Bent pipe forming process prediction method based on truncated Gaussian function virtual sample expansion

Abstract

The invention discloses a bent pipe forming process prediction method based on truncated Gaussian function virtual sample expansion. Comprising the following steps: constructing an original sample data set, and establishing an expansion domain range; training and fusing the plurality of agent models by using the original sample data set to obtain a mixed agent prediction model; generating a virtual sample data set through a multi-objective optimization algorithm according to the extended domain range and the mixed agent model; after the virtual sample data set and the original sample data set are combined, a hybrid agent prediction model is trained, and a transmission-based hybrid agent model is obtained; and predicting the cross section distortion rate according to the to-be-predicted process parameters by using a transmission-based mixed agent model. According to the method, the defects of small sample size and low prediction precision in the field of mechanical engineering are effectively overcome, the sample size and the prediction stability are increased by generating the virtual sample data set, and the prediction accuracy of the prediction model on the cross section distortion rate is improved by applying the high-precision transmission hybrid agent model.

Description

Technical Field

[0001] The invention relates to the technical field of pipe bending, and in particular to a pipe bending process prediction method based on virtual sample expansion of truncated Gaussian function. Background Art

[0002] In recent years, with the rapid development of advanced manufacturing technology and the increase in personalized needs, pipe fitting design has been increasingly used in engineering structures. However, in the actual pipe fitting design process, due to the limitations of experimental conditions and the increase in manufacturing costs, the data samples that can be used for design optimization are often very limited. Especially in complex mechanical environments, the mechanical properties analysis and optimization of pipe fittings require a large amount of high-precision experimental data, and obtaining these data often requires a lot of time and resources.

[0003] At present, predictive modeling under small sample conditions mainly relies on surrogate models, such as response surface models, support vector machines, and neural networks. These models predict the performance of pipe fittings by extracting rules from limited experimental data. However, when the sample size is insufficient, the prediction accuracy of the surrogate model may be greatly affected. By generating virtual samples, the deficiency of the original samples in the feature space can be compensated, thereby improving the prediction performance of the surrogate model. However, most of the existing virtual sample expansion methods rely on traditional interpolation or extrapolation techniques, and the quality of the generated samples is difficult to guarantee, and they cannot accurately reflect the true distribution of the original data. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a method for predicting the bending process of pipes based on virtual sample expansion of truncated Gaussian function. The method of the present invention combines the virtual sample expansion method based on truncated Gaussian function with the transfer hybrid proxy model for predicting the performance of small samples in the production of bent pipes.

[0005] The technical solution adopted by the present invention is as follows:

[0006] The method for predicting the bending process of a pipe comprises the following steps:

[0007] S1. Obtain different process parameter combinations and corresponding cross-sectional distortion data during the tube bending process to construct the original sample data set.

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

[0009] S1.1. Select at least one process parameter in the pipe bending process, pre-set the value range of each process parameter, and generate a number of process parameter combination samples by Latin hypercube sampling method according to the value range of all process parameters, each process parameter combination sample is composed of specific values ​​of all process parameters;

[0010] S1.2. Establishing respective corresponding simulation models according to the process parameter combination samples, and obtaining corresponding cross-sectional distortion data by simulating the simulation models corresponding to the process parameter combination samples;

[0011] Specifically, the cross-sectional distortion data includes cross-sectional distortion rates at a plurality of different positions of the curved portion of the pipe;

[0012] 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, and all sample pairs constitute the original sample data set.

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

[0014] Preferably, the cross-sectional distortion data include cross-sectional distortion rates at five positions of 0, 1 / 4, 1 / 2, 3 / 4 and the bending end of the pipe bending portion.

[0015] S2. Obtain an extended domain range according to the original sample data set; use the original sample data set to train multiple proxy models respectively, fuse all the trained proxy models, and 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; merge 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 as follows:

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

[0018] S2.2, using the truncated-empirical 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, using the normalized sample data set to train multiple proxy models respectively, and fusing all the 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 as follows: using an error verification method to obtain error verification results of each trained proxy model as weights, and performing weighted summation on all trained proxy models;

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

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

[0023]

[0024] In the formula, F HSM represents the hybrid proxy 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, R3 represents the mean square error of the prediction error of the trained support vector machine function;

[0025] S2.4. Use a multi-objective optimization algorithm, with the hybrid proxy prediction model obtained in step S2.3 as the fitness function, minimizing the cross-sectional distortion rate at each position as the goal, and the expanded domain range obtained in step S2.2 as the new sampling space for each process parameter. Generate multiple virtual process parameter combination samples through non-dominated sorting. All virtual process parameter combination samples constitute a virtual sample data set. Merge the virtual sample data set and the normalized sample data set into a hybrid sample data set.

