Sequential test design method for multi-response modeling based on consideration of response correlation

Abstract

The invention relates to a sequential test design method based on multi-response modeling considering response correlation. The method comprises the steps of constructing a response model capable of simultaneously predicting multiple responses based on a multi-response Bayesian support vector machine model, performing iterative sampling by combining a local development criterion based on expanded nearest neighbor gradient projection and a global exploration criterion based on a variance matrix diagonal product, and continuously updating the response model, and until the sequential test design stop criterion is met. By adopting the method, the mapping relationship between a plurality of factors and a plurality of responses in a complex application system can be constructed at the same time, the correlation between the responses is represented, available information is increased through information transmission between the responses, the modeling precision of the response model is improved, and the modeling efficiency is improved. And thus, the effectiveness of sequential test design of a complex application system with multiple responses is improved.

Description

Technical Field

[0001] This application relates to the technical field of complex system test evaluation, and particularly to a sequential test design method based on multi-response modeling considering response correlation. Background Art

[0002] Test design is an efficient sample selection method based on statistics. It can scientifically and reasonably select representative sample points, obtain as much information as possible with as few test samples as possible, ensure the effectiveness of the test, and be used to support the analysis and modeling of data. Among them, the result of the test is called "response", the variable to be investigated in the test is called "factor", and the test design is to combine different values of the factor to obtain test sample points. Test design methods are widely used in many fields such as agriculture, industry, and social systems. In practical engineering problems such as equipment test and evaluation, test design is often combined with response models to establish the relationship between the key performance indicators (responses) of the equipment and the factors, realize the prediction of responses in the entire test space, analyze the changes in responses under different combinations of influencing factors, determine the performance boundaries and key areas of the equipment, and provide a basis for subsequent test design and evaluation of the combined internal and external fields of the equipment.

[0003] For a complex test space, a large number of test samples are required to establish a response model that can accurately predict the entire space. Using traditional test design methods to implement the above sample point selection process requires huge costs and there will be many difficulties in implementation. Therefore, the sequential test design method is considered, which uses the current model information to guide the selection of the next sample point, selects sample points with more information, saves the sample size and improves the model accuracy at the same time. Sequential test design can obtain more information with a smaller sample size, and thus has become a commonly used method in the field of engineering design optimization.

[0004] However, most current sequential experimental design methods are designed for single-response problems. In engineering applications, however, there are often multiple responses in a single complex system, and multiple response models need to be constructed. If single-response sequential experimental designs are carried out separately for multiple responses, more sample points are required, resulting in greater computational overhead. Moreover, in the problem of multi-response sequential experimental design, there are complex interaction relationships between responses and between responses and factors. Therefore, when conducting sequential experimental design, the local development and global exploration requirements of different responses may conflict, and the sample spaces that sequential experimental design criteria need to focus on will also be different. Therefore, comprehensive optimization is required. Especially when there is a certain correlation between responses, how to utilize this correlation to improve the modeling accuracy and further enhance the experimental design effect is also a problem that needs to be solved. Existing single-response sequential experimental design methods do not consider the correlations between responses. If sequential experimental designs are carried out separately for multiple responses, it will result in a waste of computational overhead and is not suitable for solving such complex problems. Summary of the Invention

[0005] Based on this, in view of the above technical problems, it is necessary to provide a sequential experimental design method, a computer device, and a storage medium based on multi-response modeling considering response correlations.

[0006] A sequential experimental design method based on multi-response modeling considering response correlations, the method comprising:

[0007] Step 1, sampling from a candidate sample pool of an application system to obtain an initial sample set; wherein, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of an anthropomorphic robotic arm;

[0008] Step 2, constructing an experimental plan for anthropomorphic robotic arm inverse dynamics modeling based on the initial samples, and simulating to generate multiple responses of the experimental plan; wherein, the responses are the torques of multiple joints of the anthropomorphic robotic arm;

[0009] Step 3, using a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses. The response model constructs a kernel function that describes the correlations between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization;

[0010] Step 4, determining whether a preset stop criterion is reached. If so, stop the sequential experimental design; if not, proceed to the next step;

[0011] Step 5: Calculate the local development criterion based on extended nearest neighbor gradient projection. The local development criterion calculation includes gradient solution, nearest neighbor determination, and index calculation. Among them, the gradient solution includes constructing an extended gradient vector composed of gradients of multiple responses.

[0012] Step 6: Calculate the global exploration criterion based on the variances of multiple responses.

[0013] Step 7: Construct a comprehensive score function according to the local development criterion and the global exploration criterion, and select new sample points in the candidate sample pool to add to the initial sample set according to the comprehensive score ranking, construct a new experimental plan, and input the new responses generated by simulation into the response model until the predicted output of the response model reaches the stop criterion, thus completing the sequential experimental design for the inverse dynamics modeling of the anthropomorphic manipulator.

[0014] In one embodiment, a multi-response Bayesian support vector machine is used to construct a response model that can predict multiple responses simultaneously. The response model constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization. The specific steps include:

[0015] Use a multi-response Bayesian support vector machine to construct a response model that can predict multiple responses simultaneously. The response model consists of a semi-parametric latent factor structure and a trade-off parameter. Among them, the semi-parametric latent factor structure is used to quantitatively describe the variances between responses. For each response, there is a corresponding trade-off parameter, and the trade-off parameter is used to balance the complexity and error of the response model.

[0016] The semi-parametric latent factor structure constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization. Among them, the parameters to be optimized include the trade-off parameter, the kernel function parameter, and the hyperparameters in the covariance between multiple responses.

[0017] The means and variances of multiple responses predicted and output by the response model are respectively expressed as

[0018]

[0019] Among them, represents the mean of the response predicted by the response model when the input is the sample x to be predicted, represents the variance of the response predicted by the response model when the input is the sample x to be predicted. X is the training set of the response model constructed from the current sample set, is the covariance matrix between x and X, k M (X,X) and k M(x,x) represents the covariance matrices of X itself and x itself respectively. The covariance matrix consists of kernel functions that describe the correlations between multiple responses, C h is a variant of the trade-off parameter and satisfies C h ⊙ diag(C M ) = I, where ⊙ is the Hadamard product and is used to represent element-by-element multiplication in the matrix, is the identity matrix, is the trade-off parameter in the response model. diag(C M ) is the diagonal form of the parameter matrix C composed of the trade-off parameters of multiple responses M and is a matrix with all elements being 1 and is used to transform the trade-off parameter C, is the identity matrix, is a multi-response Gaussian process. The superscript T represents matrix transpose, is the mean vector of N f elements, represents the mean vector of N f × N elements. b is either one of b0 and b. N f and N are the number of responses and the number of samples respectively, is the state space.

