A search method for multi-objective optimization solution set based on small samples

Abstract

The present invention discloses a search method for a multi-objective optimization solution set based on small samples. It uses the Halton sampling method to sample the sample space, so that the initial samples have a high filling rate in the sample space. Then, the true value is obtained by using an integrated calculation method. Next, the method of surrogate model is used to establish a non-linear regression equation to train and predict the data set. And for the multi-objective optimization problem, the NSGA-II fast non-dominated sorting algorithm is used to find the better solutions. Finally, the Pareto front is obtained through non-dominated sorting, and it is judged whether the points on the Pareto front all meet the target constraint requirements. After screening the points through K-means clustering, the population individuals are searched and screened in each category. This method can quickly search for different types of better solutions while ensuring the diversity of solutions, and the classification can be verified through integrated calculation. After multiple iterations of adding points, it finally meets the actual requirements.

Description

Technical Field

[0001] The present invention relates to the optimization design technology of hydrogen combustion chambers, and particularly to a search method for multi-objective optimization solution sets based on small samples. Background Art

[0002] Currently, in the structural design engineering problems in the aerospace field, the engineering requirements faced in the design are how to find more design solutions that meet the design constraints with fewer samples. The constrained optimization design of a hydrogen combustion chamber refers to taking a micro-mixed hydrogen combustion chamber as the research object and studying the influence laws of its five main geometric parameters (the aperture D of the hydrogen injection hole j , the height h of the air deflector gate , the width b of the air deflector gate , the height D of the combustion chamber AGP , and the circumferential distance s of the air jet holes) on NO x and η. In engineering, the Latin hypercube sampling is mainly used to find a better solution scheme, and parametric modeling is performed on all the sampled data, resulting in a high calculation cost and a complex calculation process; at the same time, when the Latin hypercube method collects samples, it cannot judge the position of the better solution region, and the sample points generated in each round are distributed in the entire sample space. Although it ensures a high space filling rate of the sampling points, the overhead is large, resulting in an increase in calculation time. Summary of the Invention

[0003] Aiming at the above deficiencies in the prior art, the search method for multi-objective optimization solution sets based on small samples provided by the present invention solves the problems of large sample collection quantity and long sample calculation time faced when finding a better solution scheme in the constrained optimization design of hydrogen combustion chambers.

[0004] To achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0005] Provide a search method for multi-objective optimization solution sets based on small samples, which includes the steps of:

[0006] S1. Use the Halton sampling method to collect samples in the Space sample space composed of five geometric parameters of the hydrogen combustion chamber, and stop sampling when the samples meet the preset conditions to obtain a data set;

[0007] S2. Use the data set to train two Gaussian regression models respectively to obtain a first surrogate model for generating the output parameter NO x and a second surrogate model for generating the output parameter η;

[0008] S3. Randomly generate several individuals with the same dimension as the samples in the multi-objective decision space as the initial population, and use the first surrogate model and the second surrogate model to generate the output parameters NO x and η of the initial population;

[0009] S4. Based on the initial population, use the NSGA-II multi-objective genetic algorithm to find the optimal solutions, and retain the non-dominated solutions at level 1 in the population as the Pareto frontiers.

[0010] S5. Input the points on the Pareto frontiers into the first surrogate model and the second surrogate model respectively to obtain the output parameters NO x and η, and retain the points that satisfy NO x ≤2.5 and η≥95% for K-means clustering.

[0011] S6. Randomly select two points in each cluster obtained by clustering, and perform hydrogen combustion chamber modeling for each point to calculate the real output parameters NO x and η.

[0012] S7. Calculate the relative error between the output parameter NO x predicted by the first surrogate model for each point and the real output parameter NO x , and calculate the average value of the relative errors of all points.

[0013] S8. Judge whether the average value is less than the preset error. If so, add the selected points to the dataset for output; otherwise, after adding the selected points to the dataset, return to step S2.

