently, based on the cutoff line CL... i Left skewness sk iL and right skewness sk iU The value of is combined with the minimum value min and the maximum value max of the domain range X to obtain point A(CL). i 1) Point B(min, sk) iL ) and point C(max, sk iU The coordinates of the points are determined, and then the shape of the Gaussian function curve is determined based on the three points A, B, and C.

[0071] Using the cutoff line of the sampling set corresponding to each process parameter as the offset variable of the Gaussian function, the cutoff line and offset variable of the sampling set corresponding to each process parameter are processed using the following formula to obtain the upper and lower bounds of the sampling set corresponding to each process parameter:

[0072]

[0073] In the formula, UB i Represents the sample set X i The upper bound, LB i Represents the sample set X i The lower bound, CL i Represents the sample set X i The cutoff line, β i Represents the sample set X i The offset variable, c i1 Represents the sample set X i The corresponding first coefficient, c i2 Represents the sample set X i The corresponding second coefficient.

[0074] Wherein, the sampling set X i The corresponding first coefficient c i1 Second coefficient c i2 They are obtained using the following formulas respectively:

[0075]

[0076] In the formula, c1 represents the sampling set X i The corresponding first coefficient, c2, represents the sample set X. i The corresponding second coefficient, min, represents the sample set X. i The minimum value, where max represents the sample set X. iThe maximum value of CL i Represents the sample set X i The cut-off line, sk iL Represents the sample set X i The left skewness, sk iU Represents the sample set X i The rightward skewness.

[0077] Obtain the sampling set X corresponding to each process parameter. i Upper UB i and lower bound LB i The intersection with the preset range of the process parameters yields the upper and lower bounds of the domain expansion for each process parameter within the preset range, thus obtaining the expanded domain range of each process parameter.

[0078] S2.3. Train multiple proxy models using a normalized sample dataset, and then fuse all the trained proxy models based on their prediction results to obtain a hybrid proxy prediction model.

[0079] Specifically, the process of integrating all trained surrogate models involves using an error verification method to obtain the error verification results of each trained surrogate model as weights, and then using the weights of the trained surrogate models to perform a weighted summation of all trained surrogate models. This results in the highly fitted surrogate model having a larger weight in the hybrid surrogate model, thereby best fitting the original data.

[0080] In this embodiment, three surrogate models are used: Kriging, Radial Basis Function (RBF), and Support Vector Regression (SVR). The Kriging model has a hyperparameter θ = 10 and a domain range of [0.1, 20]. The RBF has a mean squared error target of 10^(-20), a maximum of 100 hidden neurons, and a spread factor of 1.0. The SVR uses an exponential kernel function, and both the gamma function and the SVR loss function are powers of 2. The SVR loss function is 0.1. In this embodiment, the reciprocal of the square of the distance between the prediction result and the true value of each surrogate model is used as the weight, so that the smaller the error, the greater the weight.

[0081] Step S2.3 specifically involves: training three surrogate models using the training set from the normalized sample dataset, resulting in three trained surrogate models (f1(x), f2(x), and f3(x)). Prediction results from the three trained surrogate models are obtained using the test set from the normalized sample dataset. The reciprocal of the squared mean square of the prediction error is used as the weight value to perform a weighted summation of the three trained surrogate models, yielding a Hybrid Surrogate Prediction Model (HSM model). The expression for the Hybrid Surrogate Prediction Model (HSM model) is as follows:

[0082]

[0083] In the formula, F HSM Let f1(x) represent the expression for the hybrid surrogate prediction model, f2(x) represent the trained Kriging model, R1 represent the mean squared error of the trained Kriging model, f2(x) represent the trained radial basis function, R2 represent the mean squared error of the trained radial basis function, f3(x) represent the trained support vector machine function, and R3 represent the mean squared error of the trained support vector machine function.

[0084] S2.4 Using a multi-objective optimization algorithm, the hybrid surrogate prediction model obtained in step S2.3 is used as the fitness function, the objective is to minimize the cross-sectional distortion rate at different locations of the pipe bending section, and the extended domain range obtained in step S2.2 is used as the new sampling space for each process parameter. Multiple virtual process parameter combination samples are generated through non-dominated sorting. All virtual process parameter combination samples form a virtual sample dataset. The virtual sample dataset and the normalized sample dataset are merged into a hybrid sample dataset.