[0020] In one embodiment, the gradient solution in the local development criterion includes: constructing an extended gradient vector composed of the gradients of N f responses, and the expression is where G(xc) represents the extended gradient vector corresponding to a certain candidate sample point xc in the candidate sample pool, and g i represents the gradient of the i-th response;

[0021] is calculated according to the means of the N f responses output by the response model. The extended gradient vector can also be expressed as

[0022]

[0023] where, represents the mean of the responses predicted by the response model when the input is xc, and k M (xc, X) is the covariance matrix between xc and X, represents the covariance matrix of Y, where Y is multiple responses generated based on X in the simulation system or dataset.

[0024] In one embodiment, the determination of the nearest neighbor in the local development criterion includes:

[0025] Assume that the current sample set X consists of N sample points, and the expression is where, xi Denote the i-th sample point as \(x_i\), and D represents the dimension of the sample space;

[0026] Divide the D-dimensional sample space into N units according to the number of samples N, and the expression is \(C = \{c_1, c_2, \ldots, c_N\}\); where, N} ; Among them, Denote the j-th cell as \(c_j\), \(j = 1, 2, \ldots, N\), and each cell is composed of the sample points in the current sample set X and the candidate sample pool \(X_c\);

[0027] Determine the sample center of each candidate sample point in the candidate sample pool through Voronoi partitioning, so that the cell \(c_j\) j contains all the points in the candidate sample pool that are closer to the center \(x_j\) of the cell \(c_j\) j than to other sample points, and is specifically defined as j where \(dom(x_j, x_j')\) is a closed half-plane, bounded by the perpendicular bisector of \(x_j\) and \(x_j'\), and this perpendicular bisector separates the points in the region closer to \(x_j\) to form \(c_j\), and all the points in \(c_j\) are closest to \(x_j\). The cell \(c_j\) is specifically expressed as

[0028]

[0029] where \(x_{i,j}\) j , \(x_{i,j}'\) j is a closed half-plane, bounded by the perpendicular bisector of \(x_j\) and \(x_j'\), and this perpendicular bisector separates the points in the region closer to \(x_j\) to form \(c_j\), and all the points in \(c_j\) are closest to \(x_j\). The cell \(c_j\) is specifically expressed as j and j 's perpendicular bisector divides the region, separating the points closer to \(x_j\) to form \(c_j\), and all the points in \(c_j\) are closest to \(x_j\). The cell \(c_j\) is specifically expressed as j where \(dom(x_j, x_j')\) is a closed half-plane, bounded by the perpendicular bisector of \(x_j\) and \(x_j'\), and this perpendicular bisector separates the points in the region closer to \(x_j\) to form \(c_j\), and all the points in \(c_j\) are closest to \(x_j\). The cell \(c_j\) is specifically expressed as j and j all the points in \(c_j\) are closest to \(x_j\), and the cell \(c_j\) is specifically expressed as j where \(x_{i,j}\) j specifically represents where, denotes the i-th candidate sample point in the cell \(c_j\), j and \(n_j\) is the number of candidate sample points in the cell \(c_j\), and \(x_{i,j}'\) j is another sample point different from \(x_{i,j}\). j ' is another sample point different from \(x_{i,j}\). j In one embodiment, the calculation of the indicators in the local development criterion includes:

[0030] Calculate the nearest neighbor gradient projection of the candidate sample point \(x_c\) according to the projection angle \(\theta\) between the expansion gradient vector \(G(x_c)\) of the candidate sample point \(x_c\) and the expansion gradient vector \(G(x)\) of the center point \(x\), and is expressed as

[0031] where \(x\) * and \(G(x)\) * ) is the expansion gradient vector of the center point \(x\), and is expressed as

[0032]

[0033] where \(x\) *is the center point of the cell where the candidate sample point xc obtained by Voronoi partitioning is located.

[0034] In one embodiment, the global exploration criterion is expressed as

[0035]

[0036] where represents the variance of the i-th response predicted by the response model when the input is the candidate sample point xc, and N f is the number of responses.

[0037] In one embodiment, a comprehensive scoring function is constructed according to the local development criterion and the global exploration criterion, and new sample points in the candidate sample pool are selected according to the comprehensive score ranking and added to the initial sample set, including:

[0038] The multiplication criterion is used to combine the local development criterion and the global exploration criterion to construct a comprehensive scoring function, which is expressed as

[0039]

[0040] where local(xc) = Gp(xc) represents the local development criterion, and Gp(xc) is the nearest neighbor gradient projection of the candidate sample point xc, represents the global exploration criterion, represents the variance of the i-th response predicted by the response model when the input is the candidate sample point xc, and N f is the number of responses, and RC(local(xc), global(xc)) represents the combination of the local development criterion and the global exploration criterion;

[0041] New sample points in the candidate sample pool are selected according to the ranking of score(xc) and added to the initial sample set, where the screening rule for the new sample points is expressed as

[0042]

[0043] where x new is the new sample point, and Xc is the candidate sample pool.

[0044] In one embodiment, the above method further includes: the stopping criterion is determined by the modeling accuracy of the response model, and the modeling accuracy is described by the average value of multiple single-response error metrics, which are respectively expressed as

[0045]

[0046] where aR 2 represents the coefficient of determination R of the test set 2The average value, aRMAE represents the average value of the relative maximum error RMAE of the test set, N f is the number of responses, y j represents the j-th test sample x j of the true response, and j = 1, 2, …, n, represents the predicted response of the test set, n is the number of the test set, y represents the true responses of all test samples, represents the predicted responses of all test samples, y ij is the i-th response true value of the j-th test sample x j is the i-th response predicted value of the j-th test sample x j represents the mean value of the true response of the j-th test sample x j y i is the true value of the i-th response of all test samples, is the predicted value of the i-th response of all test samples.