[0014] Further, step S1 further includes:

[0015] S11. Use the Halton sampling method to sample in the Space sample space composed of five geometric parameters of the hydrogen combustion chamber to obtain multiple initial samples.

[0016] S12. Input the five geometric parameters corresponding to each sample into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber.

[0017] S13. Use Fluent Mesh to automatically mesh the geometric configuration, and then use the Fluent software to calculate the output parameters NO x and η of each initial sample.

[0018] S14. Judge whether the number of the output parameters NO x and η of all initial samples that satisfy NO x ≤2.5 and η≥95% is greater than the preset quantity. If so, go to step S15; otherwise, return to step S11.

[0019] S15. Use the initial samples and their corresponding output parameters NO x and the output parameter η as the dataset.

[0020] Further, step S2 further includes:

[0021] S21. Select 50 random seeds from 1 to 50, and use the random seeds to divide the data set into a training set and a test set respectively;

[0022] S22. Use the training set and the test set corresponding to the same random seed to train two Gaussian regression models respectively to generate the output parameter NO x and generate the output parameter η;

[0023] S23. Select the Gaussian regression model with the highest accuracy among all the Gaussian regression models that generate the output parameter NO x as the first surrogate model, and select the Gaussian regression model with the highest accuracy among all the Gaussian regression models of the output parameter η as the second surrogate model.

[0024] Further, step S4 further includes:

[0025] S41. According to the output target values of the initial population, perform non-dominated sorting on the initial population according to the output target values; after non-dominated sorting, perform evolutionary operations including selection, crossover, and mutation on the initial population;

[0026] S42. Perform non-dominated sorting on the individuals in the new population after the evolutionary operation, and then use the first surrogate model and the second surrogate model to predict the output parameters NO x and η of each individual;

[0027] S43. According to the output parameters NO x and η of each individual, sort each individual in ascending order, and then calculate the crowding degree for the non-dominated solutions in each layer;

[0028] S44. Sort according to the calculated crowding degree and the layer where it is located, and then select the top N individuals to form a new offspring and add them to the initial population;

[0029] S45. Determine whether the iteration times of the NSGA-II multi-objective genetic algorithm reach the maximum iteration times. If so, enter step S46; otherwise, return to step S41;

[0030] S46. Perform non-dominated sorting on the last generation population, and retain the non-dominated solutions with level 1 in the population as the Pareto front.

[0031] Further, the expression of the optimization problem of the NSGA-II multi-objective genetic algorithm is:

[0032]

[0033] where F(x) is the optimization problem; x is the variable; D jis the aperture diameter of the hydrogen injection hole; h gate is the height of the air deflector; b gate is the width of the air deflector; D AGP is the height of the combustion chamber; s is the circumferential distance of the air jet holes.

[0034] Further, in step S43, in the sorted sequence in ascending order, those with the maximum and minimum output parameters NO x and η are used as boundary solutions, and the remaining parameters in the sequence are used as intermediate solutions;

[0035] Set the crowding degree of the boundary solutions to infinity, and calculate the crowding degree of the remaining solutions in the sequence using the crowding degree calculation formula. The crowding degree calculation formula is:

[0036]

[0037] where cd[i] m is the crowding degree corresponding to the i-th solution on the m-th objective; f[i + 1] m and f[i - 1] m are the output parameters NO x or η of the (i + 1)-th and (i - 1)-th intermediate solutions on the m-th objective respectively; and are the maximum and minimum values of the output parameters NO x or η in all non-dominated solution sets on the m-th objective respectively.

[0038] Further, when the number of executions of K-means clustering is less than or equal to 2, clustering is performed using the set number of cluster centers. When the number of executions of K-means clustering is greater than 3, clustering is performed using the number of cluster centers obtained after the second clustering is completed.

