Searching method of multi-objective optimal solution set based on small samples

Abstract

The invention discloses a search method for a multi-target optimal solution set based on small samples, which adopts a Halton sampling method to sample a sample space, so that the filling rate of an initial sample in the sample space is higher. The method comprises the following steps: firstly, extracting a data set, then obtaining a true value by adopting an integrated calculation method, then establishing a nonlinear regression equation by adopting an agent model method to train and predict the data set, searching a better solution by adopting an NSGA-II fast non-dominated sorting algorithm aiming at a multi-objective optimization problem, and finally obtaining a Pareto leading edge after non-dominated sorting. According to the method, a Pareto frontier is selected, whether points on the Pareto frontier meet target constraint requirements or not is judged, after K-means clustering is carried out on the screened points, population individuals in each category are searched and screened, different types of better solutions can be quickly searched under the condition that the diversity of the solutions is guaranteed, classification can be verified through integrated calculation, and the accuracy of classification is improved. And finally, actual requirements are met after multiple times of iteration point adding.

Description

Technical Field

[0001] The invention relates to a hydrogen combustion chamber optimization design technology, and in particular to a search method for a multi-objective optimization solution set based on a small sample. Background Art

[0002] In the current structural design engineering problems in the aerospace field, the engineering requirement faced during design is how to find more design solutions that meet the design constraints with fewer samples. The constrained optimization design of hydrogen combustion chamber refers to taking the micro-mixed hydrogen combustion chamber as the research object and studying its five main geometric parameters (hydrogen nozzle aperture D j 、Height of air deflector h gate 、Width of air deflector b gate , Combustion chamber height D AGP , the circumferential distance of the air jet hole (s) to NO x The main method for finding the optimal solution in engineering is Latin hypercube sampling, and all the data after sampling are parameterized and modeled. The calculation cost is high and the calculation process is complicated. At the same time, the Latin hypercube method cannot determine the location of the optimal solution area when collecting samples. The sample points generated in each round will be distributed in the entire sample space. Although the sampling points are highly filled in the space, the overhead is large, which increases the calculation time. Summary of the invention

[0003] In view of the above-mentioned deficiencies in the prior art, the search method for a multi-objective optimization solution set based on small samples provided by the present invention solves the problems of a large number of collected samples and a long sample calculation time when looking for a better solution in the constrained optimization design of a hydrogen combustion chamber.

[0004] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0005] A method for searching a multi-objective optimization solution set based on a small sample is provided, which comprises 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, stop sampling when the sample meets the preset conditions, and obtain a data set;

[0007] S2, use the data set to train two Gaussian regression models respectively to obtain the generated output parameter NO x A first proxy model of and a second proxy model generating an output parameter η;

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

[0009] S4. Based on the initial population, the NSGA-II multi-objective genetic algorithm is used to find the optimal solution, and the non-dominated solution with level 1 in the population is retained as the Pareto frontier;

[0010] S5. Input the points on the Pareto front into the first proxy model and the second proxy model respectively to obtain the output parameter NO x and η, and retains NO x K-means clustering was performed for points with ≤2.5 and η≥95%;

[0011] 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 parameter NO x and η;

[0012] S7. Calculate the output parameter NO of each point predicted by the first proxy model x With true output parameter NO x The relative error between them is calculated, and the average relative error of all points is calculated;

[0013] S8. Determine whether the average value is less than a preset error. If so, add the selected points to the data set output. Otherwise, add the selected points to the data set and return to step S2.

[0014] Furthermore, step S1 further comprises:

[0015] S11, using the Halton sampling method to sample in the Space sample space formed by five geometric parameters of the hydrogen combustion chamber to obtain multiple initial samples;

[0016] S12, inputting the five geometric parameters corresponding to each sample into 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 Fluent software to calculate the output parameters of each initial sample NO x and η;

[0018] S14, determine the output parameters of all initial samples NO x and η satisfy NO x ≤2.5, η≥95% is greater than the preset number; if so, proceed to step S15, otherwise return to step S11;

[0019] S15, using the initial sample and its corresponding output parameter NO x and output parameter η as a data set.

