CN111625923A - Antenna electromagnetic optimization method and system based on non-stationary Gaussian process model - Google Patents

Abstract

The invention discloses an antenna electromagnetic optimization method and system based on a non-stationary Gaussian process model. First, a population is constructed, and the size of the population is initialized, wherein each individual in the population represents a training sample point respectively; After the electromagnetic simulation of the population is performed, the target value corresponding to each individual is obtained; then, the evaluated population is used as the training set, and the training data is selected from the training set; however, in the process of training the non-stationary Gaussian process model, the differential evolution is used. The algorithm performs global optimization on the parameters to be solved in the model; according to the random population obtained after differential evolution, a potential sample point is selected from the population for electromagnetic simulation through the expectation improvement strategy; the potential sample point is added to the training set, and the A stationary Gaussian process model until the number of simulations is exhausted. The method and system propose a non-stationary Gaussian process model-assisted evolutionary algorithm framework to effectively solve the antenna design optimization problem.

Description

A method and system for antenna electromagnetic optimization based on non-stationary Gaussian process model

technical field

The invention belongs to the field of antenna optimization, and in particular relates to a method and a system for improving the simulation efficiency of antenna electromagnetic optimization based on a non-stationary Gaussian model.

Background technique

Antennas are widely used in communications, radar, electronic countermeasures and other fields as a device for receiving and converting energy. Antennas can realize functions such as high-security radar, electronic warfare, and wireless communication. Therefore, antenna research has been widely recognized. In the practice of antenna design, the design of the antenna comes down to the optimization problem, and the optimization algorithm is an effective way to solve this kind of problem. The optimization algorithms usually involved include traditional optimization methods and artificial intelligence optimization methods.

Traditional optimization methods (Newton's method, conjugate function method, gradient descent method, etc.) usually use the derivative information of the correlation function, and these derivative information is determined by the limit and can only reflect the local characteristics of the function, so the traditional optimization method Either it is difficult to obtain the global optimal solution, or it cannot be used.

In order to obtain the global optimal solution, at present, the artificial intelligence method Evolutionary Algorithm (EA) is mostly considered to solve the antenna design problem; the evolutionary algorithm is to iterate the entire group (called population) according to the idea of evolution, Some evolution operators (crossover, mutation, selection) are applied to the population, so that the quality of the entire population can be continuously improved; among them, the population information used determines that the algorithm can search in a certain space in parallel without the need for a target Continuity, derivability and other conditions of the function, the global optimal solution can be found. The evolutionary algorithm is applied to some antenna design problems. A large number of experiments have found that the evolutionary algorithm is significantly better than the traditional optimization method in solving nonlinear, multi-mode, large-scale, high-constrained, and large-uncertain antenna problems. .

However, the existing research is based on the assumption that evolutionary algorithms are easy and computationally cheap to perform evaluation of objectives and constraints, and that there are explicit objective and constraint function expressions. However, for real-life problems, it is not so simple. In the actual antenna design, the evaluation of the fitness value of these antenna optimization problems comes from expensive electromagnetic simulation experiments, which will consume a lot of computational costs. It involves hundreds of optimizations, and the time consumed is even more unacceptable.

Establishing an accurate surrogate model is very important to solve the optimization problem of antenna design, so a lot of research work has been carried out to establish a more accurate surrogate model. in these numerous studies. The Gaussian process surrogate model has been widely used, because its predicted fitness value has high accuracy, and at the same time provides the confidence of the predicted fitness value; The Gaussian process model in the problem performs better than other surrogate models (polynomial, radial basis functions, artificial neural networks, support vector machines). However, a large number of current studies are based on the assumption that the Gaussian process is a stationary process, that is, a large number of studies are based on the assumption that the Gaussian process is stationary. However, smoothness is strict and limited. In practical antenna design problems, the assumption of non-stationary processes is often required, so the establishment of non-stationary surrogate models can better reflect the characteristics of actual antenna problems.

From the perspective of scientific research, the antenna design problem is a kind of nonlinear and time-consuming optimization problem of electromagnetic simulation. The mapping relationship between the antenna structure parameters and the electromagnetic field radiation distribution belongs to a non-stationary random process in the sense of probability. The research has the application value of the latest scientific research results. This patent applies the non-stationary Gaussian process model to mine the electromagnetic distribution mechanism inside the antenna, establishes the corresponding non-stationary Gaussian process surrogate model, and then uses the surrogate model to replace the expensive electromagnetic simulation to make appropriate predictions, and combines the evolutionary algorithm for fast global optimization. . Finally, the application of non-stationary Gaussian model combined with evolutionary algorithm (model-assisted evolutionary algorithm) is realized to overcome the difficulty of "time-consuming electromagnetic simulation" in antenna design. Therefore, the non-stationary model-assisted evolutionary algorithm has important research value and significance for the research of antenna design.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is to provide an antenna electromagnetic optimization method and system based on a non-stationary Gaussian process model, aiming at the defect that the existing technology is based on a stationary surrogate model and cannot reflect the actual problem of the antenna.