[0047] A computer device includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, the following steps are implemented:

[0048] Step 1, sample an initial sample set from the candidate sample pool of the application system; wherein, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of an anthropomorphic robotic arm;

[0049] Step 2, construct an experimental scheme for anthropomorphic robotic arm inverse dynamics modeling based on the initial samples, and simulate to generate multiple responses of the experimental scheme; wherein, the responses are the torques of multiple joints of the anthropomorphic robotic arm;

[0050] Step 3, use a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses. The response model constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the mean value and variance of multiple responses through Bayesian derivation and parameter optimization;

[0051] Step 4, determine whether the preset stopping criterion is reached. If so, stop the sequential experimental design; if not, proceed to the next step;

[0052] Step 5, perform local development criterion calculation based on extended nearest neighbor gradient projection. The local development criterion calculation includes gradient solution, nearest neighbor determination, and index calculation; wherein, the gradient solution includes constructing an extended gradient vector composed of gradients of multiple responses;

[0053] ​​Step 6, calculate the global exploration criterion based on the variances of multiple responses;

[0054] Step 7, construct a comprehensive score function according to the local development criterion and the global exploration criterion, and select new sample points in the candidate sample pool to be added to the initial sample set according to the comprehensive score ranking, construct a new experimental plan, and input the new responses generated by simulation into the response model until the predicted output of the response model reaches the stopping criterion, thus completing the sequential experimental design for the inverse dynamics modeling of the anthropomorphic manipulator.

[0055] A computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the following steps are implemented:

[0056] Step 1, sample from the candidate sample pool of the application system to obtain an initial sample set; wherein, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of the anthropomorphic manipulator;

[0057] Step 2, construct an experimental plan for the inverse dynamics modeling of the anthropomorphic manipulator based on the initial samples, and simulate to generate multiple responses of the experimental plan; wherein, the responses are the torques of multiple joints of the anthropomorphic manipulator;

[0058] Step 3, use a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses. The response model constructs a kernel function that describes the correlation between multiple responses, transmits the information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization;

[0059] Step 4, determine whether the preset stopping criterion is reached. If it is reached, stop the sequential experimental design; if not, proceed to the next step;

[0060] Step 5, calculate the local development criterion based on the extended nearest neighbor gradient projection. The calculation of the local development criterion includes gradient solution, nearest neighbor determination, and index calculation; wherein, the gradient solution includes constructing an extended gradient vector composed of the gradients of multiple responses;

[0061] Step 6, calculate the global exploration criterion based on the variances of multiple responses;

[0062] Step 7, construct a comprehensive score function according to the local development criterion and the global exploration criterion, and select new sample points in the candidate sample pool to be added to the initial sample set according to the comprehensive score ranking, construct a new experimental plan, and input the new responses generated by simulation into the response model until the predicted output of the response model reaches the stopping criterion, thus completing the sequential experimental design for the inverse dynamics modeling of the anthropomorphic manipulator.

[0063] The sequential experimental design method, computer device, and storage medium for multi-response modeling considering response correlation described above have the following technical effects compared to the prior art:

[0064] 1. This application uses a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses, enabling the response model to simultaneously establish the mapping relationships between multiple factors and multiple responses in a complex application system and represent the correlation between responses. By transmitting information between responses, the available information is increased, the modeling accuracy of the response model is improved, and thus the effectiveness of the sequential experimental design for complex application systems with multiple responses is enhanced.

[0065] 2. This application calculates the local development criterion based on extended nearest neighbor gradient projection. The local development criterion combines the gradients of multiple responses to form a new extended gradient vector and uses the extended gradient vector as the sample point gradient. By projecting the sample point gradient onto the gradient direction of the nearest neighbor, the local change intensity of the sample point can be measured. Moreover, the gradient projection can not only measure the gradient norm of the sample point itself but also measure the gradient change of the sample point relative to other sample points, enabling the experimental design process to converge to the local optimum faster.

[0066] 3. This application calculates the global exploration criterion based on the variances of multiple responses and constructs a comprehensive score function by combining the local development criterion and the global exploration criterion. Based on the comprehensive score, iterative sampling is performed on the basis of the initial point design according to the point addition criterion and the information such as the response model characteristics provided by the previous sample point, continuously updating the response model until the stopping criterion is met. Compared with the one-shot experimental design, the accuracy of the response model and the accuracy of the sequential experimental design are improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 It is a schematic flowchart of the sequential experimental design method for multi-response modeling considering response correlation in an embodiment;

[0068] Figure 2 It is a schematic overall architecture diagram of the sequential experimental design method for multi-response modeling considering response correlation in an embodiment;

[0069] Figure 3 It is a schematic structural diagram of the response model constructed based on the multi-response Bayesian support vector machine in an embodiment;

[0070] Figure 4 It is a schematic diagram of a Voronoi partition example in an embodiment;

[0071] Figure 5 It is a schematic flowchart of the experimental design process of random point selection and sequential space filling in an embodiment;

[0072] Figure 6 Schematic diagram for comparing error results of different experimental design methods based on the Energy dataset in one embodiment; wherein, Figure 6 (a) is the mean of multiple response RMAEs, Figure 6 (b) is the mean of multiple responses R 2 mean;

[0073] Figure 7 Schematic diagram for comparing error results of different experimental design methods based on the Broomcorn dataset in one embodiment; wherein, Figure 7 (a) is the mean of multiple response RMAEs, Figure 7 (b) is the mean of multiple responses R 2 mean;

[0074] Figure 8 Schematic diagram for comparing error results of different experimental design methods based on the Polymer dataset in one embodiment; wherein, Figure 8 (a) is the mean of multiple response RMAEs, Figure 8 (b) is the mean of multiple responses R 2 mean;

[0075] Figure 9 Schematic diagram for comparing error results of different experimental design methods based on the Sarcos dataset in one embodiment; wherein, Figure 9 (a) is the mean of multiple response RMAEs, Figure 9 (b) is the mean of multiple responses R 2 mean;

[0076] Figure 10 Internal structure diagram of a computer device in one embodiment. Detailed implementation manners

[0077] In order to make the objectives, technical solutions and advantages of the present application clearer and more understandable, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0078] In view of the situation where there is a correlation between multiple responses, the present application proposes a sequential experimental design method based on multi-response modeling considering response correlation (abbreviated as MREI-MBSVR) based on a multi-response Bayesian support vector machine model. In this method, extended nearest neighbor gradient projection is used as the local development criterion, wherein the extended gradient is to combine the gradients of multiple responses to form a new extended gradient vector. The product of the diagonal elements of the variance matrix is used as the global exploration criterion. This method is verified by function examples and datasets, as shown in Table 1.

