A constrained uniform experimental design method based on charge repulsion

Abstract

The application discloses a constrained region uniform experiment design method based on a charge repulsion method and belongs to the technical field of simulation experiment design. In order to solve the application efficiency of simulation experiment, the application comprises the following steps: obtaining a plurality of groups of output data of a first stage of calling a ToPDE algorithm to construct a test set of the constrained region uniform experiment design method based on the charge repulsion method; constructing a population based on the charge repulsion; constructing a population evolution stage method based on the charge repulsion; deleting samples with minimum normalized Euclidean distance in a design sample set based on the charge repulsion based on MD criteria; identifying corresponding samples with MD values by using the test set of the constrained region uniform experiment design method based on the charge repulsion method to obtain a supplemented design sample set based on the charge repulsion; comparing fitness functions of sample sets before and after evolution based on the charge repulsion; and judging whether to continue iteration or to obtain a simulation result.

Description

Technical Field

[0001] This invention belongs to the field of simulation experiment design technology, specifically relating to a method for designing uniform experiments in constrained regions based on the charge repulsion method. Background Technology

[0002] In the current context of technological development, high-cost simulation technologies have become key tools in industrial design, systems engineering, and scientific research. These simulation technologies play a crucial role in providing accurate predictions and risk assessments, particularly in fields such as aerospace, military defense, the automotive industry, and bioengineering. One of the major challenges currently facing us is how to effectively extract the maximum amount of information from limited experimental data within high-cost and technologically complex simulation environments.

[0003] For some complex simulation experiment design problems, due to the interactions between decision variables and the physical properties of the product itself, each decision variable not only has upper and lower bound constraints, but also often many inequalities and equality constraints. Currently, a relatively advanced algorithm for uniform experimental design problems in constrained regions is the two-phase differential evolution (ToPDE) algorithm. The ToPDE algorithm decomposes the uniform design problem in constrained regions into a two-stage optimization problem, using differential evolution as the main evolutionary method to generate a relatively ideal uniformly distributed sample point within the constrained region. Specifically, the first stage of the algorithm addresses how to generate a sufficient number of sample points that meet the constraints. This is achieved by using the degree of constraint violation by an individual as a fitness function, minimizing this function to achieve convergence of sample points towards the constrained region. Furthermore, the algorithm uses a subpopulation strategy to further enhance the diversity of samples satisfying the constraints. The second stage of the algorithm addresses how to make the samples evenly distributed within the constrained region. This is achieved by using the minimum distance between each individual and other individuals as a fitness function, maximizing this function to achieve a uniform distribution of sample points. While ToPDE can effectively solve uniform experimental design problems in constrained regions, its iterative point-elimination approach to improve sample uniformity significantly limits the direction of population evolution. Each evolutionary step involves only the coordinate changes of individual samples rather than a more refined global evolution. Furthermore, ToPDE's stopping criterion is that the algorithm fails to improve the objective function after a specified number of consecutive iterations, leading to a large amount of useless computation and resulting in low evolutionary efficiency. Summary of the Invention

[0004] The problem this invention aims to solve is to design an efficient experimental design method to improve the cost-effectiveness and application efficiency of simulation. It proposes a uniform experimental design method for constrained regions based on the charge repulsion method.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for designing uniform experiments in a confined region based on charge repulsion includes the following steps:

[0007] S1. Construct initialization conditions for the design of uniform experiments in constrained regions;

[0008] S2. Based on the initialization conditions for the uniform experimental design of the constrained region obtained in step S1, the first stage of the ToPDE algorithm is called multiple times to obtain multiple sets of output data of the first stage of the ToPDE algorithm, and a test set of the uniform experimental design method of the constrained region based on the charge repulsion method is constructed.

[0009] S3. Randomly select data from the test set of the uniform experimental design method for constrained regions based on charge repulsion obtained in step S2 to form a population based on charge repulsion;

[0010] S4. Construct a population evolution stage method based on charge repulsion, input the population based on charge repulsion obtained in step S3 into the constructed population evolution stage method based on charge repulsion for evolution, and output a design sample set based on charge repulsion.

[0011] S5. Based on the MD criterion, delete the samples with the minimum normalized Euclidean distance in the design sample set based on charge repulsion obtained in step S4; then use the test set of the uniform experimental design method for constrained region based on charge repulsion obtained in step S2 as the test set of the MD criterion to calculate the MD value, and identify the corresponding samples that obtained the MD value in the test set of the uniform experimental design method for constrained region based on charge repulsion, and add them to the design sample set based on charge repulsion to obtain the supplemented design sample set based on charge repulsion.