The technical solution adopted by the present invention to solve the technical problem is: constructing an antenna electromagnetic optimization method based on a non-stationary Gaussian process model, comprising the following steps:

S1. When designing the antenna, first build a population and initialize the size of the population, wherein each individual in the population represents a training sample point, and each training sample point represents an antenna; using After the electromagnetic simulation evaluates the population, the target value corresponding to each individual is obtained, and the target value is the function value corresponding to the target of the antenna optimization problem established under a given antenna structure;

S2, take the evaluated population as a training set, and select training data from the training set; input the training data into a non-stationary Gaussian process model, and perform model training; wherein, the training data includes N training sample points x ¹ ,…,x ^N ∈R ^d , and the target value y ¹ ,…,y ^N corresponding to each sample point; R is a real number, d is a dimension, and R ^d is a real number space;

S3. During the training of the non-stationary Gaussian process model, a differential evolution algorithm is used to globally optimize the parameters to be solved in the model; the parameters to be solved include a weight coefficient θ representing the change of the training sample point x;

S4. According to the random population obtained after differential evolution, select a potential sample point from the random population through the expectation improvement strategy for electromagnetic simulation; add the potential sample point to the training set, and update the non-stationary Gaussian process model until the number of simulations is exhausted After that, output the optimal antenna and the antenna structure corresponding to the antenna.

An antenna electromagnetic optimization system based on a non-stationary Gaussian process model disclosed by the present invention includes the following modules:

The population building module is used to first construct the population and initialize the size of the population when designing the antenna, wherein each individual in the population represents a training sample point, and each training sample point represents a an antenna; after the population is evaluated by electromagnetic simulation, the target value corresponding to each individual is obtained, and the target value is the function value corresponding to the target of the antenna optimization problem established under a given antenna structure;

The model training module is used to use the evaluated population as a training set, and select training data from the training set; input the training data into a non-stationary Gaussian process model, and perform model training; wherein, the training data includes N training data The sample points x ¹ ,...,x ^N ∈R ^d , and the target value y ¹ ,...,y ^N corresponding to each sample point; R is a real number, d is a dimension, and R ^d is a real number space;

The differential evolution module is used to globally optimize the parameters to be solved in the model by using the differential evolution algorithm during the training of the non-stationary Gaussian process model; the parameters to be solved include the weight coefficient θ representing the change of the training sample point x;

The electromagnetic simulation module is used to select a potential sample point from the random population for electromagnetic simulation according to the random population obtained after differential evolution through the expectation improvement strategy; add the potential sample point to the training set to update the non-stationary Gaussian process model, Until the number of simulations is exhausted, the optimal antenna and the antenna structure corresponding to the antenna are output.

Implementing an antenna electromagnetic optimization method and system based on a non-stationary Gaussian process model of the present invention has the following beneficial effects:

1. After theoretical analysis, this line electromagnetic optimization method and system is more flexible and general;

2. After theoretical analysis, this line electromagnetic optimization method and system can more accurately approximate antenna electromagnetic simulation (expensive optimization function);

3. This electromagnetic optimization method and system proposes an evolutionary algorithm framework assisted by a non-stationary Gaussian process model, which solves the choice of surrogate model in data-driven evolutionary optimization, and brings certain reference significance for solving data-driven optimization problems in the future;

4. The line electromagnetic optimization method and system can effectively solve the antenna design optimization problem, thereby reducing the electromagnetic simulation calculation cost, improving the antenna design optimization efficiency (speeding up the calculation efficiency), promoting rapid optimization, and helping to speed up production.

Description of drawings

The present invention will be further described below in conjunction with the accompanying drawings and embodiments, in which:

Figure 1 is a simulation diagram of the realization of stationary and non-stationary processes;

FIG. 2 is a schematic flow chart of implementing the antenna electromagnetic optimization method based on the non-stationary Gaussian process model disclosed in the present invention;

Figure 3 is a schematic diagram of a non-stationary Gaussian model training process;

Figure 4 is a data-driven-non-stationary Gaussian model-assisted evolutionary algorithm pseudocode.