[0026] S3. Use the mixed sample data set to train the hybrid proxy prediction model to obtain a transfer-based hybrid proxy model.

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

[0028]

[0029] Where f1'(x) represents the Kriging model fitted using the mixed sample data set, f2'(x) represents the radial basis function model fitted using the mixed sample data set, f3'(x) represents the support vector regression model fitted using the mixed sample data set, R1 represents the mean square error of the prediction error of the Kriging model after training, R2 represents the mean square error of the prediction error of the radial basis function after training, and R3 represents the mean square error of the prediction error of the support vector machine function after training.

[0030] S4. Using the transfer-based hybrid proxy model, predict cross-sectional distortion data according to the process parameters to be predicted.

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

[0032] 1) Improve the prediction accuracy under small sample data conditions: The present invention generates virtual samples by truncating the empirical Gaussian membership function, which makes up for the problem of insufficient original sample data, thereby improving the prediction accuracy of the model under small sample conditions. It is particularly suitable for scenarios where experimental data is difficult to obtain or costly, helping the model to better capture the distribution characteristics of data.

[0033] 2) Performance optimization for complex structure design: The present 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 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 working conditions: The hybrid proxy model of the present invention can adapt to changes in different working conditions and material parameters, and is particularly suitable for scenarios where production conditions need to be frequently switched. In the early stage of working condition switching, virtual sample expansion technology makes up for the lack of labeled data, improves the prediction ability of the model in the early stage, and ensures the stability of product quality. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is the domain expansion based on the truncated-empirical Gaussian function and the distribution diagram of observation points and virtual sample points in the present invention.

[0036] Figure 2 It is a flow chart of the method for predicting the bending process of the pipe based on the virtual sample expansion of the truncated Gaussian function in the present invention.

[0037] Figure 3 It is a schematic diagram describing the core shaft parameters in the present invention.

[0038] Figure 4 These are cross-sectional profile diagrams and cross-sectional distortion diagrams at several different positions of the elbow in the present invention. DETAILED DESCRIPTION

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

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

[0041] The embodiment of the present invention takes the rotary traction bending forming process as an example. In the rotary traction bending process, cross-sectional distortion data is usually used to measure the bending forming quality of pipe fittings. However, the cross-sectional distortion rate cannot be measured by online sensors and can only be obtained through offline detection and experimental analysis, which will produce a large time lag and lead to the production of unqualified products. At the same time, the production process of pipe fittings often requires frequent adjustment of forming parameters according to different bending radii and material properties, resulting in insufficient labeled data collected in the initial operation stage after the process parameters are switched, making it difficult to establish an accurate forming quality prediction model.

[0042] See also Figure 1 to Figure 4 In this embodiment, the specific steps of the bending tube forming process prediction method based on the virtual sample expansion of the truncated Gaussian function are as follows:

[0043] S1. Obtain different process parameter combinations and cross-sectional distortion data corresponding to each process parameter combination during the pipe bending process to construct an original sample data set. In this embodiment, the cross-sectional distortion data comes from a simulation model of the pipe rotary traction bending.

[0044] The specific process of step S1 is:

[0045] S1.1. Select at least one key process parameter in the pipe bending process, pre-set the value range of each process parameter, generate a preset number (pre-set to 120 in this embodiment) of process parameter combination samples by Latin hypercube sampling method according to the value range of all process parameters, and 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 values.

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

[0047]

[0048] S1.2. Use the simulation software ABAQUS (engineering simulation finite element software) to establish corresponding simulation models according to each process parameter combination sample, and obtain the corresponding cross-sectional distortion data by simulating the simulation models corresponding to each process parameter combination sample.