[0020] Furthermore, step S2 further comprises:

[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, respectively train two Gaussian regression models using the training set and test set corresponding to the same random seed to generate output parameter NO x and generate output parameter η;

[0023] S23, select all generated output parameters NO x The Gaussian regression model with the highest accuracy is used as the first proxy model, and the Gaussian regression model with the highest accuracy among all output parameters η is used as the second proxy model.

[0024] Furthermore, step S4 further comprises:

[0025] S41, according to the output target value of the initial population, performing non-dominated stratification on the initial population according to the output target value; after the non-dominated stratification, performing evolutionary operations including selection, crossover and mutation on the initial population;

[0026] S42, perform non-dominated stratification on the individuals in the new population after the evolution operation, and then use the first agent model and the second agent model to predict the output parameter NO of each individual x and η;

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

[0028] S44, sorting according to the calculated crowding degree and the level, and then selecting the first N individuals to form a new offspring and add them to the initial population;

[0029] S45, determining whether the number of iterations of the NSGA-II multi-objective genetic algorithm has reached the maximum number of iterations, if so, proceeding to step S46, otherwise returning to step S41;

[0030] S46. Perform non-dominated stratification on the last generation of population and retain the non-dominated solutions with level 1 in the population as the Pareto frontier.

[0031] Furthermore, the optimization problem of the NSGA-II multi-objective genetic algorithm is expressed as:

[0032]

[0033] Among them, F(x) is the optimization problem; x is the variable; D jis the diameter of the hydrogen nozzle; 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 sequence after ascending sorting, the maximum and minimum output parameters NO x and η are taken as boundary solutions, and the remaining parameters in the sequence are taken as intermediate solutions;

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

[0036]

[0037] where cd[i] m is the congestion 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 of the i+1th and i-1th intermediate solutions on the mth target respectively. x or η; and The output parameter NO is set for all non-dominated solutions on the mth target. x Or the maximum and minimum values ​​of η.

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

[0039] Furthermore, a hydrogen combustion chamber model is performed for each point to calculate the actual output parameter NO x Methods for η include:

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

[0041] Fluent Mesh is used to automatically mesh the geometric configuration, and then Fluent software is used to calculate the output parameters of each point NO x and η.

[0042] The beneficial effects of the present invention are as follows: the scheme obtains an initial sample set through Halton sampling, and then trains the proxy model based on the initial sample set, and then searches and classifies the optimal solution through NSGA-II and K-Means methods. Different low-emission combustion modes can be discovered through classification, thereby ensuring the diversity of solutions, and different types of optimal solutions can be quickly searched at a lower computing cost, and ultimately more solutions that meet the requirements can be obtained.

[0043] Compared with the existing Latin hypercube, this scheme has a stronger ability to search for the optimal solution area. When searching for the same number of optimal solutions, it can use fewer samples, have a higher diversity of solutions, and greatly improve the calculation efficiency. The sample set obtained through this scheme can obtain different combustion modes according to the influence of the geometric parameter values ​​of different hydrogen combustion chamber designs on the degree of coupling between the main combustion zone and the external recirculation zone, which provides important support and more design ideas for the design of hydrogen combustion chambers in engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 Flowchart of the search method for multi-objective optimization solution set based on small samples.

[0045] Figure 2 These are the small sample optimization effect diagrams, where (a) is the Pareto front diagram obtained in the last round of optimization; (b) is the Pareto front clustering effect diagram for the last round; and (c) is the output distribution diagram of the better solutions obtained in four rounds.

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

[0047] Figure 4 Comparison chart of finding optimal solutions for Peaks multi-objective function and Latin hypercube.