[0039] Further, the method for modeling the hydrogen combustion chamber for each point to calculate the actual output parameters NO x and η includes:

[0040] Input the five geometric parameters corresponding to each point into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber;

[0041] Automatically mesh the geometric configuration using Fluent Mesh, and then calculate the output parameters NO x and η of each point using the Fluent software.

[0042] The beneficial effects of the present invention are as follows: In this solution, an initial sample set is obtained through Halton sampling, and then the surrogate model is trained based on the initial sample set. After that, the NSGA-II and K-Means methods are used to search for and classify the optimal solutions. Through classification, different low-emission combustion modes can be discovered, thus ensuring the diversity of solutions. It is possible to quickly search for different types of optimal solutions at a relatively low computational cost, and finally obtain more solutions that meet the requirements.

[0043] Compared with the existing Latin hypercube, this solution has a stronger ability to search for the optimal solution region. When the same number of optimal solutions is found, fewer samples are used, the diversity of solutions is higher, and the computational efficiency is also greatly improved. Based on the sample set obtained by this solution, different combustion modes can be obtained according to the influence of the geometric parameter values of different hydrogen combustion chamber designs on the coupling degree between the main combustion zone and the external recirculation zone, providing important support and more design ideas for the design of hydrogen combustion chambers in engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a flowchart of a search method for a multi-objective optimization solution set based on small samples.

[0045] Figure 2 It is an effect diagram of small sample optimization, where (a) is a Pareto front diagram obtained in the last round of optimization; (b) is a clustering effect diagram of the last-round Pareto front; (c) is a distribution diagram of the optimal solution schemes obtained in four rounds in total.

[0046] Figure 3 It is a schematic diagram of adding points on the two-dimensional plane of the Peaks multi-objective function.

[0047] Figure 4 It is a comparison diagram of finding the optimal solution between the Peaks multi-objective function and the Latin hypercube.

[0048] Figure 5 It is a comparison diagram of the method of this study and the small sample method for the Feilei five-dimensional multi-objective problem. DETAILED DESCRIPTION OF THE INVENTION

[0049] The following describes the specific implementation manners of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation manners. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are within the scope of protection.

[0050] Refer to Figure 1 , Figure 1 shows a flowchart of a search method for a multi-objective optimization solution set based on small samples; asFigure 1 As shown, the method S includes steps S1 to S8.

[0051] In step S1, the Halton sampling method is used to collect samples in the Space sample space composed of five geometric parameters of the hydrogen combustion chamber, and sampling stops when the samples meet the preset conditions, obtaining a data set.

[0052] In implementation, this solution preferably further includes in step S1:

[0053] S11. Use the Halton sampling method to sample in the Space sample space composed of five geometric parameters of the hydrogen combustion chamber to obtain a plurality of initial samples; when sampling, the initial sample size n initial is set to N*dimD, where the parameter N is generally set to about 2 - 10 according to the empirical ratio, and dimD is set to 5.

[0054] n initial ∝dimD = N

[0055] where dim D represents the dimension of the input parameters, and N represents the scaling factor;

[0056]

[0057] where Space represents the sample space; D j is the hydrogen injection hole diameter; h gate is the air deflector height; b gate is the air deflector width; D AGP is the combustion chamber height; s is the circumferential distance of the air jet holes.

[0058] S12. Input the five geometric parameters corresponding to each sample into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber;

[0059] S13. Use Fluent Mesh to automatically mesh the geometric configuration, and then use the Fluent software to calculate the output parameters NO x and η of each initial sample;

[0060] S14. Judge whether the number of output parameters NO x and η of all initial samples that satisfy NO x ≤2.5 and η≥95% is greater than the preset quantity; if so, enter step S15, otherwise return to step S11; the preset quantity in this solution is preferably 2.

[0061] S15. Use the initial samples and their corresponding output parameters NO x and the output parameter η as the data set.

[0062] Using the above method to select the initial samples can ensure a high initial sampling space filling rate, which is helpful for exploring the region satisfying the constraint conditions in the space and training the surrogate model.