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

[0050] Among them, the cross-sectional distortion rate, that is, the cross-sectional minor-axis major-axis distortion rate, is mainly obtained by processing the minor-axis distortion rate and the major-axis distortion rate through the following formula:

[0051] Cross-sectional distortion rate = short-axis distortion rate × long-axis distortion rate

[0052]

[0053] The minor axis is the minor axis of the ellipse after the cross section of the pipe is distorted, and the major axis is the major axis of the ellipse after the cross section of the pipe is distorted.

[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, and all sample pairs constitute the original sample data set.

[0055] The original sample data set is represented as: {x train ,y train},in, is the input variable, which consists of the specific values ​​of the six process parameters, is the output variable, namely the cross-sectional distortion data, which is composed of the cross-sectional distortion rates at five positions. N1 is the number of sample pairs. In this embodiment, N1=120.

[0056] S2. Obtain an extended domain based on the original sample data set; use the original sample data set to train multiple proxy models respectively, and fuse all the trained proxy models according to the prediction results of each trained proxy model to obtain a hybrid proxy prediction model; generate a virtual sample data set based on the extended domain and the hybrid proxy model through a multi-objective optimization algorithm; merge the virtual sample data set and the normalized sample data set into a hybrid sample data set.

[0057] The specific process of step S2 is:

[0058] S2.1. Divide the original sample data set into a training set and a test set in a ratio of 8:2, and normalize the original sample data set to obtain a normalized sample data set. Normalization can speed up the model convergence and reduce the model training time. Normalization is performed according to the following formula:

[0059]

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

[0061] S2.2, use the truncated-empirical Gaussian membership function as a domain range expansion tool to expand the normalized sample data set to obtain an expanded domain range. The truncated Gaussian membership function is composed of a Gaussian membership function and a truncation line. The truncation line limits the range of the Gaussian function to form an expanded domain range; while the truncated-empirical Gaussian membership function intersects with the empirical data range on the basis of this expanded domain range to further optimize the applicable scope of the function.

[0062] Step S2.2 is specifically as follows:

[0063] Set the sampling sets corresponding to the six process parameters respectively Among them, 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. 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 truncation line CL i , cutoff line CL i Set it according to the following formula:

[0065]

[0066] In the formula, CL i Denotes the sampling set X i The truncation line, N1 represents the number of virtual sample points in each sampling set X ij Denotes the sampling set X i The jth sample point in .

[0067] The left skewness 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 Denotes the sampling set X i The left skewness, sk iU Denotes the sampling set X i The right skewness of iL Denotes the sampling set X i Less than the cutoff line CL i The number of sample points, N iU Denotes the sampling set X i Greater than the cutoff line CL i The number of sample points, s p Represents the adaptivity parameter.

[0070] For each sampling set corresponding to each process parameter, the truncation line CL is calculated based on the 120 sets of sampling point data initially sampled. i , left skewness sk iL and right skewness sk iU The values ​​of the cutoff line, left skewness and right skewness of different process parameters are different. Then, according to the cutoff line CL i , left skewness sk iL and right skewness sk iU The value of , combined with the minimum value min of the domain range X and the maximum value max of the domain range X, obtains point A (CL i ,1), point B(min,sk iL ) and point C(max,sk iU ) coordinates, and then determine the shape of the Gaussian function curve based on the three points ABC.

[0071] The truncation line of the sampling set corresponding to each process parameter is used as the offset variable of the Gaussian function. The truncation 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] Where, UB i Denotes the sampling set X i The upper bound of LB i Denotes the sampling set X i The lower bound of CL i Denotes the sampling set X i The cutoff line, β i Denotes the sampling set X i The offset variable, c i1 Denotes the sampling set X i The corresponding first coefficient, c i2 Denotes the sampling set X i The corresponding second coefficient.

[0074] Among them, the sampling set X i The corresponding first coefficient c i1 and the second coefficient c i2 They are obtained by the following formulas:

[0075]

[0076] In the formula, c1 represents the sampling set X i The corresponding first coefficient, c2 represents the sampling set X i The corresponding second coefficient, min represents the sampling set X i The minimum value of max represents the sampling set X iThe maximum value, CL i Denotes the sampling set X i The cutoff line, sk iL Denotes the sampling set X i The left skewness of iU Denotes the sampling set X i The right skewness of .

[0077] Get the sampling set X corresponding to each process parameter i The upper bound UB i and lower bound LB i The intersection between the preset range of the process parameters is used to obtain the domain expansion upper and lower bounds of each process parameter within the preset range, and then the expanded domain range of each process parameter is obtained.