[0012] S6. Input the supplemented design sample set based on charge repulsion obtained in step S5 into the population evolution stage model based on charge repulsion to obtain the updated design sample set based on charge repulsion.

[0013] S7. Compare the fitness functions of the design sample set based on charge repulsion and the updated design sample set based on charge repulsion. If the fitness function of the updated design sample set based on charge repulsion is greater than that of the design sample set based on charge repulsion, return to step S5 for iterative calculation. If the fitness function of the updated design sample set based on charge repulsion is less than or equal to that of the design sample set based on charge repulsion, end the iteration and obtain the final population calculation result based on charge repulsion.

[0014] Furthermore, the initialization conditions obtained in step S1 for the uniform experimental design of the constrained region include all inequality constraints, all equality constraints, the number of factors in the simulation experiment, and the size of the sample set in the simulation experiment.

[0015] Furthermore, in step S2, the first stage of the ToPDE algorithm is called 30 times, and the output data of the first stage of the i-th call to the ToPDE algorithm is P. i The test set for the constrained region uniform experimental design method based on charge repulsion is K. .

[0016] Furthermore, the specific implementation method of step S3 is to obtain the test set of the constrained region uniform experimental design method based on charge repulsion method from step S2. Random selection Each sample constitutes a population based on charge repulsion. Suppose a population based on charge repulsion All samples in the matrix constitute a numerical matrix X.

[0017] Furthermore, the specific implementation method of step S4 includes the following steps:

[0018] S4.1. Propose the premise for the population evolution stages based on charge repulsion:

[0019] S4.1.1. It is proposed that each sample is only affected by the repulsive force of the nearest sample;

[0020] S4.1.2. It is proposed that setting 2 is used to update the position of the sample only based on the direction of the force and the evolution step size L;

[0021] S4.2. The specific method for setting and updating the sample position is as follows: For populations based on charge repulsion Any sample in the sample is only subject to the repulsive force of its nearest neighbor. Impact, according to The direction and evolution step size L are used to update the sample position, and the expression is:

[0022]

[0023]

[0024] in, For the sample The change in displacement, x i 'for The corresponding updated sample;

[0025] S4.3. Normalize the sample set X obtained in step S3, and let the number of iterations be... ,counter The formula for normalization is:

[0026]

[0027] in, For normalization , For the sample The value at the i-th level, This is the lower bound for the i-th level. This is the upper bound of the i-th level;

[0028] S4.4. For each Update the sample according to the method in step S4.2. ;

[0029] S4.5. For each obtained in step S4.4 Calculate the degree of constraint violation And make a judgment if Then its coordinates are restored to The formula for calculating the degree of constraint violation is:

[0030]

[0031]

[0032] Where l represents the number of inequality constraints, and m represents the number of equality constraints. For the k-th inequality constraint, For the k-th equality constraint, This indicates the degree of violation of the k-th inequality constraint. This indicates the degree of violation of the k-th equality constraint. This indicates the degree of relaxation of the equality constraints, taking... ;

[0033] S4.6. Order ,if If so, return to step S4.4;

[0034] S4.7. Let the sample set before 10 iterations be... Calculate X and The F-norm M of the difference, i.e., calculating If the calculated value of M is not less than the previously calculated value of M, then set the counter... ;

[0035] S4.8. If or If the result is positive, the sample set X is normalized, ending a population evolution stage based on charge repulsion; otherwise, return to step S4.4. The threshold for the counter count. This represents the maximum number of iterations.

[0036] Furthermore, the specific implementation method of step S5 includes the following steps:

[0037] S5.1. Based on the MD criterion, reset the samples with the minimum Euclidean distance and delete the samples with the minimum normalized Euclidean distance in the sample set X.

[0038] S5.2. Calculate the MD value using the test set K. The calculation formula is as follows:

[0039]

[0040] in, For the sample With population distance, For population The MD metric value, where N is the total number of samples in the test set K, and m is any value in N. In order to make The corresponding sample in the test set that achieves the maximum value. The population for the MD index to be tested;

[0041] Then Samples that achieve the maximum value The supplemented sample set is obtained by adding it to the sample set X, which is based on charge repulsion.