[0063] In step S2, the two Gaussian regression models are trained respectively using the data set to obtain the first surrogate model for generating the output parameter NO x and the second surrogate model for generating the output parameter η;

[0064] In implementation, this solution preferably further includes in step S2:

[0065] S21. Select 50 random seeds from 1 to 50, and use the random seeds to divide the data set into a training set and a test set respectively; the ratio of the samples in the training set and the test set is 9:1, and the random seeds can ensure the consistency of the randomness in each division.

[0066] S22. Use the training set and the test set corresponding to the same random seed to train the two Gaussian regression models respectively to generate the output parameter NO x and the output parameter η;

[0067] S23. Select the Gaussian regression model with the highest accuracy among all the Gaussian regression models generating the output parameter NO x as the first surrogate model, and select the Gaussian regression model with the highest accuracy among all the Gaussian regression models of the output parameter η as the second surrogate model.

[0068] When selecting the first surrogate model and the second surrogate model, it is mainly achieved by using the following several evaluation criteria:

[0069] It mainly includes the mean relative error (MRE), the root mean square error (RMSE) and the coefficient of determination (R2_score), and their formulas are as follows:

[0070]

[0071] In the formula, n is the number of samples in the test set; y i is the calculated value of the test set sample; y i_pred is the model prediction value; is the mean value of the calculated values of the test set samples; the calculated values of the test set samples are obtained by the methods in steps S22 - S23.

[0072] By comparing the error indicators under different division methods, the optimal division method of the sample set can be found to train the model, so as to ensure the accuracy of the generated output parameters of the finally selected first surrogate model and second surrogate model.

[0073] In step S3, a number of individuals with the same dimension as the sample are randomly generated within the multi-objective decision space as the initial population, and the output parameters NO x and η of the initial population are generated using the first surrogate model and the second surrogate model; the sample dimension refers to the 5 geometric parameters included in the sample.

[0074] In step S4, based on the initial population, the NSGA-II multi-objective genetic algorithm is used to find the optimal solution, and the non-dominated solutions at level 1 in the population are retained as the Pareto front.

[0075] In an embodiment of the present invention, step S4 further includes:

[0076] S41. According to the output target values of the initial population, the initial population is non-dominated stratified according to the output target values; after non-dominated stratification, an evolutionary operation including selection, crossover, and mutation is performed on the initial population; the detailed implementation process of step S41 is as follows:

[0077] According to the population output target values, the initial population is non-dominated stratified according to the output target values. Non-dominated stratification is based on the non-dominated relationship. Domination means that for any two population individuals p and q in the genetic algorithm, all output target values of individual p are greater than or equal to q, and there is at least one output target on which individual p is strictly greater than individual q, then it is said that p > q, or p dominates q. Non-domination means that no output target can be found such that individual p is strictly greater than individual q.

[0078] In the design of the hydrogen combustion chamber, the output target maximum-minimum mode is set to the output parameter NO x to be the minimum, and the output parameter efficiency η to be the maximum. If p > q, it means that for individuals p and q in NO x , the output target value of individual p is strictly less than that of individual q, and on η, the output target value of individual p is strictly greater than that of individual q. Stratification is based on the domination and non-domination relationships between individuals, and each individual in the population is hierarchically divided.

[0079] Each individual first identifies all solutions that are not dominated by any other individual except itself to form the first layer. Then, this process is repeated among the remaining individuals except the first layer until all individuals are assigned to the corresponding levels. The individuals in each layer are considered equivalent, that is, the individuals in each layer are non-dominated by each other.

[0080] The mathematical expression for If f k (p) ≤ f k (q), where k ∈ {1, 2,..., r} and st f l (p) < f l(q) It is said that p > q or p dominates q, where r represents the number of output targets, Pop represents the evolutionary population of the genetic algorithm, and f k (x) and f l (p) refer to the target values obtained by model prediction for individual x and individual p on a certain output target.