[0079] Table 1 Function example and dataset information

[0080]

[0081] Among them, Forrester and Branin are 1D function examples and 2D function examples respectively, and Energy, Polymer, Broomcorn, and Sarcos are real test data sets. The specific information is as follows:

[0082] Energy: This data set contains 2 responses, namely the heating load and cooling load requirements of building energy efficiency, and 8 factors such as daylighting area, roof area, and overall height. It contains a total of 768 sample points.

[0083] Polymer: This data set is a polymer test data set. The polymer test contains 10 factors such as temperature and feed rate, and 4 responses. This data set contains a total of 61 sample points.

[0084] Broomcorn: This data set is composed of sorghum samples from the Crop Seed Resources Institute of the Chinese Academy of Agricultural Sciences, containing 128 sample points. The responses are the component contents of protein, lysine, and starch in sorghum samples. It contains 19 factors and 3 responses. It contains a total of 128 sample points.

[0085] Sarcos: This data set is a high-dimensional large-scale data set. This data set is for the inverse dynamics modeling problem of a 7-degree-of-freedom anthropomorphic robotic arm. The robotic arm has 21 factors (7 joint positions, 7 joint velocities, and 7 joint accelerations) and the corresponding torques of 7 joints as responses. For the convenience of display, only the first 6 responses are selected to display the modeling results.

[0086] In one embodiment, as Figure 1 and Figure 2 shown, a sequential experimental design method for multi-response modeling based on considering response correlation is provided. This method is applied to a complex system weight based on the Sarcos data set, including the following steps:

[0087] Step 1, sample from the candidate sample pool of the application system to obtain an initial sample set; among them, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of the anthropomorphic robotic arm.

[0088] Specifically, when applying this step, since the number of some data sets is limited and to reduce the total number of test samples, for different data sets, 45 sample points are required, calculated according to 0.35N max It is calculated that the number of initial sample points is 15, which are randomly selected from the sample pool. Among them, the Polymer data set has a small sample size, and only 20 sample points are sequentially added. N max is the maximum value of the number of sample points.

[0089] Step 2: Based on the initial samples, construct an experimental plan for the inverse dynamics modeling of the anthropomorphic robotic arm, and simulate to generate multiple responses of the experimental plan; where the responses are the torques of multiple joints of the anthropomorphic robotic arm.

[0090] Step 3: Use the multi-response Bayesian support vector machine to construct a response model that can predict multiple responses simultaneously. The response model constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization.

[0091] Step 4: Determine whether the preset stopping criterion is met. If it is met, stop the sequential experimental design; if not, proceed to the next step.

[0092] Step 5: Calculate the local development criterion based on the extended nearest neighbor gradient projection. The calculation of the local development criterion includes gradient solution, nearest neighbor determination, and index calculation; where the gradient solution includes constructing an extended gradient vector composed of the gradients of multiple responses.

[0093] Step 6: Calculate the global exploration criterion based on the variances of multiple responses.

[0094] Specifically, in the response model, the final output is the covariance matrix of multiple responses, and the diagonal of the matrix corresponds to the variance of a single response. Use the product of the diagonal elements of the matrix, that is, the product of the variances of single responses, to characterize the global uncertainty of multiple responses.

[0095] Step 7: Construct a comprehensive scoring function according to the local development criterion and the global exploration criterion, and select new sample points from the candidate sample pool to add to the initial sample set according to the comprehensive score ranking, construct a new experimental plan, and input the newly generated responses obtained by simulation into the response model until the predicted output of the response model reaches the stopping criterion, thus completing the sequential experimental design for the inverse dynamics modeling of the anthropomorphic robotic arm.

[0096] In one embodiment, the specific steps of using the multi-response Bayesian support vector machine to construct a response model that can predict multiple responses simultaneously include:

[0097] Assume that a multi-response modeling problem contains N f responses and N sample points. Define a vector Y to represent the responses of the sample points. The vector Y contains N f ×N elements. The multi-response Bayesian support vector machine aims to simultaneously approximate N f responses {g i (x)} 1≤i≤Nf, by considering the correlation between responses, a more accurate response model that can simultaneously predict multiple responses is established. The response model consists of a semiparametric latent factor model (SLFM) and trade-off parameters, as shown in Figure 3 . Among them, a g(x) includes a linear combination of Q implicit functions. According to the SLFM structure, the variance between g(x) can be quantitatively described. Figure 3 The rightmost column represents the trade-off parameters. For each response, there is a corresponding trade-off parameter, which is used to balance the complexity and error of the response model. When the trade-off parameter is large, the model complexity is high, which may lead to overfitting and poor generalization ability; when the trade-off parameter is small, it is prone to underfitting. Therefore, the trade-off parameter needs to be further optimized.

[0098] In the multi-response problem, consider assuming the Gaussian process as a multi-response Gaussian process. By describing the correlation between responses, a new kernel function is constructed, and according to Bayesian derivation, the prior distribution and posterior distribution of the new multi-response are obtained. In the multi-response Gaussian process, the core part is the construction of the covariance matrix for correlation description, which is mainly carried out through the SLFM model. The SLFM model samples the latent variables of the Gaussian process and constructs a new kernel function to describe the correlation between responses. The SLFM structure is introduced below.

[0099] The SLFM structure assumes that there are Q shared latent Gaussian processes, and generally the number of latent Gaussian processes is less than the number of responses. The SLFM represents the response as a linear combination of Q Gaussian processes. Taking two responses as an example, select two Gaussian process implicit functions, that is, N f =Q=2. Assume that u1(x) and u2(x) are sampled from two Gaussian processes respectively, where g1(x) and g2(x) are obtained by linearly transforming u1(x) and u2(x):

[0100] g1(x)=a 1,1 u1(x)+a 2，2 u2(x);

[0101] g2(x)=a 2，1 u1(x)+a 2，2 u2(x);

[0102] Among them, u1(x) and u2(x) respectively represent following a Gaussian process distribution, having different covariance functions, and being independent of each other. That is, in the case of q≠q', u q (x)⊥u q' (x'), cov(u q (x),u q'(x')) = 0. Since a new covariance function can be obtained by linearly combining several covariance functions, g1(x) and g2(x) can be written as:

[0103] g(x) = a1u1(x) + a2u2(x);

[0104] where g(x) = [g1(x), g2(x)] Τ , a1 = [a 1,1 , a 2,1 Τ , a2 = [a 1,2 , a 2,2 Τ , the covariance of the multi-response outputs g(x) and g(x') at two different sample points x and x' can be calculated as:

[0105]

[0106] Define B1 = a1(a1) Τ , B2 = a2(a2) Τ We can get:

[0107] k M (x, x') = B1k1(x, x') + B2k2(x, x');

[0108]

[0109] where, is the Kronecker product, which is an operation between two matrices of arbitrary sizes and is a special form of the tensor product. Further, considering multiple responses a more general form is:

[0110]

[0111] The above equation can be described using matrices as:

[0112] g(x) = Au(x);

[0113] where: u(x) = [u1(x), u2(x), …, u Q (x)] Τ ;

[0114]

[0115] Since multiple Gaussian processes are independent, for multiple responses the correlation of multiple responses corresponding to different x can be expressed as:

[0116]

[0117]

[0118] Among them, The element corresponding to the i-th response is is the Kronecker product, which is a special form of the tensor product. To further describe the multi-response Gaussian process, it is necessary to determine the number Q of implicit functions. It is found that the larger Q is, the more flexible the model is and more variability can be described. However, the increase of Q will lead to an increase in the parameters to be optimized, resulting in an increase in computational overhead. Therefore, in order to balance the flexibility and accuracy of the response model as well as the computational overhead, the value of Q is set to N f . Then the correlation can be further expressed as:

[0119] k M (x, x') = R diag[k1(x, x'), …, k Q (x, x')] R Τ ;

[0120] A q = r q r q Τ ;

[0121] R = [r1, r2, …, r Q ;

[0122]

[0123] And Q = N f , that is, the kernel function of a Gaussian process has a set of parameters to be optimized If D is the sample dimension, then for Q implicit functions, each implicit function has a set of θ, which is used to measure the importance of the input relative to the specified response. Then the kernel function parameters to be optimized include which is used to describe the covariance between multiple responses. Among them, A is an upper triangular matrix and is also a set of unknown hyperparameters to be optimized.

[0124] On this basis, the relationship between the response and the factor can be expressed as:

[0125]

[0126] Among them, is independent and identically distributed random error, and its distribution form is usually unknown. Y is the value of the multi-response. g(x) is the SVR (Support Vector Regression) model, which is a multi-response Gaussian process. Since there are multiple responses, multiple responses satisfy:

[0127]

[0128] According to the SLFM principle, ∑0 = AA T , where A is the parameter to be optimized. N f Gaussian process responses can be expressed as:

[0129]

[0130] To make the response model satisfy the Gaussian process assumption and facilitate the solution, the Gaussian process responses are stored in a stack manner as: For the convenience of formula derivation, it is expressed as N f and N represent the number of responses and the number of samples respectively.

[0131] Through Bayesian derivation and parameter optimization solution, the mean and variance of multiple responses finally predicted and output by the response model are respectively expressed as

[0132]

[0133] Among them, represents the mean of the response predicted by the response model when the input is the sample x to be predicted, represents the variance of the response predicted by the response model when the input is the sample x to be predicted, X is the training set of the response model constructed from the current sample set, is the covariance matrix between x and X, k M (X,X) and k M (x,x) represent the covariance matrices of X itself and x itself respectively. The covariance matrix is composed of kernel functions describing the correlation between multiple responses, C h is the variant of the trade-off parameter and satisfies C h ⊙diag(C M ) = I, where ⊙ is the Hadamard product and ⊙ is used to represent element-by-element multiplication in the matrix, is the identity matrix, is the trade-off parameter in the response model, diag(C M ) is the diagonal form of the parameter matrix C composed of the trade-off parameters of multiple responses M and is a matrix with all elements being 1, which is used to transform the trade-off parameter C, is the identity matrix, is the multi-response Gaussian process, and the superscript T represents the matrix transpose, is the mean vector of N f elements, represents the mean vector of N f ×N elements, b is either b0 or b, N f and N are the number of responses and the number of samples respectively, is the state space.

[0134] In one embodiment, based on modeling multiple responses using a response model, a local development criterion is further calculated, and its main steps include gradient solution, nearest neighbor determination, and metric calculation.

[0135] Among them, the gradient solution in the local development criterion includes: constructing an extended gradient vector composed of the gradients of N responses, and the expression is where G(xc) represents the extended gradient vector corresponding to a certain candidate sample point xc in the candidate sample pool, and g i represents the gradient of the i-th response.

[0136] Calculated according to the mean of the N f responses output by the response model, the extended gradient vector can also be expressed as

[0137]

[0138] where, represents the mean of the responses predicted by the response model when the input is xc, and k M (xc,X) is the covariance matrix between xc and X, represents the covariance matrix of Y, and Y is multiple responses generated based on X in the simulation system or dataset.

[0139] The nearest neighbor determination in the local development criterion includes: the nearest neighbor is determined by Voronoi partitioning, and an example of Voronoi partitioning is shown Figure 4 as shown. Voronoi diagrams are usually used in computational geometry for global exploration and path search. The Voronoi partitioning method is used to determine the corresponding sample centers, and then the gradient projection is calculated according to the sample centers of the candidate samples. The gradients of the candidate points belonging to the same Voronoi will be projected onto the same center point.

[0140] Assume that the current sample set X consists of N sample points, and the expression is where x i represents the i-th sample point, and D represents the dimension of the sample space;

[0141] The D-dimensional sample space is divided into N cells according to the number of samples N, and the expression is C = {c1, c2,..., c N}; where, represents the j-th cell, j = 1, 2,..., N, and each cell consists of the sample points in the current sample set X and the candidate sample pool Xc;

[0142] Determine the sample centers of each candidate sample point in the candidate sample pool through Voronoi partitioning, such that the cell cj Contains all cells c in the candidate sample pool j The center of x j A point closer than other sample points, specifically defined as

[0143]

[0144] Among them, dom(x j ,x j ') is a closed half plane, with x j and x j ' is bounded by the perpendicular bisector of j The points are separated to form c j , and c j All points in the j , cell c j Specifically expressed as in, Represents cell c j The i-th candidate sample point in It is cell c j The number of candidate sample points in x j 'To distinguish from x j Another center point.

[0145] After determining the nearest neighbor, further index calculation is performed, including: according to the extended gradient vector G(xc) of the candidate sample point xc and the center point x * The extended gradient vector G(x * ), calculate the nearest neighbor gradient projection of the candidate sample point xc, expressed as

[0146]

[0147] Among them, x * It is the center point of the cell where the candidate sample point xc is located obtained by Voronoi division.