Fig. 5 is the algorithm result comparison diagram of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization;

6 is a performance comparison diagram of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization algorithms in a d=2-dimensional test problem set;

7 is a performance comparison diagram of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization algorithms in a d=5-dimensional test problem set;

8 is a performance comparison diagram of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization algorithms in a d=10-dimensional test problem set;

Figure 9 is the initial geometry of the antenna design;

Fig. 10 is the gain curve diagram of the optimal antenna optimized by the algorithm;

Fig. 11 is the standing wave curve diagram of the optimal antenna optimized by the algorithm;

12 is a structural diagram of an antenna electromagnetic optimization system based on a non-stationary Gaussian process model disclosed in the present invention.

Detailed ways

In order to have a clearer understanding of the technical features, objects and effects of the present invention, the specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

Since the antenna design optimization problem has received extensive attention, and data-driven surrogate model-assisted evolutionary optimization has yielded satisfactory results in solving practical antenna design problems. In the evolutionary optimization assisted by the on-line data-driven surrogate model, first of all, the most critical scientific problem is how to choose a reasonable surrogate model and improve the accuracy of the surrogate model. It can characterize the original function, so that the misleading search converges to the wrong region rather than the region of the optimal solution of the original problem; since establishing an accurate surrogate model is very important for solving the antenna design optimization problem, a lot of research work has been carried out one after another. Accurate proxy model.

in these numerous studies. The Gaussian process surrogate model has been widely used, because its predicted fitness value has high accuracy, and at the same time provides the confidence of the predicted fitness value; The Gaussian process model in the problem performs better than other surrogate models (polynomial, radial basis functions, artificial neural networks, support vector machines). However, a large number of current studies are based on the assumption that the Gaussian process is a stationary process, that is, a large number of studies are based on the assumption that the Gaussian process is stationary. However, smoothness is strict and limited. In the actual problem of antenna design, the assumption of non-stationary process is often required. Please refer to Figure 1, which is a simulation diagram of the realization of stationary and non-stationary processes. In Figure 1, the solid line represents the realization of the non-stationary process, and the dotted line represents the realization of the non-stationary process is the realization of a stationary process, in which the horizontal and vertical coordinates represent different indicators in different application scenarios, for example, in the application scenario of the transmission signal sequence: the horizontal axis represents time, and the vertical axis represents intensity; from this figure It can also be further clarified in that the characteristics of most antenna design problems actually solved are non-stationary characteristics, so the establishment of non-stationary surrogate models can better reflect the characteristics of actual antenna design problems.

Please refer to FIG. 2 , which is a schematic flowchart of implementing the method for electromagnetic optimization of an antenna based on a non-stationary Gaussian process model disclosed in the present invention, including the following steps:

S1. Construct a population when designing the antenna, and initialize the size of the population, wherein each individual in the population represents a training sample point, and the training sample point is a single antenna; electromagnetic simulation is used for this After the population is evaluated, the target value corresponding to each individual is obtained, and the target value is the function value corresponding to the target of the antenna optimization problem established under a given antenna structure; wherein:

The initializing setting of the population size, specifically, using the method of Latin square sampling to initialize the population size;

In the current embodiment, the size of the initial population is defined as: 11×d-1; d≥1, where d is the dimension of the problem to be solved in the optimization process.

S2, take the evaluated population as a training set, and select training data from the training set; input the training data into a non-stationary Gaussian process model, and perform model training; wherein, the training data includes N training sample points x ¹ ,…,x ^N ∈R ^d , and the target value y ¹ ,…,y ^N corresponding to each sample point; R is a real number, d is a dimension, and R ^d is a real number space;

The current step is also a key point in the entire program, and the following explanations can be made in detail:

First, for the establishment of the model:

The current key point is the construction of the non-stationary surrogate model. The established non-stationary Gaussian process model (NGP) has the following form:

Among them, f(x) is the regression function, β is the weight coefficient of the regression function f(x) to be solved; p is the total number of predefined regression functions; Z(x)～N(0,σ ² _z ) is stationary term, where the mean value of the stationary term is 0, and the variance term to be solved is σ ² _z ;

The correlation measurement between any two acquired training sample points x and x' is determined by the Gaussian kernel function, and the mathematical form of the Gaussian kernel function is defined as:

Among them, θ is the weight coefficient to be solved that represents the change of the sample point x; the value range of i is [1, d], and d represents the dimension of x.