[0042] Furthermore, in step S7, the objective is to minimize the normalized Euclidean distance between individuals. The formula for calculating the fitness function of the population is:

[0043]

[0044] in, Let be the fitness function of the population. The minimum distance between the i-th sample and other samples. For population Size.

[0045] The beneficial effects of this invention are:

[0046] This invention discloses a constrained region uniform experimental design method based on charge repulsion. It treats sample points within the constrained region as equal-mass spheres with the same charge. The method evolves based on the physical property that objects with the same charge repel each other, aiming to maximize the minimum distance between sample points. Each iteration of this evolutionary approach involves updating the positions of all samples, resulting in a more refined global evolution capability. Furthermore, the constrained region uniform experimental design method (CRA) described in this invention uses the failure of the objective function to improve even once as a stopping condition, avoiding a large amount of time-consuming and useless computation.

[0047] The present invention discloses a constrained region uniform experimental design method based on charge repulsion, which is capable of obtaining a more uniform sample set than ToPDE and can guarantee good algorithm stability. The present invention can be used for constrained region uniform experimental design in aerospace, military defense, automotive industry and bioengineering. It is capable of dealing with various strongly constrained and high-dimensional constrained region uniform experimental problems, and can guarantee both high computational efficiency and obtain a highly uniform experimental sample set. Attached Figure Description

[0048] Figure 1 This is a flowchart of a method for designing a uniform confined region experiment based on the charge repulsion method, as described in this invention.

[0049] Figure 2 This is a flowchart of the CRA algorithm of the present invention;

[0050] Figure 3 This is the verification result based on MD for this invention;

[0051] Figure 4 This is the verification result of the present invention based on minimum distance;

[0052] Figure 5 This is a diagram showing the sample layout effect obtained by CRA in this invention;

[0053] Figure 6 The image shows the sample layout effect obtained by ToPDE for comparison. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only for explaining the invention and are not intended to limit the invention; that is, the described specific embodiments are merely a part of the embodiments of the invention, and not all of them. The components of the specific embodiments of the invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations, and the invention may also have other embodiments.

[0055] Therefore, the following detailed description of specific embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected specific embodiments of the invention. All other specific embodiments obtained by those skilled in the art based on these specific embodiments without inventive effort are within the scope of protection of this invention.

[0056] To further understand the invention's content, features, and effects, the following specific embodiments are provided, along with accompanying drawings. Figure 1 - Appendix Figure 6 Detailed explanation is as follows:

[0057] Example 1:

[0058] A method for designing uniform experiments in a confined region based on charge repulsion includes the following steps:

[0059] S1. Construct initialization conditions for the design of uniform experiments in constrained regions;

[0060] Furthermore, the initialization conditions obtained in step S1 for the uniform experimental design of the constrained region include all inequality constraints, all equality constraints, the number of factors in the simulation experiment, and the size of the sample set in the simulation experiment.

[0061] S2. Based on the initialization conditions for the uniform experimental design of the constrained region obtained in step S1, the first stage of the ToPDE algorithm is called multiple times to obtain multiple sets of output data of the first stage of the ToPDE algorithm, and a test set of the uniform experimental design method of the constrained region based on the charge repulsion method is constructed.

[0062] S3. Randomly select data from the test set of the uniform experimental design method for constrained regions based on charge repulsion obtained in step S2 to form a population based on charge repulsion;

[0063] Furthermore, the specific implementation method of step S3 is to obtain the test set of the constrained region uniform experimental design method based on charge repulsion method from step S2. Random selection Each sample constitutes a population based on charge repulsion. Suppose a population based on charge repulsion The numerical matrix formed by all samples in the matrix is ​​X.

[0064] S4. Construct a population evolution stage method based on charge repulsion, input the population based on charge repulsion obtained in step S3 into the constructed population evolution stage method based on charge repulsion for evolution, and output a design sample set based on charge repulsion.

[0065] Furthermore, the specific implementation method of step S4 includes the following steps:

[0066] S4.1. Propose the premise for the population evolution stages based on charge repulsion:

[0067] S4.1.1. It is proposed that each sample is only affected by the repulsive force of the nearest sample;

[0068] Furthermore, considering that the objective function of the evolution is to maximize the minimum distance between samples, the force source of each sample is set to the sample with which it has the minimum normalized Euclidean distance. This is because, theoretically, if all samples are used to calculate the resultant force, the uniformity of the sample distribution is not ideal due to the influence of the complex electric field on the samples;

[0069] S4.1.2. It is proposed that setting 2 is used to update the position of the sample only based on the direction of the force and the evolution step size L;

[0070] Furthermore, if the velocity and acceleration of each sample point are analyzed and calculated according to the strict physical motion formula, it may lead to some sample points that are very close to each other in the early stage of evolution gaining extremely large velocities and accelerations, making their position updates difficult to control. In addition, due to the existence of setting 1, some samples in the later stage of evolution may fall into a periodic motion state, causing the population to fail to converge to a stable state.