[0148] It can be understood that the local development criterion based on the nearest neighbor gradient projection measures the drastic degree of local changes of the sample point by projecting the sample point gradient and the gradient direction of the nearest neighbor with the help of the sample point gradient obtained by the extended gradient vector. Gradient projection can not only measure the gradient norm of the sample point itself, but also measure the gradient change of the sample point relative to other sample points. For areas with large gradient norm and large gradient direction changes, the local changes are more drastic and the uncertainty is greater.

[0149] In one of the embodiments, since the response model can simultaneously perform model construction on multiple responses, the mean and variance of multiple responses can be obtained through one model. And since the correlation of multiple responses has been considered during modeling, when conducting sequential experimental design, the differences existing in multiple responses do not need to be considered. Therefore, the product of the variances of multiple responses is used as the global exploration criterion, and the extended nearest neighbor gradient projection of multiple responses is used as the local development criterion. The two are combined through multiplication to obtain the sequential experimental design criterion.

[0150] Among them, the global exploration criterion is expressed as

[0151]

[0152] Among them, represents the variance of the i-th response predicted by the response model when the input is the candidate sample point xc, and N f is the number of responses.

[0153] The multiplication criterion is used to combine the local development criterion and the global exploration criterion to construct a comprehensive score function, which is expressed as

[0154]

[0155] Among them, local(xc) = Gp(xc) represents the local development criterion, Gp(xc) is the nearest neighbor gradient projection of the candidate sample point xc, and RC(local(xc), global(xc)) represents the combination of the local development criterion and the global exploration criterion.

[0156] According to the sorting of score(xc), new sample points in the candidate sample pool are selected and added to the initial sample set. Among them, the screening rule for new sample points is expressed as

[0157]

[0158] Among them, x new is the new sample point, and Xc is the candidate sample pool.

[0159] Furthermore, this method also includes: the stopping criterion is determined by the modeling accuracy of the response model, and the modeling accuracy is described by the average value of multiple single-response error metrics, which are respectively expressed as

[0160]

[0161] Among them, aR 2 represents the average value of the fitting coefficient R 2 of the test set, aRMAE represents the average value of the relative maximum error RMAE of the test set, N f is the number of responses, and y j represents the j-th test sample xj The true response, and j = 1, 2, …, n, represents the predicted response of the test set, n is the number of the test set, y represents the true response of all test samples, represents the predicted response of all test samples, y ij is the true value of the i-th response of the j-th test sample x j is the predicted value of the i-th response of the j-th test sample x j represents the mean of the true response of the j-th test sample x j y i is the true value of the i-th response of all test samples, is the predicted value of the i-th response of all test samples.

[0162] Among them, the fitting coefficient R 2 and the relative maximum absolute error (RMAE) characterize the selection effect of sample points. R 2 is used as a measure of model accuracy, applicable to comparing the accuracy of regression analysis, and is a measure of global accuracy. RMAE is used as a measure of local accuracy. R 2 The expressions are as follows:

[0163]

[0164] R 2 The larger it is, the higher the model accuracy. The expression of RMAE is:

[0165]

[0166] Furthermore, for ease of understanding, the corresponding means in the following text are respectively denoted as R 2 and RMAE. Different error metrics provide different perspectives for measuring the accuracy of the model.

[0167] To verify the effectiveness of the method proposed in this application, the sequential experimental design method for multi-response modeling considering response correlation is compared with the random sampling method (Random) and the SSS (sequential space filling) method. The experimental design processes of the two are as Figure 5 shown. Among them, the random sampling method randomly selects new sample points in the sample pool and adds them to the sample set. There are many SSS methods. One of them is the method based on the maximum-minimum distance criterion. This method selects the newly added sample points from the candidate sample pool, and its purpose is to select the point with the largest minimum distance from the candidate sample pool to the current sample point, Figure 5 ​​Among them, \(x\) represents the sample point of the index to be calculated, and \(x\) t represents other sample points in the candidate pool, and \(D\) min (x) represents the index value.

[0168] When comparing various methods, sequential experimental design is carried out under the condition of the initial sample point. Four datasets, namely Energy, Broomcorn, Polymer, and Sarcos, are used to compare different experimental design methods. A fixed test set is used for calculating the model error index. Finally, the change results of the mean values of the error indexes of multiple responses under different experimental designs are as Figures 6 to 9 shown.

[0169] Generally speaking, among the three methods, the method proposed in this application has great advantages on the three test datasets. Among them, it has great advantages throughout the whole process of sequential experimental design of the Broomcorn dataset and the Sarcos dataset, as well as in the later stage of the experimental design of the Energy dataset and the Polymer dataset.

[0170] Comparing the other two methods, SSS is better than the random sampling method. In the Broomcorn dataset and the Sarcos dataset, SSS is better than the random sampling method in the error indexes RMAE and R 2 . Theoretically speaking, based on the input SSS, the uniformity and filling of the sample point distribution are considered, and it has a better space filling effect than random sampling. However, in the Energy dataset, SSS has great advantages in the early stage, but as the number of sample points increases, this advantage is gradually weakened, which is contrary to the conclusion of theoretical analysis. This may be related to the poor space filling of the dataset itself or the response distribution.

[0171] Taking the datasets Energy, Broomcorn, and Sarcos as examples, 30 sample points are added to these three datasets. Taking the model accuracy when randomly selecting 30 sample points as the benchmark, the number of sample points when the method proposed in this application and the SSS method reach the same error accuracy is statistically counted. Calculate the number of saved sample points, expressed as a percentage. The specific results are shown in Table 2.

[0172] Table 2 The accuracy improvement effects of the method proposed in this application and the SSS method relative to random sampling

[0173]

[0174] Table 3 The accuracy improvement effect of the method proposed in this application relative to SSS

[0175] Dataset RMAE <![CDATA[R 2 > Sarcos 18(40％) 21(30％) Energy 19(36.67％) 20(33.33％) Broomcorn 28(6.67％) 27(10％)

[0176] In the dataset Energy, 15 (50%) respectively indicate that for SSS to reach the accuracy when randomly selecting 30 sample points, only 15 sample points are required, saving 50% of the number of sample points. It can be seen from the quantization results that both of these two methods can effectively save the number of sample points. Except for the error metric R in the dataset Energy 2 in, in other datasets, the sample quantity can be reduced by 40% - 50%. Further, taking the accuracy when SSS obtains 30 sample points as the benchmark, the number of sample points required for the method proposed in this application to reach this accuracy is statistically counted, and the results are shown in Table 3. It can be seen that compared with SSS, the method proposed in this application can effectively save the sample volume. Especially in the datasets Sarcos and Energy, more than 30% of the sample volume is saved when reaching the same error threshold.