[0078] S2.3. Use the normalized sample data set to train multiple proxy models separately, and fuse all the trained proxy models according to the prediction results of each trained proxy model to obtain a hybrid proxy prediction model.

[0079] The specific steps of fusing all the trained proxy models are as follows: the error verification results of each trained proxy model are obtained as weights using the error verification method, and the weights of the trained proxy models are used to perform weighted summation on all the trained proxy models, so that the high-fitting proxy model occupies a larger weight in the mixed proxy model, thereby best fitting the original data.

[0080] In this embodiment, three proxy models are used: Kriging model (Kriging), Radial Basis Function (RBF) and Support Vector Regression (SVR). Among them, the hyperparameter of the Kriging model is θ=10, and the domain range is [0.1,20]; the mean square error target of the radial basis function is 10^(-20), the maximum number of hidden layer neurons is 100, and the expansion coefficient of the radial basis function is 1.0; the kernel function of the support vector machine uses an exponential function, and the gamma function and the loss function of the SVC both use an exponential power of 2, and the loss function of the support vector machine uses 0.1. In this embodiment, the square inverse of the distance between the prediction results of each proxy model and the true value is used as the weight, so that the smaller the error, the greater the weight.

[0081] Step S2.3 is specifically as follows: use the training set in the normalized sample data set to train the three proxy models respectively, and obtain three trained proxy models (f1(x), f2(x), f3(x)). Use the test set in the normalized sample data set to obtain the prediction results of the three trained proxy models respectively, use the square inverse of the mean square value of the prediction error as the weight value, and perform weighted summation on the three trained proxy models to obtain a hybrid proxy prediction model (HSM model). The expression of the hybrid proxy prediction model (HSM model) is as follows:

[0082]

[0083] In the formula, F HSM represents the expression of the hybrid proxy 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.

[0084] S2.4. Use a multi-objective optimization algorithm, with the hybrid proxy prediction model obtained in step S2.3 as the fitness function, minimizing the cross-sectional distortion rate at different positions of the bending part of the pipe as the goal, and the expanded domain range obtained in step S2.2 as the new sampling space for each process parameter. Generate multiple virtual process parameter combination samples through non-dominated sorting. All virtual process parameter combination samples constitute a virtual sample data set. The virtual sample data set and the normalized sample data set are merged into a hybrid sample data set.

[0085] The role of domain expansion in multi-objective optimization algorithms is to enhance the coverage of the solution space, so that the algorithm can explore a wider solution space and obtain more potential Pareto optimal solutions. This strategy is crucial for dealing with complex multi-objective optimization problems, especially when the scope and complexity of the objective space are large, expanding the domain can significantly improve the effect of 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 is composed of specific values ​​of six process parameters. After the individual is input into the fitness function, the cross-sectional distortion rates at five positions are obtained. Non-dominated sorting is performed based on the cross-sectional distortion rates at these five positions. After reaching the 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 data set. In this embodiment, a virtual sample data set consisting of 400 groups of high-quality virtual samples is obtained by the NSGA-III optimization algorithm.

[0087] In this embodiment, the preset conditions include that the average values ​​of the cross-sectional distortion rates are all less than a preset threshold value and / or the average value of the cross-sectional distortion rates after a certain iteration is almost not reduced compared with the average value before the iteration.

[0088] S3. Establish a transfer-based hybrid proxy model: Use the mixed sample data set to train the hybrid proxy prediction model obtained in step S2.3 to obtain a transfer-based hybrid proxy model (THSM model).

[0089] The expression of the transfer-based hybrid agent model (THSM model) is as follows:

[0090]

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

[0092] Step S3 retains the weights of the three single proxy models based on the transfer strategy, and transfers the weights to the final prediction model, that is, the hybrid proxy model based on transfer. Through this process, the weights with high fitness of the original data are transferred to the fitting of the mixed sample data set, ensuring that the weight of the proxy model with good fitness for the data set accounts for a high proportion, and the weight of the proxy model with poor fitness accounts for a low proportion.

[0093] S4. Use the transfer-based hybrid proxy model obtained in step S3 to predict cross-sectional distortion data according to the process parameters to be predicted.