[0071] Furthermore, regarding the sample , ,set up for by The force, for by The forces acting on each other can be calculated using the following formula:

[0072]

[0073] In the formula, l is the proportionality coefficient, which can be left as follows: . Direction vector and The direction vectors are the same, that is:

[0074]

[0075] Combining the two equations, we get:

[0076]

[0077] S4.2. The specific method for setting and updating the sample position is as follows: For populations based on charge repulsion Any sample in the sample is only subject to the repulsive force of its nearest neighbor. Impact, according to The direction and evolution step size L are used to update the sample position, and the expression is:

[0078]

[0079]

[0080] in, For the sample The change in displacement, x i 'for The corresponding updated sample;

[0081] S4.3. Normalize the sample set X obtained in step S3, and let the number of iterations be... ,counter The formula for normalization is:

[0082]

[0083] in, For normalization , For the sample The value at the i-th level, This is the lower bound for the i-th level. This is the upper bound of the i-th level;

[0084] S4.4. For each Update the sample according to the method in step S4.2. ;

[0085] S4.5. For each obtained in step S4.4 Calculate the degree of constraint violation And make a judgment if Then its coordinates are restored to The formula for calculating the degree of constraint violation is:

[0086]

[0087]

[0088] Where l represents the number of inequality constraints, and m represents the number of equality constraints. For the k-th inequality constraint, For the k-th equality constraint, This indicates the degree of violation of the k-th inequality constraint. This indicates the degree of violation of the k-th equality constraint. This indicates the degree of relaxation of the equality constraints, taking... ;

[0089] S4.6. Order ,if If so, return to step S4.4;

[0090] S4.7. Let the sample set before 10 iterations be... Calculate X and The F-norm M of the difference, i.e., calculating If the calculated value of M is not less than the previously calculated value of M, then set the counter... ;

[0091] S4.8. If or If the result is positive, the sample set X is normalized, ending a population evolution stage based on charge repulsion; otherwise, return to step S4.4. The threshold for the counter count. This represents the maximum number of iterations.

[0092] S5. Based on the MD criterion, delete the samples with the minimum normalized Euclidean distance in the design sample set based on charge repulsion obtained in step S4; then use the test set of the uniform experimental design method for constrained region based on charge repulsion obtained in step S2 as the test set of the MD criterion to calculate the MD value, and identify the corresponding samples that obtained the MD value in the test set of the uniform experimental design method for constrained region based on charge repulsion, and add them to the design sample set based on charge repulsion to obtain the supplemented design sample set based on charge repulsion.

[0093] Furthermore, the specific implementation method of step S5 includes the following steps:

[0094] S5.1. Based on the MD criterion, reset the samples with the minimum Euclidean distance and delete the samples with the minimum normalized Euclidean distance in the sample set X.

[0095] S5.2. Calculate the MD value using the test set K. The calculation formula is as follows:

[0096]

[0097] in, For the sample With population distance, For population The MD metric value, where N is the total number of samples in the test set K, and m is any value in N. In order to make The corresponding sample in the test set that achieves the maximum value. The population for the MD index to be tested;

[0098] Then Samples that achieve the maximum value The supplemented sample set is obtained by adding it to the sample set X, which is based on charge repulsion.

[0099] S6. Input the supplemented design sample set based on charge repulsion obtained in step S5 into the population evolution stage model based on charge repulsion to obtain the updated design sample set based on charge repulsion.

[0100] S7. Compare the fitness functions of the design sample set based on charge repulsion and the updated design sample set based on charge repulsion. If the fitness function of the updated design sample set based on charge repulsion is greater than that of the design sample set based on charge repulsion, return to step S5 for iterative calculation. If the fitness function of the updated design sample set based on charge repulsion is less than that of the design sample set based on charge repulsion, end the iteration and obtain the final population calculation result based on charge repulsion.