When there are N training sample points x ¹ ,...,x ^N ∈R ^d and the corresponding target values y=y ¹ ,...,y ^N , a non-stationary Gaussian process model is established, but in this model , first need to determine some hyperparameters in the model. The hyperparameters include the weight coefficient β of the regression function f(x), the weight coefficient θ representing the variation of the sample point x, and the variance term to be solved as σ ² _z , however, these hyperparameters are estimated by maximizing the likelihood function. Obtained, the log form of the maximum likelihood function is:

Among them, C is the covariance matrix of N*N, N is the number of input sample points, y=(y ¹ ,...,y ^N ) ^T is an N-dimensional column vector; G=(g _j (x _i )), i=1,...,N,j=1,...,p, G is the regression function matrix of N*p dimension, and g(x) is the regression function vector of p*1 dimension;

The above hyperparameters are calculated as:

1. Estimation of parameter β:

其计算公式为：In this embodiment, the estimated value is obtained by estimating β through the least squares method

Its calculation formula is:

2. Estimation of variance term σ ² _z :

带入下述计算公式(5)，计算第二项参数σ² _z，计算公式为：The estimated value obtained from the previous calculation

Bring in the following calculation formula (5) to calculate the second parameter σ ² _z , the calculation formula is:

3. Estimate the weight coefficient θ representing the change of the sample point x:

和σ² _z代入上述定义的最大似然函数及公式(3)中，得到用于评估θ的似然函数：Calculated above

and σ ² _z are substituted into the above-defined maximum likelihood function and formula (3) to obtain the likelihood function for evaluating θ:

Since the maximum likelihood function is only related to the parameters in the covariance matrix, the above formula (6) can be abbreviated as follows:

-N logσ ² _z -log(det(C)); (7)

Finally, when solving the parameter θ, based on the formula (7), the differential evolution algorithm is used to solve the θ;

The above is the calculation process of the hyperparameters to be solved. After obtaining the above hyperparameters, the non-stationary Gaussian process model can be determined; then, the prediction of any untested sample point x can pass the best linear unbiased estimation get, for example:

When an uninformative prior is distributed, the mean of the untested sample points x is:

The corresponding variance is:

Here h is:

h=g(x)-G ^T C ^-1 r; (10)

是p*1维回归函数的系数矩阵，r是N*1由未测试点x和训练数据组成的协方差矩阵。Among them, g(x) is the regression function vector of p*1 dimension,

is the coefficient matrix of the p*1 dimensional regression function, and r is the N*1 covariance matrix consisting of the untested points x and the training data.

However, the regression function used in the above algorithm adopts radial basis function (RBF) as the regression function in this embodiment, and its form is:

表示定义的回归函数形式，‖*‖³表示x与c之间距离的三次方。当前，根据柯尔莫哥洛夫定理RBF的个数设为2d+1；上述的聚类中心c通过Kmeans聚类的方法确定。Among them, c is the cluster center of the input N training sample points,

Represents the defined regression function form, ‖*‖ ³ represents the cube of the distance between x and c. Currently, according to Kolmogorov's theorem, the number of RBFs is set to 2d+1; the above-mentioned cluster center c is determined by the method of Kmeans clustering.

S3. After constructing the non-stationary Gaussian process model based on step S2, input the training data to the non-stationary Gaussian process model to perform model training, and in the training process, adopt the differential evolution algorithm to solve the parameters in the model. Global optimization (refer to Figure 4, which is a data-driven-non-stationary Gaussian model-assisted evolutionary algorithm pseudo-code); wherein:

The parameter to be solved includes a weight coefficient θ representing the change of the training sample point x;

The input of the model is: in step S1, the population evaluated by the expensive original function (ie electromagnetic simulation) is used as the training set, and the training data is selected from the training set; the data set is generated based on the individual obtained after evaluating each individual in the population;

The output of the model is: the optimal solution in the data set, that is, the optimal antenna and the structure corresponding to the optimal antenna.

The differential evolution algorithm is used to evolve a population, that is, in the training process, the non-stationary Gaussian process model is used to replace the expensive evaluation function to re-evaluate the individuals in the population, and a random population is obtained.

S4. According to the random population obtained after differential evolution, select a potential sample point from the population for electromagnetic simulation (ie expensive evaluation) through the expectation improvement strategy; add the potential sample point to the training set, and update the non-stationary Gaussian process model, until the number of simulations (ie expensive evaluations) is exhausted. (For the current training process, please refer to Figure 4, which is a pseudo-code of a data-driven-non-stationary Gaussian model-assisted evolutionary algorithm) where:

After using the non-stationary Gaussian process model to replace the expensive evaluation function to re-evaluate the individuals in the population, select the most potential solution promising point in the population (that is, the solution with the best target value of the training sample point), and use the expensive evaluation function ( That is, electromagnetic simulation) evaluation, and this solution is added to the training data set, and the data set is updated; until the expensive evaluation times (that is, the number of simulation times) are exhausted, the optimal solution in the current data set is output.