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

[0095] The mean absolute error (MAE) mainly represents the average deviation of the prediction error and reduces the sensitivity to outliers. The smaller the mean absolute error, the better the prediction performance. The mean absolute error is set according to the following formula:

[0096]

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

[0098] The mean square error (MSE) mainly represents larger errors, so it is suitable for scenarios that are sensitive to large errors. The smaller the mean square error (MSE), the more the model improves performance and the better the method is. The mean square error (MSE) is set according to the following formula:

[0099]

[0100] In the formula, num2 represents the total number of test set samples, y i Represents the input process parameter x i The corresponding cross-sectional distortion value is, Represents the input process parameter x i The smaller the MSE, the more the THSM model improves performance, which shows the superiority of the method.

[0101] The method of the present invention avoids domain extension distortion caused by linear domain extension, solves the weighted mixing problem of hybrid proxy models, combines multiple individual proxy models, and therefore has a higher computational / model complexity compared to individual proxy models and other proxy models. This fusion can fully utilize the advantages of each individual model, thereby enhancing the prediction ability and more comprehensively reflecting the basic data.

[0102] The above specific implementation modes are used to explain the present invention rather than to limit the present invention. Any modification and change made to the present invention within the spirit of the present invention and the protection 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, so all equivalent changes or modifications made according to the structure, characteristics and principles described in the scope of the patent application of the present invention are included in the scope of the patent application of the present invention.

Claims

1. A method for predicting tube bending process based on virtual sample expansion of truncated Gaussian function, characterized by: The following steps are involved: S1. Obtain different process parameter combinations and corresponding cross-sectional distortion data during the tube bending process to construct an original sample data set; S2, obtaining an extended domain range according to the original sample data set; using the original sample data set to train multiple proxy models respectively, fusing all trained proxy models to obtain a hybrid proxy prediction model; then generating a virtual sample data set according to the extended domain range and the hybrid proxy model through a multi-objective optimization algorithm; merging the virtual sample data set and the original sample data set into a hybrid sample data set; S3, using the mixed sample data set to train the hybrid proxy prediction model to obtain a transfer-based hybrid proxy model; S4. Using the transfer-based hybrid proxy model, predict cross-sectional distortion data according to the process parameters to be predicted.

2. The method for predicting the bending process of a pipe based on virtual sample expansion of a truncated Gaussian function according to claim 1 is characterized in that: The process of step S1 is specifically as follows: S1.

1. Select at least one process parameter in the pipe bending process, pre-set the value range of each process parameter, and generate a number of process parameter combination samples by Latin hypercube sampling method according to the value range of all process parameters, each process parameter combination sample is composed of specific values ​​of all process parameters; S1.

2. Establish corresponding simulation models according to each process parameter combination sample, and obtain corresponding cross-sectional distortion data by simulating the simulation models corresponding to each process parameter combination sample; the cross-sectional distortion data includes cross-sectional distortion rates at multiple different positions of the bending 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, and all sample pairs constitute the original sample data set.

3. The method for predicting the bending process of a pipe based on virtual sample expansion of truncated Gaussian function according to claim 2 is characterized in that: The process of step S2 is specifically as follows: S2.1, normalizing the original sample data set to obtain a normalized sample data set; S2.2, using the truncated-empirical Gaussian membership function as a domain range expansion tool to expand the normalized sample data set to obtain an expanded domain range; S2.3, using the normalized sample data set to train multiple proxy models respectively, and fusing all the trained proxy models according to the prediction results of each trained proxy model to obtain a hybrid proxy prediction model; S2.

4. Use a multi-objective optimization algorithm, with the hybrid proxy prediction model obtained in step S2.3 as the fitness function, minimizing the cross-sectional distortion rate at each position as the goal, and the expanded domain range obtained in step S2.2 as the new sampling space for each process parameter. Generate multiple virtual process parameter combination samples through non-dominated sorting. All virtual process parameter combination samples constitute a virtual sample data set. Merge the virtual sample data set and the normalized sample data set into a hybrid sample data set.

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

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

6. The method for predicting the bending process of a pipe based on virtual sample expansion of truncated Gaussian function according to claim 5 is characterized in that: In step S2.3, all trained proxy models are fused according to the following formula to obtain a hybrid proxy prediction model: In the formula, F HSM represents the hybrid proxy 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.

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

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

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

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