[0101] Furthermore, in step S7, the objective is to minimize the normalized Euclidean distance between individuals. The formula for calculating the fitness function of the population is:

[0102]

[0103] in, Let be the fitness function of the population. The minimum distance between the i-th sample and other samples. For population Size.

[0104] The effects of implementing this invention are as follows:

[0105] For the need to quickly obtain simulation results, designing efficient simulation experiments is an important approach. The effectiveness of CRA is demonstrated using seven different test problems, including one two-dimensional visual EXAMPLE problem, five standard test problems (G04, G05, G09, G18, G21) selected from the 2006 IEEE Evolutionary Computation Conference "Special Session and Competition on Evolutionary Constrained Real-Parameter Single-Objective Optimization", and one crash box structural design problem from engineering practice.

[0106] In the test problem, the parameter values ​​of ToPDE and CRA (Charge repulsion algorithm for uniform experimental design in constrained regions based on charge repulsion) are shown in Tables 1 and 2, respectively. The value 30 / 100 means that the value is 30 in the EXAMPLE problem and 100 in other problems.

[0107] Table 1. Experimental values ​​of various parameters of ToDPE

[0108]

[0109] Table 2. Values ​​of CRA parameters in the experiment.

[0110]

[0111] The EXAMPLE problem includes three inequality constraints within a constrained region consisting of an ellipse, a parabola, and a straight line. The goal is to generate 30 uniformly distributed sample points within this region. First, the ToPDE and CRA algorithms were applied to the EXAMPLE problem, with each algorithm run independently 50 times. The mean and variance of the MD values ​​obtained from each run are statistically analyzed in Table 3. Figure 3 and Figure 4 Box plots of the performance metrics of the two algorithms are then presented, where Figure 3 This is a box plot of the MD index. Figure 4 For the objective function The box plots are shown in Table 3. Based on the results, it can be seen that the mean MD of CRA is significantly smaller than that of ToPDE, with an optimization rate of 10.75%. For the objective function based on the minimum distance maximization criterion, the mean of CRA is significantly larger than that of ToPDE, with an optimization rate of 34.37%. The experimental results show that CRA achieves significantly better results than ToPDE in both uniformity evaluation indices, and has a very significant advantage in maximizing the minimum distance between samples.

[0112] Table 3. Experimental results of the two algorithms on the EXAMPLE problem.

[0113]

[0114] also, Figure 5 and Figure 6 The diagram also shows the sample set distribution after CRA and ToPDE. It is clear from the diagram that the sample set obtained by CRA is more uniform and orderly than that obtained by ToPDE, with the minimum distance between all samples being almost identical. In addition, CRA has an interesting characteristic: the resulting sample set closely adheres to the constraint boundaries and is evenly distributed along the boundaries.

[0115] Tables 4 to 8 present the experimental results of CRA and ToPDE for five standard test cases under 50 repeated experiments, including the mean and variance of the MD index. The better results are bolded in the tables. The experimental results show that, compared to ToPDE, CRA achieves a lower mean MD and a higher mean objective function across all five test problems. From the perspective of algorithm stability, CRA achieves a smaller MD variance across all five test problems compared to ToPDE, but the objective function variance is slightly larger in problems G04, G05, and G18.

[0116] Table 4. Experimental results of the two algorithms on the G04 problem.

[0117]

[0118] Table 5. Experimental results of the two algorithms on the G05 problem.

[0119]

[0120] Table 6. Experimental results of the two algorithms on the G09 problem.

[0121]

[0122] Table 7. Experimental results of the two algorithms on the G18 problem.

[0123]

[0124] Table 8. Experimental results of the two algorithms on the G21 problem.

[0125]

[0126] Table 9 presents the statistical data of the experimental results of the two algorithms on the Crash Box problem. The results show that although CRA is better than ToPDE in maximizing the objective (optimization ratio of 2.02%), it is slightly worse than ToPDE in MD (decrease ratio of 2.75%). Therefore, there is no significant difference in the solution quality obtained by the two algorithms on the Crash Box problem.

[0127] A comprehensive analysis of the overall solution performance for the seven test problems was conducted, using the fitness function... When using the MD criterion as the evaluation metric, CRA achieved superior scores across all problems, with a significant improvement evident in the box plots. This demonstrates CRA's substantial advantage in maximizing the minimum value between two samples. Furthermore, when using the MD criterion, CRA outperformed ToPDE in all six test problems, only slightly lagging behind on the high-dimensional Crash_box problem. In conclusion, CRA is the superior algorithm, capable of obtaining a more uniform sample set than ToPDE while maintaining good algorithmic stability.