Please refer to Figure 3, which is a schematic diagram of the non-stationary Gaussian model training process. When the model training is performed, it is divided into the following steps:

First, obtain the initial sample point and data set, and judge whether the stopping criterion is reached (the stopping criterion is to judge whether the number of simulations is exhausted). When the stopping criterion is reached, the best solution in the output data set is the optimal antenna corresponding structure; otherwise, go to the next step;

Second, select training samples, use the non-stationary Gaussian process model as a surrogate model, and train the surrogate model;

Secondly, adopt the differential evolution algorithm to evolve the population, and use the surrogate model as an expensive evaluation function;

Finally, according to the random population obtained after differential evolution, a potential sample point (promising point) is selected from the population for electromagnetic simulation through the expectation improvement strategy, and the data set is updated until the stopping criterion is reached, and then the training is stopped, and the output data set is the best solution.

The disclosed algorithm in the present invention mainly solves the problem of high cost of electromagnetic simulation calculation in antenna design. The invention proposes a non-stationary Gaussian-assisted evolution algorithm to solve the high cost of electromagnetic simulation in antenna design. The difference between this algorithm and other model-assisted evolutionary optimization algorithms is the establishment of the model. In the current research field, the Gaussian surrogate model is better than other approximation techniques in solving problems with dimensions below 15 dimensions. In the present invention, the proposed Data Driven-NGP Model Assisted Evolutionary Algorithm (DD-NGP-MAEA) is compared with the Evolutionary Algorithm Assisted with Stationary Gaussian Processes (DD-SGP-MAEA), and the objective in this experiment is minimization. Among them, the CEC2014 expensive optimization problem is used as the test problem set, which is used to test the performance of the algorithm. The data-driven-NGP model-assisted evolutionary algorithm (DD-NGP-MAEA) and the evolutionary algorithm assisted by a stationary Gaussian process were tested and compared at d=2, 5, and 10, respectively, and the following conclusions were obtained (please refer to Fig. 5):

1. The performance comparison of the algorithm performance in the d=2 test problem set:

As shown in Figure 6, it represents the results of the mean ± variance, the best value and the worst value obtained by running the two algorithms DD-NGP-MAEA and DD-SGP-MAEA independently on the test set problem for 25 times; And FriedmanRank and Wlicoxon Rank sum test statistical tests were performed on these results.

By comparison, it is obvious that the algorithm proposed in this embodiment is superior to the existing evolutionary algorithm assisted by a stationary Gaussian model. Specifically reflected in two aspects:

a. Average:

The average value of DD-NGP-MAEA is smaller than that of DD-SGP-MAEA;

b. Statistical test comparison:

The Friedman Rank of DD-NGP-MAEA is smaller than that of DD-SGP-MAEA.

By Wlicoxon Rank sum test, there is a significant difference between DD-NGP-MAEA and DD-SGP-MAEA at the significance index α=0.05.

2. The performance comparison of the algorithm performance in the d=5 test problem set:

As shown in Figure 7, it represents the results of the mean±variance, the best value and the worst value obtained by running the two algorithms DD-NGP-MAEA and DD-SGP-MAEA independently on the test set problem for 25 times; And FriedmanRank and Wlicoxon Rank sum test statistical tests were performed on these results.

By comparison, it is obvious that the algorithm proposed in this embodiment is superior to the existing evolutionary algorithm assisted by a stationary Gaussian model. It is reflected in two aspects:

a. Average:

The average value of DD-NGP-MAEA is mostly smaller than that of DD-SGP-MAEA, except for F5, because the F7 function increases with the dimension, and the valley of the fitness landscape of the function becomes narrower, which brings great impact to all algorithms. challenge;

b. Statistical test comparison:

The Friedman Rank of DD-NGP-MAEA is smaller than that of DD-SGP-MAEA.

By Wlicoxon Rank sum test, there is a significant difference between DD-NGP-MAEA and DD-SGP-MAEA at the significance index α=0.1.

3. The performance comparison of the algorithm performance in the d=10 test problem set:

As shown in Figure 8, it represents the results of the mean±variance, the best value and the worst value obtained by running the two algorithms DD-NGP-MAEA and DD-SGP-MAEA independently on the test set problem for 25 times; And FriedmanRank and Wlicoxon Rank sum test statistical tests were performed on these results.

By comparison, it is obvious that the algorithm proposed in this embodiment performs better than the existing stationary Gaussian model-assisted evolution algorithm on the d=10-dimensional test problem. It is reflected in two aspects:

a. Average:

The average value of DD-NGP-MAEA is smaller than that of DD-SGP-MAEA;

b. Statistical test comparison:

The Friedman Rank of DD-NGP-MAEA is smaller than that of DD-SGP-MAEA.

By Wlicoxon Rank sum test, there is a significant difference between DD-NGP-MAEA and DD-SGP-MAEA at the significance index α=0.05.