[0177] For the specific limitations of the sequential experimental design device for multi-response modeling considering response correlation, reference can be made to the limitations of the sequential experimental design method for multi-response modeling considering response correlation in the above text, which will not be elaborated here. Each module in the above sequential experimental design device for multi-response modeling considering response correlation can be implemented in whole or in part through software, hardware, and their combination. The above-mentioned modules can be embedded in the processor in the computer device in hardware form or be independent of it, or can be stored in the memory in the computer device in software form, so as to facilitate the processor to call and execute the operations corresponding to the above respective modules.

[0178] In one embodiment, a computer device is provided. This computer device can be a terminal, and its internal structure diagram can be as Figure 10 shown. This computer device includes a processor, a memory, a network interface, a display screen, and an input device connected through a system bus. Among them, the processor of this computer device is used to provide computing and control capabilities. The memory of this computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The network interface of this computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, it realizes a sequential experimental design method for multi-response modeling considering response correlation. The display screen of this computer device can be a liquid crystal display screen or an electronic ink display screen. The input device of this computer device can be a touch layer covering the display screen, or can be a button, a trackball, or a touchpad set on the shell of the computer device, or can also be an external keyboard, a touchpad, or a mouse, etc.

[0179] Those skilled in the art can understand, Figure 10The structure shown is only a block diagram of some structures related to the solution of this application, and does not constitute a limitation on the computer device to which the solution of this application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine some components, or have a different component layout.

[0180] In one embodiment, a computer device is provided, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the following steps are implemented:

[0181] Step 1, sample and obtain an initial sample set from the candidate sample pool of the application system; wherein, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of the anthropomorphic robotic arm.

[0182] Step 2, construct an experimental scheme for anthropomorphic robotic arm inverse dynamics modeling based on the initial samples, and simulate to generate multiple responses of the experimental scheme; wherein, the response is the torques of multiple joints of the anthropomorphic robotic arm.

[0183] Step 3, use a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses. The response model constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization.

[0184] Step 4, determine whether a preset stopping criterion is reached. If so, stop the sequential experimental design; if not, proceed to the next step.

[0185] Step 5, calculate the local development criterion based on extended nearest neighbor gradient projection. The local development criterion calculation includes gradient solving, nearest neighbor determination, and index calculation; wherein, the gradient solving includes constructing an extended gradient vector composed of gradients of multiple responses.

[0186] Step 6, calculate the global exploration criterion based on the variances of multiple responses.

[0187] Step 7, construct a comprehensive scoring function according to the local development criterion and the global exploration criterion, and select new sample points from the candidate sample pool to add to the initial sample set according to the comprehensive score ranking, construct a new experimental scheme, and input the newly simulated responses into the response model until the predicted output of the response model reaches the stopping criterion, completing the sequential experimental design of anthropomorphic robotic arm inverse dynamics modeling.

[0188] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the following steps are implemented:

[0189] Step 1: Sample from the candidate sample pool of the application system to obtain an initial sample set; wherein, the application system is an inverse dynamics modeling system, and the initial samples include the positions, velocities, and accelerations of multiple joints of the anthropomorphic manipulator.

[0190] Step 2: Construct an experimental scheme for the inverse dynamics modeling of the anthropomorphic manipulator based on the initial samples, and simulate to generate multiple responses of the experimental scheme; wherein, the responses are the torques of multiple joints of the anthropomorphic manipulator.

[0191] Step 3: Use a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses. The response model constructs a kernel function that describes the correlation between multiple responses, transmits information between responses according to the kernel function, and predicts the means and variances of multiple responses through Bayesian derivation and parameter optimization.

[0192] Step 4: Determine whether the preset stopping criterion is met. If it is met, stop the sequential experimental design; if not, proceed to the next step.

[0193] Step 5: Calculate the local development criterion based on extended nearest neighbor gradient projection. The local development criterion calculation includes gradient solution, nearest neighbor determination, and index calculation; wherein, the gradient solution includes constructing an extended gradient vector composed of the gradients of multiple responses.

[0194] Step 6: Calculate the global exploration criterion based on the variances of multiple responses.

[0195] Step 7: Construct a comprehensive score function according to the local development criterion and the global exploration criterion, and select new sample points from the candidate sample pool to add to the initial sample set according to the comprehensive score ranking. Construct a new experimental scheme and input the newly simulated responses into the response model until the predicted output of the response model reaches the stopping criterion, and complete the sequential experimental design of the inverse dynamics modeling of the anthropomorphic manipulator.

[0196] Those of ordinary skill in the art can understand that all or part of the processes in the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above methods. Among them, any reference to a memory, storage, database, or other medium used in the embodiments provided in the present application can include non-volatile and / or volatile memories. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM), etc.

[0197] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in this specification.

[0198] The above-described embodiments merely represent several implementation manners of the present application. The description thereof is relatively specific and detailed, but it should not be construed as a limitation on the scope of the present application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.

Claims

1. A sequential experimental design method based on multi-response modeling considering response correlation, characterized in that: The method comprises: Step 1, sampling from a candidate sample pool of an application system to obtain an initial sample set; wherein the application system is an inverse dynamics modeling system, and the initial samples include positions, velocities, and accelerations of multiple joints of an anthropomorphic robotic arm; Step 2, constructing a test scheme for inverse dynamics modeling of an anthropomorphic robotic arm based on the initial sample, and simulating and generating multiple responses of the test scheme; wherein the responses are torques of multiple joints of the anthropomorphic robotic arm; Step 3, using a multi-response Bayesian support vector machine to construct a response model that can simultaneously predict multiple responses, wherein the response model constructs a kernel function that describes the correlation between multiple responses, transfers information between the responses according to the kernel function, and predicts the means and variances of the multiple responses by using Bayesian derivation and parameter optimization solution; Step 4: Determine whether the preset stop criteria are met. If so, stop the sequential test design; if not, proceed to the next step. Step 5, performing local development criterion calculation based on extended nearest neighbor gradient projection, wherein the local development criterion calculation includes gradient solution, nearest neighbor determination and index calculation; wherein the gradient solution includes constructing an extended gradient vector composed of gradients of multiple responses; Step 6, calculating the global exploration criterion based on the variance of multiple responses; Step 7, construct a comprehensive score function based on the local development criterion and the global exploration criterion, and select new sample points in the candidate sample pool according to the comprehensive score ranking to add to the initial sample set, construct a new test plan and input the new response generated by simulation into the response model until the predicted output of the response model reaches the stopping criterion, completing the sequential experimental design of inverse dynamics modeling of the anthropomorphic robotic arm.