[0048] Figure 5 This is a comparison chart between this research method and the small sample method for Feilei's five-dimensional multi-objective problem. DETAILED DESCRIPTION

[0049] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0050] refer to Figure 1 , Figure 1 FIG. 4 is a flowchart showing a method for searching a multi-objective optimization solution set based on a small sample; Figure 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 formed by five geometric parameters of the hydrogen combustion chamber, and the sampling is stopped when the sample meets the preset conditions to obtain a data set;

[0052] During implementation, the preferred step S1 of this solution further includes:

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

[0054] n initial ∝dimD=N

[0055] Among them, dim D represents the dimension of the input parameter, and N represents the scale factor;

[0056]

[0057] Where Space represents the sample space; D j is the diameter of the hydrogen nozzle; 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.

[0058] S12, inputting the five geometric parameters corresponding to each sample into 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 Fluent software to calculate the output parameters of each initial sample NO x and η;

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

[0061] S15, using the initial sample and its corresponding output parameter NO x and output parameter η as a data set.

[0062] The above method is used to select the initial samples, which can ensure a high initial sampling space filling rate, and is of great help to the exploration of the area that meets the constraint conditions in the space and the training of the proxy model.

[0063] In step S2, two Gaussian regression models are trained using the data set to generate output parameters NO x A first proxy model of and a second proxy model generating an output parameter η;

[0064] During implementation, the preferred step S2 of this solution further includes:

[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 samples in the training set and the test set is 9:1, and the random seeds can ensure that the randomness of each division is consistent.

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

[0067] S23, select all generated output parameters NO x The Gaussian regression model with the highest accuracy is used as the first proxy model, and the Gaussian regression model with the highest accuracy among all output parameters η is used as the second proxy model.

[0068] When selecting the first proxy model and the second proxy model, the following evaluation criteria are mainly used:

[0069] It mainly includes mean relative error (MRE), root mean square error (RMSE) and determination coefficient (R2_score). The formula is as follows:

[0070]

[0071] Where n is the number of test set samples; y i Calculate the value for the test set sample; y i_pred is the model prediction value; is the mean of the calculated values ​​of the test set samples; the calculated values ​​of the test set samples are calculated through steps S22 to 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, thereby ensuring the accuracy of the generated output parameters of the first proxy model and the second proxy model finally selected.

[0073] In step S3, several individuals with the same dimensions as the sample are randomly generated in the multi-objective decision space as the initial population, and the output parameters NO of the initial population are generated by the first agent model and the second agent model. x and η; the sample dimension refers to the five 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 search for a better solution, and the non-dominated solution with level 1 in the population is retained as the Pareto frontier.

[0075] In one embodiment of the present invention, step S4 further comprises:

[0076] S41, according to the output target value of the initial population, the initial population is non-dominated and stratified according to the output target value; after the non-dominated stratification, the initial population is subjected to evolutionary operations including selection, crossover and mutation; the detailed implementation process of step S41 is:

[0077] According to the population output target value, the initial population is non-dominated and stratified according to the output target value. Non-dominated stratification is based on the non-dominated relationship. Domination refers to any two population individuals p and q in the genetic algorithm, where all output target values ​​of individual p are greater than or equal to q, and there is at least one output target where individual p is strictly greater than individual q, then p>q, or p dominates q. Non-domination means that it is impossible to find an output target that makes individual p strictly greater than individual q.

[0078] In the hydrogen combustion chamber design, the output target maximum and minimum modes are set to output parameter NO x minimum, the output parameter efficiency η is maximum, if p>q, it means that individual p and individual q are in NO x On η, the output target value of individual p is strictly smaller than that of individual q, and on η, the output target value of individual p is strictly larger than that of individual q. Stratification is based on the dominance and non-domination relationship between individuals, and each individual in the population is divided into levels.

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

[0080] Mathematical expressions for If f k (p)≤f k (q), where k∈{1,2,…,r} and f l (p) <f l(q), then p>q or p dominates q, where r represents the number of output targets, Pop represents the genetic algorithm evolution population, and f k (x) and f l (p) refers to the target value of individual x and individual p on a certain output target obtained through model prediction.

[0081] After the non-dominated stratification, the population begins to be evolved, including selection, crossover and mutation. Selection is mainly to retain individuals with higher fitness (for the first generation population, the inverse of the level corresponding to each individual in the non-dominated stratification is directly used as the fitness), crossover is to exchange the parameter characteristics between different excellent individuals to help produce more excellent individuals, and mutation can expand the scope of the target space and increase the diversity of solutions. The evolved population can be obtained by setting the crossover and mutation probabilities in the genetic algorithm.