[0128] Table 9. Experimental results of the two algorithms on the G04 problem.

[0129]

[0130] Example 2:

[0131] This embodiment of the constrained region uniform experimental design method based on charge repulsion further introduces relevant definitions used in the CRA algorithm compared to Embodiment 1, including the description of the constrained region uniform experimental design problem, the MD criterion, operators related to the differential evolution algorithm, and the evolutionary objective function. The description of the constrained region uniform experimental design problem is a detailed mathematical description of the entire problem, a classic framework widely used in constrained region uniform experimental design problems, and it runs through the entire algorithm flow. The MD criterion is a uniformity evaluation index, used as a point deletion strategy in charge repulsion-based evolutionary methods, and also as an evaluation index for experimental results. The operators related to the differential evolution algorithm are a classic evolutionary method, representing the evolutionary approach of the first stage of the existing ToPDE algorithm. The evolutionary objective function... It is the objective function in the evolution method based on charge repulsion, and also a uniformity evaluation index.

[0132] 1.1 Description of Experimental Design Problems in Uniform Constraint Regions

[0133] The uniform experimental design problem within the constrained region is modeled as a constrained optimization problem. By solving this optimization problem, a sample set uniformly distributed within the constrained region can be obtained. The specific definition is as follows:

[0134] (1)

[0135] In the formula, X represents an n×d dimensional matrix consisting of n sample points, where each sample point represents a d-dimensional vector, and d represents the number of experimental factors. This refers to a metric used to measure the uniformity of a sample set, and maximizing this metric aims to achieve a more uniform distribution of sample points. Let represent the i-th inequality in l inequality constraints, and Let j represent the j-th equality among m equality constraints. Let X represent the k-th sample in the sample set X. D is the decision space consisting of the upper and lower bounds of each experimental factor, i.e.:

[0136] (2)

[0137] In the formula This represents the value of the k-th sample in the i-th experimental factor. and These represent the upper and lower bounds of the corresponding factors, respectively.

[0138] The degree to which each sample violates the constraints is defined as follows:

[0139] (3)

[0140] (4)

[0141] In the formula, This indicates the degree of violation of the k-th inequality constraint. This indicates the degree of violation of the k-th equality constraint. It is a very small positive number that represents the degree of relaxation of the equality constraints, and is usually taken as... .

[0142] 1.2MD Criterion

[0143] To quantitatively evaluate the results of uniform experimental designs, it is necessary to introduce evaluation criteria for the uniformity of design points. The maximum distance (MD) criterion is a commonly used distance-based criterion, and its definition is as follows:

[0144] set up Let D be a d-dimensional point set containing n sample points. A subspace and for any They all Then the distance between point set P and region D is defined as:

[0145] (5)

[0146] (6)

[0147] In the formula, The operator used to calculate the normalized Euclidean distance is defined by equations (7) and (8):

[0148] (7)

[0149] (8)

[0150] In practical calculations, a test set K containing a large number of samples within region D can be constructed to approximate all points within region D. Let... The MD index can be approximately calculated using the following formula:

[0151] (9)

[0152] In the formula, N is the size of the test set K, which is usually taken as... .

[0153] 1.3 Operators Related to Differential Evolution Algorithm

[0154] The relevant operators of the differential evolution algorithm are given by equations (10) to (12).

[0155] (10)

[0156] (11)

[0157] (12)

[0158] Equation (10) is the mutation operator. and These represent the original individual and the mutated individual, respectively. , and Three distinct integers are randomly selected from 1 to NP, rand is a random number uniformly distributed in [0,1], and F is a scaling factor with a value of 0.9. Equation (11) is the crossover operator. This represents the individuals after crossover. Similarly, CR represents a random number uniformly distributed in [0,1], where CR is the crossover control parameter with a value of 0.9. This is an integer randomly selected from 1 to d. The operator's function is to update the values ​​of each dimension of the mutated individual to the same dimension of the original individual with a 90% probability. This ensures that at least one dimension of the original individual is updated to the mutated value. Equation (12) is the selection operator, from... and The individual with the better fitness function is selected as the new individual after the update.

[0159] 1.4 Evolutionary Objective Function

[0160] With the goal of minimizing the normalized Euclidean distance between individuals, the fitness function of the population is defined as equation (13):

[0161] (13)

[0162] Example 3:

[0163] The experimental design method for a uniform confined region based on the charge repulsion method described in this embodiment also introduces the method of the first stage of ToPDE compared with Embodiment 1 and Embodiment 2.