2. The method according to claim 1, characterized in that A response model that can simultaneously predict multiple responses is constructed using a multi-response Bayesian support vector machine. The response model constructs a kernel function that describes the correlation between multiple responses, transfers information between responses according to the kernel function, and predicts the mean and variance of multiple responses by using Bayesian derivation and parameter optimization, including: A multi-response Bayesian support vector machine is used to construct a response model that can simultaneously predict multiple responses. The response model consists of a semi-parametric latent factor structure and a trade-off parameter. The semi-parametric latent factor structure is used to quantitatively describe the variance between the responses. For each response, there is a corresponding trade-off parameter, which is used to weigh the complexity and error of the response model. The semi-parametric latent factor structure constructs a kernel function that describes the correlation between multiple responses, transfers information between the responses according to the kernel function, and predicts the means and variances of multiple responses by using Bayesian derivation and parameter optimization. The parameters to be optimized include trade-off parameters, kernel function parameters, and hyperparameters in the covariance between multiple responses. The means and variances of the multiple responses predicted by the response model are expressed as in, It represents the mean of the response predicted by the response model when the input is the sample x to be predicted. It represents the variance of the response predicted by the response model when the input is the sample to be predicted x, X is the training set of the response model constructed by the current sample set, is the covariance matrix between x and X, k M (X,X) and k M (x,x) represent the covariance matrix of X and x respectively. The covariance matrix is ​​composed of a kernel function that describes the correlation between multiple responses. h is a variation of the trade-off parameter that satisfies C h ⊙diag(C M )=I, ⊙ is the Hadamard product, and ⊙ is used to represent the element-by-element multiplication in the matrix, is the identity matrix, is the trade-off parameter in the response model, diag(C M ) is the parameter matrix C composed of multiple response trade-off parameters M The diagonal form of is a matrix whose elements are all 1, used to transform the trade-off parameter C. is the identity matrix, is a multi-response Gaussian process, the superscript T represents the matrix transpose, N f The mean vector of elements, N f The mean vector of ×N elements, b is any element between b0 and b, N f and N are the number of responses and the number of samples respectively, is the state space.

3. The method according to claim 2, characterized in that Gradient solution in local development criterion includes: constructing N f The extended gradient vector is composed of the gradients of the responses, and the expression is Among them, G(xc) represents the extended gradient vector corresponding to a candidate sample point xc in the candidate sample pool, g i represents the gradient of the i-th response; According to the response model output N f The mean of the responses is calculated, and the extended gradient vector can also be expressed as in, represents the mean of the response predicted by the response model when the input is xc, k M (xc,X) is the covariance matrix between xc and X, Represents the covariance matrix of Y, where Y is the multiple responses generated from X in a simulation system or dataset.

4. The method according to claim 3, characterized in that Nearest neighbor determination in local development criteria, including: Assuming that the current sample set X consists of N sample points, the expression is Among them, x i represents the i-th sample point, and D represents the dimension of the sample space; According to the number of samples N, the D-dimensional sample space is divided into N units, and the expression is C = {c1, c2, ..., c N };in, represents the jth cell, j = 1, 2, ..., N, and each cell consists of the sample points in the current sample set X and the candidate sample pool Xc; The sample center of each candidate sample point in the candidate sample pool is determined by Voronoi division, so that the cell c j Contains all cells c in the candidate sample pool j The center of x j A point closer than other sample points, specifically defined as Among them, dom(x j ,x j ') is a closed half plane, with x j and x j ' is bounded by the perpendicular bisector of j The points are separated to form c j , and c j All points in the closest to x j , cell c j Specifically expressed as in, Represents cell c j The i-th candidate sample point in It is cell c j The number of candidate sample points in x j 'To distinguish from x j Another center point.

5. The method according to claim 4, characterized in that Calculation of indicators in local development criteria, including: According to the extended gradient vector G(xc) of the candidate sample point xc and the center point x * The extended gradient vector G(x * ), calculate the nearest neighbor gradient projection of the candidate sample point xc, expressed as Among them, x * It is the center point of the cell where the candidate sample point xc is located obtained by Voronoi division.

6. The method according to claim 1, characterized in that The global exploration criterion is expressed as in, represents the variance of the i-th response predicted by the response model when the input is the candidate sample point xc, N f The number of responses.

7. The method according to claim 1, characterized in that A comprehensive score function is constructed according to the local development criterion and the global exploration criterion, and new sample points in the candidate sample pool are selected according to the comprehensive score ranking to be added to the initial sample set, including: The multiplication criterion is used to integrate the local development criterion and the global exploration criterion to construct a comprehensive score function, which is expressed as Among them, local(xc)=Gp(xc) represents the local development criterion, Gp(xc) is the nearest neighbor gradient projection of the candidate sample point xc, represents the global exploration criterion, represents the variance of the i-th response predicted by the response model when the input is the candidate sample point xc, N f For the number of responses, RC(local(xc),global(xc)) represents the integration of the local exploitation criterion and the global exploration criterion; According to the ranking of score(xc), new sample points in the candidate sample pool are selected to be added to the initial sample set, where the screening rule for new sample points is expressed as Among them, x new is a new sample point, and Xc is a candidate sample pool.

8. The method according to claim 1, characterized in that The method further includes: the stopping criterion is determined by the modeling accuracy of the response model, and the modeling accuracy is described by the average value of multiple single response error indicators, which are respectively expressed as Among them, aR 2 Represents the fitting coefficient R of the test set 2 The average value of RMAE is the average value of the relative maximum error RMAE of the test set, N f is the number of responses, y j represents the jth test sample x j The true response of , and j = 1, 2, ..., n, represents the predicted response of the test set, n is the number of test sets, and y represents the true response of all test samples. represents the predicted response of all test samples, y ij is the jth test sample x j The true value of the ith response, is the jth test sample x j The predicted value of the ith response, represents the jth test sample x j The mean of the true response, y i is the true value of the ith response of all test samples, is the predicted value of the i-th response for all test samples.

9. A computer device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 8 are implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.

Images