4. Taking the design of an elliptical slot microstrip patch antenna as an example, the initial geometric structure of the antenna design is shown in Figure 9. Through experimental comparison, the algorithm proposed by the present invention has excellent performance in antenna design, and its advantages are specific It is reflected in the gain of the antenna and the standing wave ratio of the antenna:

First of all, when the algorithm proposed in the present invention designs the antenna, the gain of the antenna is all greater than 0 (please refer to Fig. 10):

Antenna gain is used to measure the ability of an antenna to send and receive signals in a specific direction, and it is one of the most important parameters for selecting a base station antenna. In general, the increase in gain mainly depends on reducing the lobe width of the radiation in the vertical plane, while maintaining the radiation performance in the omnidirectional plane. Antenna gain is extremely important to the operational quality of a mobile communication system because it determines the signal level at the edge of the cell. Increasing the gain can increase the coverage of the network in a certain direction, or increase the gain margin within a certain range. Any cellular system is a two-way process, and increasing the gain of the antenna reduces the gain budget headroom in the two-way system at the same time.

Secondly, when the algorithm proposed in the present invention is used for antenna design, the standing wave ratio of the antenna is less than 2.0 (please refer to Figure 11):

Generally, the standing wave ratio of an antenna is less than 2.0, which is a good indicator. Many finished antennas require a standing wave ratio of less than 2.0, and some even reach 2.5.

In summary:

a. The computational cost of the antenna designed by the DD-NGP-MAEA algorithm is reduced by more than 10 times;

b. The antenna designed by the DD-NGP-MAEA algorithm has excellent performance (specifically reflected in the gain of the antenna and the standing wave ratio of the antenna).

Please refer to FIG. 12 , which is a structural diagram of an antenna electromagnetic optimization system based on a non-stationary Gaussian process model disclosed by the present invention. The system includes a population building module L1, a model training module L2, a differential evolution module L3 and an electromagnetic simulation module L4, Among them, the functions of each of the above modules are:

The functional role of the population building block L1:

Build a population and initialize the size of the population, where each individual in the population represents a training sample point, and each sample point represents an antenna; after evaluating the population, the target value corresponding to each individual is obtained , the target value is the function value corresponding to the target of the antenna optimization problem established under a given antenna structure.

The function of the model training module L2:

Use the evaluated population as a training set, and select training data from the training set; input the training data into a non-stationary Gaussian process model, and perform model training; wherein, the training data includes N training sample points x ¹ , . . . ,x ^N ∈R ^d , and the target value y ¹ ,...,y ^N corresponding to each sample point; R is a real number, d is a dimension, and R ^d is a real number space.

Please refer to FIG. 12 , the model training module further includes a β value estimation submodule L21, a σ ² _z value calculation submodule L22, a maximum likelihood function construction submodule L23, a maximum likelihood function simplification submodule L24, a θ The value calculation sub-module L25 and the non-stationary Gaussian process model building module L26, and the functions of each of the above modules are:

其计算公式为：The β value estimation sub-module L21 is used to estimate β through the least squares method to obtain the estimated value

Its calculation formula is:

Among them, C is the covariance matrix of N*N, N is the number of input sample points, y=(y ¹ ,...,y ^N ) ^T is an N-dimensional column vector; G=(g _j (x _i )), i=1,...,N,j=1,...,p, G is the regression function matrix of N*p dimension, and g(x) is the regression function vector of p*1 dimension;

带入下述计算公式，计算第二项参数σ² _z，计算公式为：The σ ² _z value calculation sub-module L22 is used to calculate the estimated value calculated by the β value estimation module

Bring in the following calculation formula to calculate the second parameter σ ² _z , the calculation formula is:

The maximum likelihood function building submodule L23 is used to build the maximum likelihood function:

Among them, det(C) is the determinant of the covariance matrix C;

和σ² _z代入最大似然函数构建子模块构建的最大化似然函数中，并对上述最大化似然函数进行简化，得到用于评估θ的似然函数：The maximizing likelihood function simplification submodule L24 is used to convert

and σ ² _z are substituted into the maximum likelihood function constructed by the maximum likelihood function building submodule, and the above maximum likelihood function is simplified to obtain the likelihood function for evaluating θ:

-N logσ ² _z -log(det(C));

The θ value calculation sub-module L25 is used to obtain θ based on the likelihood function for evaluating θ by using differential evolution algorithm;

σ² _z以及参数θ带入到非平稳高斯过程模型，完成所述非平稳高斯过程模型的构建。The non-stationary Gaussian process model building module L26 is used to calculate the estimated value of the obtained parameter β

σ ² _z and parameter θ are brought into the non-stationary Gaussian process model to complete the construction of the non-stationary Gaussian process model.

The role of differential evolution module L3:

During the training of the non-stationary Gaussian process model, a differential evolution algorithm is used to globally optimize the parameters to be solved in the model; the parameters to be solved include a weight coefficient θ representing the change of the training sample point x.