[0085] In multi-objective optimization algorithms, expanding the domain range enhances the coverage of the solution space, enabling the algorithm to explore a wider range of solutions and thus obtain more potential Pareto optimal solutions. This strategy is crucial for handling complex multi-objective optimization problems, especially when the scope and complexity of the objective space are large, as expanding the domain range can significantly improve the optimization process.

[0086] In this embodiment, NSGA-III (Non-dominated Sorting Genetic Algorithm III) is used as the optimization algorithm. During the optimization process, each individual consists of specific values ​​of six process parameters. After inputting the individual into the fitness function, the cross-sectional distortion rate at five locations is obtained. Non-dominated sorting is performed based on these five cross-sectional distortion rates. After reaching a preset number of iterations or the Pareto front solution set meets the preset iteration conditions, the Pareto front solution set is used as the final virtual sample dataset. In this embodiment, a virtual sample dataset consisting of 400 high-quality virtual samples was obtained using the NSGA-III optimization algorithm.

[0087] In this embodiment, the preset conditions include that the average distortion rate of each cross section is less than a preset threshold and / or that the average distortion rate of the cross section after a certain iteration is almost no less than that before the iteration.

[0088] S3. Establish a transitive hybrid agent model: Train the hybrid agent prediction model obtained in step S2.3 using the mixed sample dataset to obtain the transitive hybrid agent model (THSM model).

[0089] The expression for the transitive hybrid proxy model (THSM model) is as follows:

[0090]

[0091] In the formula, f1'(x) represents the Kriging model fitted using a mixed sample dataset, f2'(x) represents the radial basis function model fitted using a mixed sample dataset, f3'(x) represents the support vector regression machine model fitted using a mixed sample dataset, R1 represents the mean squared error of the prediction error of the Kriging model after training, R2 represents the mean squared error of the prediction error of the radial basis function after training, and R3 represents the mean squared error of the prediction error of the support vector machine function after training.

[0092] Step S3, based on a transfer strategy, retains the weights of the three individual surrogate models and transfers these weights to the final prediction model, i.e., the transfer-based hybrid surrogate model. This process transfers the weights with high fitness in fitting the original data to the fitting of the mixed sample dataset, ensuring that surrogate models with good fitness for the dataset have a high weight ratio, while surrogate models with poor fitness have a low weight ratio.

[0093] S4. Using the transitive hybrid proxy model obtained in step S3, predict the cross-sectional distortion data based on the process parameters to be predicted.

[0094] In this embodiment, mean absolute error (MAE) and mean squared error (MSE) are used to characterize the prediction performance of the transmission-based hybrid surrogate model.

[0095] Mean Absolute Error (MAE) primarily characterizes the average deviation of prediction errors, reducing sensitivity to outliers. A smaller MAE indicates better prediction performance. The MAE is set according to the following formula:

[0096]

[0097] In the formula, num1 represents the total number of samples in the test set, and y i Indicates the input process parameter x i The corresponding cross-sectional distortion value, Indicates the input process parameter x i The predicted cross-sectional distortion value.

[0098] Mean squared error (MSE) primarily characterizes larger errors, making it suitable for scenarios sensitive to large errors. A smaller MSE indicates greater performance improvement from the model and better method performance. The MSE is set according to the following formula:

[0099]

[0100] In the formula, num2 represents the total number of samples in the test set, and y i Indicates the input process parameter x i The corresponding cross-sectional distortion value, Indicates the input process parameter x i The predicted cross-sectional distortion value. The smaller the MSE, the more the THSM model improves performance, and the better the method is demonstrated.

[0101] This invention avoids the domain expansion distortion caused by linear domain expansion, solves the weighted mixing problem of hybrid surrogate models, and combines multiple individual surrogate models. Therefore, its computational / model complexity is higher compared to individual surrogate models and other surrogate models. This fusion can fully utilize the advantages of each individual model, thereby enhancing predictive power and more comprehensively reflecting the underlying data.

[0102] The above specific embodiments are used to explain and illustrate the present invention, but not to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims shall fall within the protection scope of the present invention.

[0103] The above description is only a preferred embodiment of the present invention. Therefore, all equivalent changes or modifications made to the structure, features and principles described in the claims of this patent application are included in the scope of this patent application.