[0082] S42, perform non-dominated stratification on the individuals in the new population after the evolution operation, and then use the first agent model and the second agent model to predict the output parameter NO of each individual x and η;

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

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

[0085]

[0086] where cd[i] m is the congestion 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 of the i+1th and i-1th intermediate solutions on the mth target respectively. x or η; and The output parameter NO is set for all non-dominated solutions on the mth target. x Or the maximum and minimum values ​​of η.

[0087] S44, sorting according to the calculated crowding degree and the level, and then selecting the first N individuals to form a new offspring and add them to the initial population;

[0088] S45, determining whether the number of iterations of the NSGA-II multi-objective genetic algorithm has reached the maximum number of iterations, if so, proceeding to step S46, otherwise returning to step S41;

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

[0090] This scheme combines the advantages of NSGA-II to set the initial population size N to 500, the maximum number of iterations to 500, and the RI encoding method to avoid the conversion of individuals from phenotype to genotype, and can directly perform crossover and mutation operations on individuals.

[0091] In step S5, the points on the Pareto front are input into the first proxy model and the second proxy model respectively to obtain the output parameter NO x and η, and retains NO x K-means clustering was performed for points with ≤2.5 and η≥95%;

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

[0093] In step S6, two points are randomly selected from each cluster obtained by clustering, and a hydrogen combustion chamber model is performed on each point to calculate the actual output parameter NO x and η:

[0094] The five geometric parameters corresponding to each point were input into Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber;

[0095] Fluent Mesh is used to automatically mesh the geometric configuration, and then Fluent software is used to calculate the output parameters of each point NO x and η.

[0096] In step S7, the output parameter NO of each point predicted by the first proxy model is calculated. x With true output parameter NO x The relative error between them is calculated, and the average relative error of all points is calculated;

[0097] In step S8, it is determined whether the average value is less than a preset error. If so, the selected points are added to the data set output. Otherwise, the selected points are added to the data set and the process returns to step S2.

[0098] During implementation, the optimization problem expression of the NSGA-II multi-objective genetic algorithm selected in this scheme is:

[0099]

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

[0101] The effect of the search method of the multi-objective optimization solution set based on a small sample in this scheme is described below in conjunction with the embodiments:

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

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

[0104] Table 1. Accuracy comparison of four-wheel point-adding proxy models

[0105]

[0106] From Table 1, we can see that as the number of rounds of adding points increases, the average relative error of the prediction results of the optimal solution found by the model decreases. In the third round, the average relative error drops to 15.02%. To avoid contingency, another round of adding points is added on the basis of the third round. It is found that the average relative error of the prediction results is still less than 20%, which shows the effectiveness of this solution. There are 28 points in total in the four rounds of adding points, and 13 of them meet the design requirements after calculation verification.

[0107] Figure 2 (a) shows the Pareto front diagram obtained in the fourth round of optimization. The blue dots represent the points on the Pareto front, and the red dots in the figure are the 70 points found on the Pareto front that meet the constraint design requirements; (b) shows the clustering effect diagram of the fourth round of Pareto front, and (c) shows the output distribution diagram of the better solutions obtained in the four rounds. Figure 2 It can be seen that with four rounds of additional points in the case of an initial sample number of 40, three different types of solution sets that meet the constraints are found with a smaller total number of samples, and these solutions can be used to guide the low-emission design of hydrogen combustors.