The role of electromagnetic simulation module L4:

According to the random population obtained after differential evolution, a potential sample point is selected from the population for electromagnetic simulation (ie, expensive evaluation) through the expectation improvement strategy; the potential sample point is added to the training set, and the non-stationary Gaussian process model is updated until the electromagnetic The number of simulations (expensive evaluations) is exhausted.

Based on the above analysis results, the antenna electromagnetic optimization method and system based on the non-stationary Gaussian process model disclosed in the present invention has the following two obvious advantages:

1. The antenna electromagnetic optimization method and system are more flexible and general;

2. The antenna electromagnetic optimization method and system can more accurately approximate the expensive optimization function.

In addition, the antenna electromagnetic optimization method and system also proposes a general non-stationary Gaussian process model-assisted evolutionary algorithm framework, which solves the high computational cost and difficulty encountered in antenna design, and solves the problem of proxy in data-driven optimization. The difficulty of model selection is the selection of surrogate models, which brings certain reference significance for solving data-driven optimization problems in the future.

In addition, the antenna electromagnetic optimization method and system can effectively solve the time-consuming problem of electromagnetic simulation of antenna design, thereby reducing the computational cost of electromagnetic simulation, improving the efficiency of antenna design (speeding up computing efficiency), promoting rapid optimization, and helping to speed up production speed.

The embodiments of the present invention have been described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific embodiments, which are merely illustrative rather than restrictive. Under the inspiration of the present invention, without departing from the scope of protection of the present invention and the claims, many forms can be made, which all belong to the protection of the present invention.

Claims (9)

1. An antenna electromagnetic optimization method based on a non-stationary Gaussian process model is characterized by comprising the following steps:

s1, when designing the antenna, firstly constructing a population, and initializing the scale of the population, wherein each individual in the population represents a training sample point, and each training sample point represents an antenna; evaluating the population by adopting electromagnetic simulation to obtain a target value corresponding to each individual, wherein the target value is a function value corresponding to an antenna optimization problem target established under a given antenna structure;

s2, taking the evaluated population as a training set, and selecting training data from the training set; inputting the training data into a non-stationary Gaussian process model for model training; wherein the training data comprises N training sample points x¹,…,x^N∈R^dAnd a target value y corresponding to each sample point¹,…,y^N(ii) a R is a real number, d is a dimension, R^dA real number space;

s3, in the process of non-stationary Gaussian process model training, global optimization is carried out on the parameters to be solved in the model by adopting a differential evolution algorithm; the parameter to be solved comprises a weight coefficient theta representing the change of the training sample point x;

s4, selecting a potential sample point from the random population to perform electromagnetic simulation through an expected lifting strategy according to the random population obtained after differential evolution; and adding the potential sample point into a training set, updating a non-stable Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.

2. The antenna electromagnetic optimization method based on the non-stationary Gaussian process model according to claim 1, characterized in that in step S1, the size of the population is initialized by a Latin square sampling method;

the size of the initial population was: 11 xd-1; d is more than or equal to 1, and d represents the dimension of the problem solved in the optimization process.

3. The method for optimizing the antenna electromagnetism based on the non-stationary Gaussian process model according to the claim 1 or 2, wherein in the step S2, the mathematical form of the non-stationary Gaussian process model is as follows:

wherein f (x) is a regression function, β is a weight coefficient of the regression function f (x) to be solved, p is a predefined total number of regression functions, Z (x) N (0, sigma)² _z) Is a stationary term, wherein the mean of the stationary term is 0 and the variance term to be solved is σ² _z；

The correlation measure between any two acquired training sample points x and x' is determined by a gaussian kernel whose mathematical form is defined as:

wherein, theta is a weight coefficient to be solved representing the change of the sample point x; the value range of i is [1, d ], and d represents the dimension of x.

4. The method for optimizing the electromagnetic property of the antenna based on the non-stationary Gaussian process model as claimed in claim 3, wherein the solution β is obtained₁,…,β_p、θ₁,…,θ_dAnd σ² _zThe method for constructing the non-stationary Gaussian process model comprises the following steps:

s21, estimating β through a least square method to obtain an estimated value

The calculation formula is as follows:

where C is a covariance matrix of N x N, N is the number of input sample points, and y is (y)¹,…,y^N)^TIs an N-dimensional column vector; g ═ G (G)_j(x_i) I 1, …, N, j 1, …, p, G is a regression function matrix in dimensions N × p, G (x) is a regression function vector in dimensions p × 1;

s22, calculating the estimated value obtained in the step S21

Substituting the following calculation formula to calculate a second term parameter sigma² _zThe calculation formula is as follows:

s23, constructing a maximum likelihood function:

where det (C) is the determinant of the covariance matrix C;

s24, mixing

And σ² _zSubstituting into the maximized likelihood function constructed in step S23, and simplifying the maximized likelihood function to obtain a likelihood function for estimating θ:

-N logσ² _z-log(det(C))；

s25, solving by adopting a differential evolution algorithm to obtain theta based on a likelihood function for evaluating the theta;

s26, obtaining the result based on the step S21The obtained estimated value of the parameter β

σ obtained based on step S22² _zAnd substituting the parameter theta obtained in the step S25 into the non-stationary Gaussian process model to finish the construction of the non-stationary Gaussian process model.