Claims

1. A method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian functions, characterized in that: Includes the following steps: S1. Obtain different combinations of process parameters and corresponding cross-sectional distortion data during the bending and forming of pipe fittings, and construct the original sample dataset; S2. Obtain the extended domain range based on the original sample dataset; simultaneously train multiple agent models using the original sample dataset, merge all trained agent models to obtain a hybrid agent prediction model; then generate a virtual sample dataset using a multi-objective optimization algorithm based on the extended domain range and the hybrid agent model; merge the virtual sample dataset and the original sample dataset into a hybrid sample dataset. S3. Train the hybrid agent prediction model using the mixed sample dataset to obtain a transmission-based hybrid agent model; S4. Using the aforementioned transmission-based hybrid proxy model, predict the cross-sectional distortion data based on the process parameters to be predicted.

2. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian function according to claim 1, characterized in that: The process of step S1 is as follows: S1.1 Select at least one process parameter in the bending and forming process of a pipe fitting, pre-set the value range of each process parameter, and generate several process parameter combination samples by means of the Latin hypercube sampling method based on the value range of all process parameters. Each process parameter combination sample is composed of the specific values ​​of all process parameters. S1.

2. Establish corresponding simulation models based on each combination of process parameters, and obtain corresponding cross-sectional distortion data by simulating the simulation models corresponding to each combination of process parameters; the cross-sectional distortion data includes the cross-sectional distortion rate at multiple different locations of the bent part of the pipe fitting. S1.

3. Combine the specific values ​​of all process parameters and the corresponding cross-sectional distortion data in each process parameter combination sample into a sample pair. All sample pairs constitute the original sample dataset.

3. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian functions according to claim 2, characterized in that: The process of step S2 is as follows: S2.1 Normalize the original sample dataset to obtain a normalized sample dataset; S2.

2. Use the truncated-empirical Gaussian membership function as a domain range expansion tool to expand the normalized sample dataset to obtain the expanded domain range; S2.

3. Train multiple proxy models using a normalized sample dataset, and fuse all the trained proxy models based on the prediction results of each trained proxy model to obtain a hybrid proxy prediction model. S2.4 Using a multi-objective optimization algorithm, the hybrid surrogate prediction model obtained in step S2.3 is used as the fitness function, the objective is to minimize the cross-sectional distortion rate at each location, and the extended domain range obtained in step S2.2 is used as the new sampling space for each process parameter. Multiple virtual process parameter combination samples are generated through non-dominated sorting. All virtual process parameter combination samples form a virtual sample dataset. The virtual sample dataset and the normalized sample dataset are merged into a hybrid sample dataset.

4. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian function according to claim 3, characterized in that: In step S2.3, the fusion of all trained agent models specifically involves: using an error verification method to obtain the error verification results of each trained agent model as weights, and then performing a weighted summation on all trained agent models.

5. The method for predicting pipe bending forming process based on virtual sample augmentation using truncated Gaussian function according to claim 3, characterized in that: In step S2.3, the surrogate model includes the Kriging model, radial basis function, and support vector machine.

6. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian function according to claim 5, characterized in that: In step S2.3, all trained agent models are fused according to the following formula to obtain a hybrid agent prediction model: In the formula, F HSM Let f1(x) represent the hybrid surrogate prediction model, f2(x) represent the trained Kriging model, R1 represent the mean squared error of the trained Kriging model, f2(x) represent the trained radial basis function, R2 represent the mean squared error of the trained radial basis function, f3(x) represent the trained support vector machine function, and R3 represent the mean squared error of the trained support vector machine function.

7. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian function according to claim 6, characterized in that: In step S3, the expression for the transmission-based hybrid proxy model is as follows: In the formula, f1'(x) represents the Kriging model fitted using a mixed sample dataset, f2'(x) represents the radial basis function model fitted using a mixed sample dataset, f3'(x) represents the support vector regression machine model fitted using a mixed sample dataset, R1 represents the mean squared error of the prediction error of the Kriging model after training, R2 represents the mean squared error of the prediction error of the radial basis function after training, and R3 represents the mean squared error of the prediction error of the support vector machine function after training.

8. The method for predicting pipe bending forming process based on virtual sample augmentation using truncated Gaussian function according to claim 1, characterized in that: The process parameter combination consists of core ball diameter, core ball width, core ball spacing, initial elongation, number of core balls, and bending radius.

9. The method for predicting pipe bending forming process based on virtual sample expansion using truncated Gaussian function according to claim 1, characterized in that: The cross-sectional distortion data includes the cross-sectional distortion rates at five locations: 0, 1 / 4, 1 / 2, 3 / 4 of the bent portion of the pipe fitting, and the end of the bend.

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