[0108] This solution combines the second embodiment and the third embodiment to illustrate the feasibility of the search method of this solution from a mathematical perspective:

[0109] Example 2: Using Peaks 2D multi-objective function to verify the feasibility of this solution

[0110] The Peaks two-dimensional multi-objective function, combined with the traditional single-objective Peaks function, has a local minimum and two local maximums. The Peaks multi-objective function is formed by stacking two single-objective Peaks functions, in which 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, the five geometric parameters (five-dimensional parameters) of the hydrogen combustion chamber of this scheme are replaced by two-dimensional input, and the sample space is also replaced by two-dimensional. When using the NSGA-II multi-objective genetic algorithm, its optimization problem is replaced by a two-dimensional optimization problem. The output of this scheme is replaced by two output constraints: output 1 is greater than 5, and output 2 is greater than 7. The 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 based on a small sample of multi-objective optimization solutions is used. The output function is set to the maximum of f1(x) and f2(x), and it needs to satisfy (f1(x)≥5,f2(x)≥7). Then the above-mentioned replacement and adjusted solution search method is run. The search results are as follows: Figure 3 and Figure 4 shown.

[0113] Figure 3 Figure 1 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 in this study. Figure 3 It can be seen that this research method found more solutions that met the requirements when the initial number of points was small, and points in three different areas were discovered through clustering.

[0114] Figure 4 A comparison chart of finding optimal solutions for Peaks multi-objective function and Latin hypercube. Figure 4 The vertical axis represents the number of better solutions, and the horizontal axis represents the current number of samples. Because the Latin hypercube scatter points are more random, we repeated the drawing several times to get its envelope. From the figure, we can intuitively find that under the condition of finding the same number of better solutions, the adjusted solution uses fewer samples.

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

[0116] Example 3: Using Feilei's five-dimensional multi-objective function to verify the feasibility of this solution

[0117] Feilei five-dimensional multi-objective function uses a sine function when constructing the function, and the composition of the hyperbolic function makes the output and input have a strong nonlinear relationship. When using Feilei's five-dimensional multi-objective function, the five geometric parameters (five-dimensional parameters) of the hydrogen combustion chamber of this scheme are replaced by five-dimensional input, and the output of this scheme is replaced by two output constraints: 1 is greater than 18, and output 2 is greater than 80. The other method steps of this scheme remain unchanged; the optimization constraint equation of the five-dimensional mathematical problem is set as follows:

[0118]

[0119] After the equation is established, a search method based on a small sample of multi-objective optimization solutions is used. The output function is set to the maximum of f1(x) and f2(x), and it needs to satisfy (f1(x)≥18,f2(x)≥80); then the above-mentioned replacement and adjusted solution search method is run, and the search results are as follows Figure 5 shown.

[0120] Figure 5 This is a comparison chart of the optimization results of Feilei's 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 this research method and the number of points found. In order to reduce the randomness of Latin hypercube sampling itself and avoid the influence of accidental results, an envelope diagram is used to display the results of multiple samplings. The red curve in the figure represents the average value of the number of optimal solution sets found in multiple samplings, and the blue envelope is composed of the maximum and minimum values ​​of the number of optimal solution sets obtained from each sampling, showing the range of data variation. Figure 5 It can be concluded that this research method has stronger optimal solution search capability in high-dimensional problems compared with the Latin hypercube sampling method.

[0121] The effectiveness of this method is demonstrated in engineering by Example 1. The method is compared with the Latin hypercube sampling method in 2D and 5D mathematical problems in Examples 2 and 3. The results verify the effectiveness and feasibility of this method. By applying it to 2D mathematical functions, we can perform visual analysis to analyze the distribution of optimal solutions and the ability to search for different types of optimal solutions. Since the 5D mathematical function has the same dimensions as the actual engineering problem, the effectiveness of this method can be further verified on 5D mathematical problems.