[0081] After non-dominated sorting, evolutionary operations on the population are started, including selection, crossover, and mutation. Selection mainly retains individuals with higher fitness (for the initial population, the reciprocal of the level corresponding to each individual according to non-dominated sorting is directly used as the fitness). Crossover is to exchange the parameter characteristics between different excellent individuals, which is beneficial to generating more excellent individuals. Mutation can expand the range of the target space and increase the diversity of solutions. By setting the crossover and mutation probabilities in the genetic algorithm, the evolved population can be obtained.

[0082] S42. Perform non-dominated sorting on the individuals in the new population after evolutionary operations, and then use the first surrogate model and the second surrogate model to predict the output parameters NO x and η of each individual;

[0083] S43. According to the output parameters NO x and η of each individual, sort each individual in ascending order, and then calculate the crowding degree for the non-dominated solutions in each layer; in the sorted sequence in ascending order, the individuals with the maximum and minimum output parameters NO x and η are used as boundary solutions, and the remaining parameters in the sequence are used as intermediate solutions.

[0084] In step S43, set the crowding degree of the boundary solutions to infinity, and use the crowding degree calculation formula to calculate the crowding degree of the remaining solutions in the sequence. The crowding degree calculation formula is:

[0085]

[0086] where cd[i] m is the crowding degree corresponding to the i-th solution on the m-th target; f[i + 1] m and f[i - 1] m are the output parameters NO x or η of the (i + 1)-th and (i - 1)-th intermediate solutions on the m-th target respectively; and are the maximum and minimum values of the output parameters NO x or η in all non-dominated solution sets on the m-th target respectively.

[0087] S44. Sort according to the calculated crowding degree and the level where it is located, and then select the first N individuals to form a new offspring and add them to the initial population;

[0088] S45. Judge whether the iteration times of the NSGA-II multi-objective genetic algorithm reach the maximum iteration times. If so, go to step S46; otherwise, return to step S41.

[0089] S46. Perform non-dominated sorting on the last generation of the population and retain the non-dominated solutions with level 1 in the population as the Pareto front.

[0090] In this solution, combining the advantages of NSGA-II, the initial population size N is set to 500, the upper limit of the maximum iteration times is set to 500, and the RI coding method is adopted to avoid the conversion of individuals from phenotype to genotype, and the operations of crossover and mutation can be directly performed on the individuals.

[0091] In step S5, the points on the Pareto front are respectively input into the first surrogate model and the second surrogate model to obtain the output parameters NO x and η, and retain the points that satisfy NO x ≤2.5 and η≥95% for K-means clustering.

[0092] When the execution times of K-means clustering are less than or equal to 2, clustering is performed using the set number of clustering centers. When the execution times of K-means clustering are greater than 3, clustering is performed using the number of clustering centers obtained after the second clustering.

[0093] In step S6, randomly select two points in each cluster obtained by clustering, and perform hydrogen combustion chamber modeling on each point to calculate the true output parameters NO x and η:

[0094] Input the five geometric parameters corresponding to each point into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber.

[0095] Automatically perform mesh generation on the geometric configuration using Fluent Mesh, and then use the Fluent software to calculate the output parameters NO x and η of each point.

[0096] In step S7, calculate the relative error between the output parameter NO x predicted by the first surrogate model for each point and the true output parameter NO x , and calculate the average value of the relative errors of all points.

[0097] In step S8, judge whether the average value is less than the preset error. If so, add the selected points to the dataset for output; otherwise, after adding the selected points to the dataset, return to step S2.

[0098] When implemented, the expression of the optimization problem of the preferred NSGA-II multi-objective genetic algorithm in this solution is:

[0099]

[0100] Among them, F(x) is the optimization problem; x is the variable.