5. The method of claim 4, wherein the radial basis function is defined mathematically as a regression function based on a non-stationary Gaussian process model:

wherein c is the clustering center of the input N training sample points,

represents a defined regression function form, | |³Representing the third power of the distance between x and c.

6. An antenna electromagnetic optimization system based on a non-stationary Gaussian process model is characterized by comprising the following modules:

the population construction module is used for firstly constructing a population and carrying out initialization setting on the population scale when the antenna design is carried out, wherein each individual in the population represents a training sample point, and each training sample point represents an antenna; evaluating the population by adopting electromagnetic simulation to obtain a target value corresponding to each individual, wherein the target value is a function value corresponding to an antenna optimization problem target established under a given antenna structure;

the model training module is used for selecting training data from a training set by taking the evaluated population as the training set; inputting the training data into a non-stationary Gaussian process model for model training; wherein the training data comprises NTraining sample point x¹,…,x^N∈R^dAnd a target value y corresponding to each sample point¹,…,y^N(ii) a R is a real number, d is a dimension, R^dA real number space;

the differential evolution module is used for carrying out global optimization on the parameters to be solved in the model by adopting a differential evolution algorithm in the process of non-stationary Gaussian process model training; the parameter to be solved comprises a weight coefficient theta representing the change of the training sample point x;

the electromagnetic simulation module is used for selecting a potential sample point from the random population to perform electromagnetic simulation through an expected lifting strategy according to the random population obtained after differential evolution; and adding the potential sample point into a training set, updating a non-stable Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.

7. The antenna electromagnetic optimization system based on the non-stationary Gaussian process model according to claim 6, wherein in the model training module, the mathematical form of the constructed non-stationary Gaussian process model is as follows:

wherein f (x) is a regression function, β is a weight coefficient of the regression function f (x) to be solved, p is a predefined number of regression functions, Z (x) N (0, sigma)² _z) Is a stationary term, wherein the mean of the stationary term is 0 and the variance term to be solved is σ² _z；

The correlation measure between any two acquired training sample points x and x' is determined by a gaussian kernel whose mathematical form is defined as:

wherein, theta is a weight coefficient to be solved representing the change of the sample point x; the value range of i is [1, d ], and d represents the dimension of x.

8. The system of claim 7, wherein the non-stationary gaussian process model based antenna electromagnetic optimization is achieved by solving β₁,…,β_p、θ₁,…,θ_dAnd σ² _zTo construct a non-stationary gaussian process model, and in the model training module, the following sub-modules are further included:

β value estimation submodule for estimating β by least square method to obtain estimation value

The calculation formula is as follows:

where C is a covariance matrix of N x N, N is the number of input sample points, and y is (y)¹,…,y^N)^TIs an N-dimensional column vector; g ═ G (G)_j(x_i) I 1, …, N, j 1, …, p, G is a regression function matrix in dimensions N × p, G (x) is a regression function vector in dimensions p × 1;

σ² _za value calculation operator module for calculating the estimated value obtained by the β value estimation module

Substituting the following calculation formula to calculate a second term parameter sigma² _zThe calculation formula is as follows:

a maximum likelihood function construction submodule for constructing a maximum likelihood function:

where det (C) is the determinant of the covariance matrix C;

a maximum likelihood function simplifying submodule for simplifying

And σ² _zSubstituting into the maximum likelihood function constructed by the maximum likelihood function construction submodule, and simplifying the maximum likelihood function to obtain a likelihood function for evaluating theta:

-N logσ² _z-log(det(C))；

the theta value calculation operator module is used for solving by adopting a differential evolution algorithm to obtain theta based on a likelihood function for evaluating the theta;

a non-stationary Gaussian process model building module for building the estimated value of the parameter β

σ² _zAnd bringing the parameter theta into the non-stationary Gaussian process model to complete the construction of the non-stationary Gaussian process model.

9. The non-stationary gaussian process model based antenna electromagnetic optimization system according to claim 8, wherein the radial basis function is defined mathematically as a regression function having the form:

wherein c is the clustering center of the input N training sample points,

represents a defined regression function form, | |³Representing the third power of the distance between x and c.

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