Claims

1. A method for searching a multi-objective optimization solution set based on a small sample, characterized in that: Includes steps: S1. Use the Halton sampling method to collect samples in the Space sample space composed of five geometric parameters of the hydrogen combustion chamber, stop sampling when the sample meets the preset conditions, and obtain a data set; S2, use the data set to train two Gaussian regression models respectively to obtain the generated output parameter NO x A first proxy model of and a second proxy model generating an output parameter η; S3. Randomly generate several individuals with the same dimensions as the sample as the initial population in the multi-objective decision space, and use the first agent model and the second agent model to generate the output parameters of the initial population NO x and η; S4. Based on the initial population, the NSGA-II multi-objective genetic algorithm is used to find the optimal solution, and the non-dominated solution with level 1 in the population is retained as the Pareto frontier; S5. Input the points on the Pareto front into the first proxy model and the second proxy model respectively to obtain the output parameter NO x and η, and retains NO x K-means clustering was performed for points with ≤2.5 and η≥95%; 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 parameter NO x and η; S7. Calculate the output parameter NO of each point predicted by the first proxy model x With true output parameter NO x The relative error between them is calculated, and the average relative error of all points is calculated; S8. Determine whether the average value is less than a preset error. If so, add the selected points to the data set output. Otherwise, add the selected points to the data set and return to step S2.

2. The search method according to claim 1, characterized in that: Step S1 further comprises: S11, using the Halton sampling method to sample in the Space sample space formed by five geometric parameters of the hydrogen combustion chamber to obtain multiple initial samples; S12, inputting the five geometric parameters corresponding to each sample into Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber; S13, use Fluent Mesh to automatically mesh the geometric configuration, and then use Fluent software to calculate the output parameters of each initial sample NO x and η; S14, determine the output parameters of all initial samples NO x and η satisfy NO x ≤2.5, η≥95% is greater than the preset number; if so, proceed to step S15, otherwise return to step S11; S15, using the initial sample and its corresponding output parameter NO x and output parameter η as a data set.

3. The search method according to claim 1, characterized in that: Step S2 further comprises: 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, respectively train two Gaussian regression models using the training set and test set corresponding to the same random seed to generate output parameter NO x and generate output parameter η; S23, select all generated output parameters NO x The Gaussian regression model with the highest accuracy is used as the first proxy model, and the Gaussian regression model with the highest accuracy among all output parameters η is used as the second proxy model.

4. The search method according to claim 1, characterized in that: Step S4 further comprises: S41, according to the output target value of the initial population, performing non-dominated stratification on the initial population according to the output target value; after the non-dominated stratification, performing evolutionary operations including selection, crossover and mutation on the initial population; S42, perform non-dominated stratification on the individuals in the new population after the evolution operation, and then use the first agent model and the second agent model to predict the output parameter NO of each individual x and η; S43, according to each individual output parameter NO x and η, sort each individual in ascending order, and then calculate the crowding degree of the non-dominated solutions in each layer; S44, sorting according to the calculated crowding degree and the level, and then selecting the first N individuals to form a new offspring and add them to the initial population; S45, determining whether the number of iterations of the NSGA-II multi-objective genetic algorithm has reached the maximum number of iterations, if so, proceeding to step S46, otherwise returning to step S41; S46. Perform non-dominated stratification on the last generation of population and retain the non-dominated solutions with level 1 in the population as the Pareto frontier.

5. The search method according to claim 1 or 4, characterized in that: The optimization problem of the NSGA-II multi-objective genetic algorithm is expressed as: Among them, F(x) is the optimization problem; x is the variable; D j is the diameter of the hydrogen nozzle; 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.

6. The search method according to claim 4, characterized in that: In step S43, in the sequence sorted in ascending order, the sequence with the largest and smallest output parameters NO x and η are taken as boundary solutions, and the remaining parameters in the sequence are taken as intermediate solutions; The crowding degree of the boundary solution is set to infinity, and the crowding degree calculation formula is used to calculate the crowding degree of the remaining solutions in the sequence. The crowding degree calculation formula is: where cd[i] m is the congestion 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 of the i+1th and i-1th intermediate solutions on the mth target respectively. x or η; and The output parameter NO is set for all non-dominated solutions on the mth target. x Or the maximum and minimum values ​​of η.

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

8. The search method according to claim 1, characterized in that: A hydrogen combustion chamber is modeled for each point to calculate the true output parameter NO x Methods for η include: The five geometric parameters corresponding to each point were input into Catia software for parametric modeling to obtain the geometric configuration of the hydrogen combustion chamber; Fluent Mesh is used to automatically mesh the geometric configuration, and then Fluent software is used to calculate the output parameters of each point NO x and η.

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