[0164] Similar to the ToPDE framework, the CRA algorithm also requires a certain number of constrained sample points as the initial population for subsequent evolution before invoking it. Unlike ToPDE, however, the CRA algorithm requires constructing a test set K containing a large number of constrained samples for calculating the MD value of the population. This method involves repeatedly calling the first stage of the ToPDE algorithm until a test set K of a specified size is obtained. To balance computational overhead and optimization performance, this method sets the size of the test set K to 30. After the test set K is constructed, it is then randomly selected. A new population was constructed from individual samples. Used for subsequent evolution. The specific steps of the first stage of the ToPDE algorithm are as follows:

[0165] Step 2.1 Randomly generate an initial population P of size NP within the decision space.

[0166] Step 2.2 Randomly generate a sample point r within the decision space, let... Here, T represents a temporary population.

[0167] Step 2.3 In population P, find the sample point z with the smallest normalized Euclidean distance to sample point r. Then, find the (NS-1) sample points in population P with the smallest normalized Euclidean distance to sample point z. These (NS-1) sample points and sample point z together constitute a subpopulation. Where NS represents the subpopulation size.

[0168] Step 2.4 Subpopulation Removed from the original population P.

[0169] Step 2.5 Subpopulations are analyzed according to equations (10) and (11). Each individual undergoes mutation and crossover to obtain a new subpopulation. .

[0170] Step 2.6 For Each individual in And the individual that corresponds to it. Using equation (3) as the fitness function, selection is performed according to equation (12) to obtain the updated subpopulation. .

[0171] Step 2.7 Let The updated subpopulation Merge into the temporary population T, that is, let .

[0172] Step 2.8 If If so, return to step 2.3; otherwise, merge the temporary population with the remaining individuals in the original population to form a new population, i.e., let .

[0173] Step 2.9 If each subpopulation At least one Individuals that meet the constraints are randomly selected from each subpopulation. A new population is formed from several feasible samples. Otherwise, return to step 2.2. For new population Size.

[0174] Example 4:

[0175] The difference between this embodiment and embodiments 1-3 is that this embodiment is applied to a constrained region uniform experimental design in aerospace, military defense, automotive industry and bioengineering. The initialization conditions obtained in step S1 for the constrained region uniform experimental design include all inequality constraints, all equality constraints, the number of factors in the simulation experiment, and the size of the sample set in the simulation experiment.

[0176] It should be noted that relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0177] Although this application has been described above with reference to specific embodiments, various modifications can be made and components can be replaced with equivalents without departing from the scope of this application. In particular, as long as there is no structural conflict, the features in the specific embodiments disclosed in this application can be combined with each other in any way. The lack of an exhaustive description of these combinations in this specification is merely for the sake of brevity and resource conservation. Therefore, this application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A method for designing a uniformly oriented confined region experiment based on the charge repulsion method, characterized in that, Includes the following steps: S1. Construct initialization conditions for the design of uniform experiments in constrained regions; S2. Based on the initialization conditions for the uniform experimental design of the constrained region obtained in step S1, the first stage of the ToPDE algorithm is called multiple times to obtain multiple sets of output data of the first stage of the ToPDE algorithm, and a test set of the uniform experimental design method of the constrained region based on the charge repulsion method is constructed. S3. Randomly select data from the test set of the uniform experimental design method for constrained regions based on charge repulsion obtained in step S2 to form a population based on charge repulsion; S4. Construct a population evolution stage method based on charge repulsion, input the population based on charge repulsion obtained in step S3 into the constructed population evolution stage method based on charge repulsion for evolution, and output a design sample set based on charge repulsion. S5. Based on the MD criterion, remove the samples with the minimum normalized Euclidean distance from the design sample set based on charge repulsion obtained in step S4. Then, the test set of the uniform experimental design method for constrained regions based on charge repulsion obtained in step S2 is used as the test set of the MD criterion to calculate the MD value. The corresponding samples that obtained the MD value are identified in the test set of the uniform experimental design method for constrained regions based on charge repulsion and added to the design sample set based on charge repulsion to obtain the supplemented design sample set based on charge repulsion. S6. Input the supplemented design sample set based on charge repulsion obtained in step S5 into the population evolution stage model based on charge repulsion to obtain the updated design sample set based on charge repulsion. S7. Compare the fitness functions of the design sample set based on charge repulsion and the updated design sample set based on charge repulsion. If the fitness function of the updated design sample set based on charge repulsion is greater than that of the design sample set based on charge repulsion, return to step S5 for iterative calculation. If the fitness function of the updated design sample set based on charge repulsion is less than or equal to that of the design sample set based on charge repulsion, end the iteration and obtain the final population calculation result based on charge repulsion.