[0101] The following is an illustration of the effect of the search method for the multi-objective optimization solution set based on small samples in this solution with reference to the embodiments:

[0102] Embodiment 1: Optimization design of hydrogen combustion chamber

[0103] The search method of this solution is used to search for sample points. The number of sample points added to the data set and the statistical results of the average relative error in the 1st to 4th rounds during the search process are shown in Table 1.

[0104] Table 1 Comparison chart of the accuracy of the surrogate model with four rounds of adding points

[0105]

[0106] It can be seen from Table 1 that as the number of rounds of adding points increases, the average relative error of the prediction results of the better solution plans found by the model decreases. The average relative error drops to 15.02% in the third round. To avoid contingency, one more round of adding points is added on the basis of the third round, and it is found that the average relative error of the prediction results is still less than 20%, indicating the effectiveness of this solution. And there are a total of 28 points added in four rounds, and 13 of them meet the design requirements after calculation and verification.

[0107] Figure 2 In (a) of [reference], the Pareto front graph obtained by the fourth-round optimization is given. The blue points represent the points on the Pareto front, and the red points in the figure are 70 points that meet the constraint design requirements found on the Pareto front; (b) gives the clustering effect diagram of the fourth-round Pareto front, and (c) gives the output distribution diagram of the better solution plans obtained in four rounds. Through Figure 2 It can be seen that with four rounds of adding points under the condition of 40 initial sample numbers, three different types of solution sets that meet the constraints are found with a relatively small total number of samples, and these plans can be used to guide the low-emission design of the hydrogen combustion chamber.

[0108] This solution will illustrate the feasibility of the search method of this solution from a mathematical perspective in combination with Embodiment 2 and Embodiment 3:

[0109] Example 2: Verify the feasibility of this solution using the Peaks two-dimensional multi-objective function

[0110] The Peaks two-dimensional multi-objective function, combined with the traditional single-objective Peaks function, has one local minimum and two local maxima. The Peaks multi-objective function is formed by stacking two single-objective Peaks functions, where the local minimum of one Peaks function corresponds to the local maximum of the other function. The function constructed in this way can meet the requirements of multi-objective optimization. When using the Peaks two-dimensional multi-objective function, two-dimensional inputs are used to replace the five geometric parameters (five-dimensional parameters) of the hydrogen combustion chamber in this scheme, and the sample space is also replaced with a two-dimensional one. In the NSGA-II multi-objective genetic algorithm, its optimization problem is replaced with a two-dimensional optimization problem, and the output of this scheme is replaced with two output constraints: output 1 is greater than 5, and output 2 is greater than 7. Other method steps of this scheme remain unchanged; the optimization constraint equation of the two-dimensional mathematical problem is set as follows in the two-dimensional variable space:

[0111]

[0112] After the equation is established, a search method for the multi-objective optimization solution set based on small samples is adopted. The output functions are set to maximize f1(x) and f2(x), and it is required to satisfy (f1(x)≥5, f2(x)≥7). Then, the search method of the above-mentioned replaced and adjusted scheme is run, and the search results are as Figure 3 and Figure 4 shown.

[0113] Figure 3 is a schematic diagram of adding points on the two-dimensional plane of the Peaks multi-objective function. The blue points represent the initial sample points, and the red points are the points added iteratively using the method of this study. From Figure 3 it can be seen that the method of this study found more qualified solutions with fewer initial points, and three different regions of points were found through clustering.

[0114] Figure 4 is a comparison chart of the Peaks multi-objective function and Latin hypercube for finding better solutions. Figure 4 The ordinate in it represents the number of better solutions, and the abscissa represents the current number of samples. Because the scatter points of the Latin hypercube are more random, its envelope is obtained by repeating the drawing several times. It can be more intuitively found from the figure that under the condition of finding the same number of better solutions, the adjusted scheme uses fewer samples.

[0115] Under the same method idea, the effect of the scheme obtained by replacing the input and output and the optimization problem is still significantly better than that of the Latin hypercube, thus verifying the effectiveness of the search scheme provided by this scheme at the mathematical level.