2. The experimental design method for a uniform confined region based on charge repulsion as described in claim 1, characterized in that, The initialization conditions obtained in step S1 for the uniform experimental design of the constrained region include all inequality constraints, all equality constraints, the number of factors in the simulation experiment, and the size of the sample set in the simulation experiment.

3. A method for designing a uniform confined region experiment based on charge repulsion as described in claim 1 or 2, characterized in that, In step S2, the first stage of the ToPDE algorithm is called 30 times. The output data of the first stage of the ToPDE algorithm during the i-th call is P. i The test set for the constrained region uniform experimental design method based on charge repulsion is K. .

4. The experimental design method for a uniform confined region based on charge repulsion as described in claim 3, characterized in that, The specific implementation method of step S3 is to obtain the test set of the constrained region uniform experimental design method based on charge repulsion method from step S2. Random selection Each sample constitutes a population based on charge repulsion. Suppose a population based on charge repulsion All samples in the matrix constitute a numerical matrix X.

5. The experimental design method for a uniform confined region based on charge repulsion as described in claim 4, characterized in that, The specific implementation method of step S4 includes the following steps: S4.

1. Propose the premise for the population evolution stages based on charge repulsion: S4.1.

1. It is proposed that each sample is only affected by the repulsive force of the nearest sample; S4.1.

2. It is proposed that setting 2 is used to update the position of the sample only based on the direction of the force and the evolution step size L; S4.

2. The specific method for setting and updating the sample position is as follows: For populations based on charge repulsion Any sample in the sample is only subject to the repulsive force of its nearest neighbor. Impact, according to The direction and evolution step size L are used to update the sample position, and the expression is: in, For the sample The change in displacement, x i 'for The corresponding updated sample; S4.

3. Normalize the sample set X obtained in step S3, and let the number of iterations be... ,counter The normalization calculation formula is as follows: in, For normalization , For the sample The value at the i-th level, This is the lower bound for the i-th level. This is the upper bound of the i-th level; S4.

4. For each Update the sample according to the method in step S4.

2. ; S4.

5. For each obtained in step S4.4 Calculate the degree of constraint violation And make a judgment if Then its coordinates are restored to The formula for calculating the degree of constraint violation is: Where l represents the number of inequality constraints, and m represents the number of equality constraints. For the k-th inequality constraint, For the k-th equality constraint, This indicates the degree of violation of the k-th inequality constraint. This indicates the degree of violation of the k-th equality constraint. This indicates the degree of relaxation of the equality constraints, taking... ; S4.

6. Order ,if If so, return to step S4.4; S4.

7. Let the sample set before 10 iterations be... Calculate X and The F-norm M of the difference, i.e., calculating If the calculated value of M is not less than the previously calculated value of M, then set the counter... ; S4.

8. If or If the result is positive, the sample set X is normalized, ending a population evolution stage based on charge repulsion; otherwise, return to step S4.

4. The threshold for the counter count, This represents the maximum number of iterations.

6. The experimental design method for a uniform confined region based on charge repulsion as described in claim 5, characterized in that, The specific implementation method of step S5 includes the following steps: S5.

1. Based on the MD criterion, reset the samples with the minimum Euclidean distance and delete the samples with the minimum normalized Euclidean distance in the sample set X. S5.

2. Calculate the MD value using the test set K. The calculation formula is as follows: in, For the sample With population distance, For population The MD metric value, where N is the total number of samples in the test set K, and m is any value in N. In order to make The corresponding sample in the test set that achieves the maximum value. The population for the MD index to be tested; Then Samples that achieve the maximum value The supplemented sample set is obtained by adding it to the sample set X, which is based on charge repulsion.

7. The experimental design method for a uniform confined region based on charge repulsion as described in claim 6, characterized in that, In step S7, the objective is to minimize the normalized Euclidean distance between individuals. The formula for calculating the fitness function of the population is: in, Let be the fitness function of the population. The minimum distance between the i-th sample and other samples. For population Size.

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