[0116] Example 3: Use the Feilei five-dimensional multi-objective function to verify the feasibility of this scheme

[0117] The Feilei five-dimensional multi-objective function uses a sine function and a hyperbolic function when constructing the function, resulting in a strong non-linear relationship between the output and the input. When using the Feilei five-dimensional multi-objective function, five-dimensional input is used to replace the five geometric parameters (five-dimensional parameters) of the hydrogen combustion chamber in this solution, and the output of this solution is replaced with two output constraints: output 1 is greater than 18, and output 2 is greater than 80. The other method steps of this solution remain unchanged; the optimization constraint equation for the five-dimensional mathematical problem is set as follows:

[0118]

[0119] After the equation is established, a search method for the multi-objective optimization solution set based on small samples is adopted, and the output functions are set to maximize f1(x) and f2(x), and it is necessary to satisfy (f1(x)≥18, f2(x)≥80); then run the above-mentioned search method after replacement and adjustment, and the search results are as Figure 5 shown.

[0120] Figure 5 It is a comparison chart of the optimization effects of the Feilei five-dimensional multi-objective problem using the small sample method and the Latin hypercube method. The blue curve in the figure represents the relationship between the number of points added by the method in this study and the number of points found. In order to reduce the randomness of the Latin hypercube sampling itself and avoid the influence of accidental results, a way of envelope diagram is adopted to display the results of multiple samplings. The red curve in the figure represents the average value of the number of relatively optimal solution sets found in multiple samplings, and the blue envelope is composed of the maximum and minimum values of the number of relatively optimal solution sets obtained from each sampling, showing the range of data changes. From Figure 5 it can be obtained that the method in this study has stronger relatively optimal solution search ability compared with the Latin hypercube sampling method in high-dimensional problems.

[0121] The effectiveness of this method is proved in engineering through Example 1. It is compared with the Latin hypercube sampling method in the 2D and 5D mathematical problems in Example 2 and Example 3, and the results verify the effect and feasibility of this method. Visual analysis can be carried out through application on the 2D mathematical function, analyzing the distribution of its relatively optimal solutions and the search ability for different types of relatively optimal solutions. Since the 5D mathematical function has the same dimension as the actual engineering problem, the effect of this method can be further verified on the 5D mathematical problem.

Claims

1. A search method for multi-objective optimization solution sets based on small samples, characterized in that, Including steps: S1. Use the Halton sampling method to collect samples within the Space sample space composed of five geometric parameters of the hydrogen combustion chamber, and stop sampling when the samples meet the preset conditions to obtain a data set; S2. Use the dataset to train two Gaussian regression models respectively to obtain the first surrogate model that generates the output parameter NO x and the second surrogate model that generates the output parameter η; S3. Randomly generate a number of individuals with the same dimension as the sample in the multi-objective decision space as the initial population, and use the first surrogate model and the second surrogate model to generate the output parameters NO x and η; S4. According to the initial population, use the NSGA-II multi-objective genetic algorithm to find the optimal solutions, and retain the non-dominated solutions with level 1 in the population as the Pareto front; S5. Input the points on the Pareto front into the first surrogate model and the second surrogate model respectively to obtain the output parameters NO x and η, and retain the points that satisfy NO x ≤ 2.5, η ≥ 95% for K-means clustering; S6. Randomly select two points in each cluster obtained by clustering, and perform hydrogen combustion chamber modeling for each point to calculate the true output parameters NO x and η; S7. Calculate the output parameter NO of each point predicted by the first surrogate model x and the true output parameter NO x to calculate the relative error therebetween, and calculate the average value of the relative errors of all points; S8. Judge whether the average value is less than the preset error. If so, add the selected points to the data set for output. Otherwise, after adding the selected points to the data set, return to step S2.

2. The search method according to claim 1, wherein Step S1 further includes: S11. Use the Halton sampling method to sample within the Space sample space composed of five geometric parameters of the hydrogen combustion chamber to obtain a number of initial samples; S12. Respectively input the five geometric parameters corresponding to each sample into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber; S13. Automatically mesh the geometric configuration using Fluent Mesh, and then calculate the output parameters NO x and η of each initial sample using Fluent software; S14. Determine the output parameter NO of all initial samples x and η that satisfy NO x ≤ 2.5, η ≥ 95% to check if the number is greater than the preset quantity; if so, proceed to step S15, otherwise return to step S11; S15. Use the initial sample and its corresponding output parameter NO x and the output parameter η as the data set.

3. The search method according to claim 1, wherein Step S2 further includes: S21. Select 50 random seeds from 1 to 50, and use the random seeds to divide the data set into a training set and a test set respectively; S22. Train two Gaussian regression models using the training set and test set corresponding to the same random seed respectively to generate the output parameter NO x and generate the output parameter η; S23. Select all the generated output parameters NO x Among the Gaussian regression models, select the one with the highest accuracy as the first surrogate model, and among the Gaussian regression models of all output parameters η, select the one with the highest accuracy as the second surrogate model.

4. The search method according to claim 1, wherein Step S4 further includes: S41. According to the output target values of the initial population, perform non-dominated stratification on the initial population according to the output target values; after non-dominated stratification, perform evolutionary operations including selection, crossover, and mutation on the initial population; S42. Perform non-dominated sorting on the individuals in the new population after the evolutionary operation, and then use the first surrogate model and the second surrogate model to predict the output parameters NO x and η; S43. According to the output parameters NO x and η of each individual, sort each individual in ascending order, and then calculate the crowding degree of the non-dominated solutions in each layer; S44. Sort according to the calculated crowding degree and the level where it is located, and then select the first N individuals to form a new offspring and add them to the initial population; S45. Judge whether the number of iterations of the NSGA-II multi-objective genetic algorithm reaches the maximum number of iterations. If so, enter step S46. Otherwise, return to step S41; S46. Perform non-dominated stratification on the last generation population, and retain the non-dominated solutions with level 1 in the population as the Pareto front.

5. The search method according to claim 1 or 4, characterized in that, The expression of the optimization problem of the NSGA-II multi-objective genetic algorithm is: Among them, F(x) is the optimization problem; x is the variable; D j is the hydrogen injection hole diameter; h gate is the air deflector height; b gate is the air deflector width; D AGP is the combustion chamber height; s is the circumferential distance of the air jet holes.

6. The search method according to claim 4, wherein In step S43, in the ascending-sorted sequence, the ones with the maximum and minimum output parameters NO x and η are used as boundary solutions, and the remaining parameters in the sequence are used as intermediate solutions; Set the crowding degree of the boundary solutions to infinity, and use the crowding degree calculation formula to calculate the crowding degree of the remaining solutions in the sequence. The crowding degree calculation formula is: Among them, cd[i] m is the crowding degree corresponding to the i-th solution on the m-th objective; f[i + 1] m and f[i - 1] m are the output parameters NO x or η of the (i + 1)-th and (i - 1)-th intermediate solutions on the m-th objective, respectively; and are the maximum and minimum values of the output parameter NO x or η in all non-dominated solution sets on the m-th objective, respectively.

7. The search method according to claim 1, wherein When the number of executions of K-means clustering is less than or equal to 2, perform clustering using the set number of cluster centers. When the number of executions of K-means clustering is greater than 3, perform clustering using the number of cluster centers obtained after the second clustering is completed.

8. The search method according to claim 1, characterized in that Modeling the hydrogen combustion chamber for each point to calculate the true output parameters NO x and η includes: Respectively input the five geometric parameters corresponding to each point into the Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber; Automatically mesh the geometric configuration using Fluent Mesh, and then calculate the output parameters NO x and η using Fluent software.

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