CN111625923B - Antenna electromagnetic optimization method and system based on non-stationary Gaussian process model - Google Patents
Antenna electromagnetic optimization method and system based on non-stationary Gaussian process model Download PDFInfo
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Abstract
The invention discloses an antenna electromagnetic optimization method and system based on a non-stationary Gaussian process model, which comprises the steps of firstly, constructing a population, and initializing the scale of the population, wherein each individual in the population represents a training sample point; after electromagnetic simulation is carried out on the population, a target value corresponding to each individual is obtained; then, taking the evaluated population as a training set, and selecting training data from the training set; in the training process of the non-stationary Gaussian process model, a differential evolution algorithm is adopted to conduct global optimization on parameters to be solved in the model; according to the random population obtained after differential evolution, selecting a potential sample point from the population through an expected lifting strategy to perform electromagnetic simulation; and adding potential sample points into the training set, and updating the non-stationary Gaussian process model until the simulation times are exhausted. The method and the system provide an evolution algorithm framework assisted by a non-stationary Gaussian process model, and effectively solve the problem of antenna design optimization.
Description
Technical Field
The invention belongs to the field of antenna optimization, and particularly relates to a method and a system for improving electromagnetic optimization simulation efficiency of an antenna based on a non-stationary Gaussian model.
Background
The antenna is used as equipment for energy transceiving and conversion, is widely applied to the fields of communication, radars, electronic countermeasures and the like, and can realize the functions of high-safety radars, electronic warfare, wireless communication and the like. The study of antennas is widely accepted. In antenna design practice, the design of an antenna is reduced to an optimization problem, and an optimization algorithm is an effective way to solve such a problem. The optimization algorithm generally involved is a traditional optimization method and an artificial intelligence optimization method.
The conventional optimization method (newton method, conjugate function method, gradient descent method, etc.) generally uses derivative information of the related function, and the derivative information is determined by limits and can only reflect local characteristics of the function, so that the conventional optimization method is difficult to obtain a globally optimal solution or cannot be used.
In order to obtain a globally optimal solution, at present, an artificial intelligence method evolution algorithm (Evolutionary Algorithm, EA) is mostly considered to solve the antenna design problem; the evolution algorithm iterates the whole population (called population) according to the thought of evolution, and the quality of the whole population is continuously improved by applying a plurality of evolution operators (hybridization cross, mutation and selection) to the population; the used population information determines that the algorithm can search in a certain space in parallel, and the global optimal solution can be found without the conditions of continuity, conductivity and the like of an objective function. The evolution algorithm is applied to some antenna design problems, and a large number of experiments find that the performance of the evolution algorithm is obviously superior to that of the traditional optimization method in solving the antenna problems of nonlinearity, multimode, large scale, high constraint and large uncertainty.
However, existing studies are based on the assumption that the evolution algorithm is easy to perform evaluation of targets and constraints, computationally inexpensive, and that there are explicit target, constraint function expressions. However, for real life problems, this is not so simple, and in practical antenna designs, the evaluation of the adaptation values of these antenna optimization problems comes from expensive electromagnetic simulation experiments, which would consume a lot of computational expense, however, the optimization process involves hundreds of optimizations at a time, and the consumed time is more unacceptable.
The establishment of accurate proxy models is important to solve the optimization problem of antenna design, so a great deal of research work has been developed successively to establish more accurate proxy models. In these extensive studies. The Gaussian process agent model is widely applied, and the predicted fitness value has higher precision and simultaneously provides the confidence coefficient of the predicted fitness value; in addition, the performance of the gaussian process model is better than other proxy models (polynomials, radial basis functions, artificial neural networks, support vector machines) in solving optimization problems with dimensions below 15 dimensions. However, in the current large amount of researches, the assumption is made that the gaussian process is a stationary process, i.e., the large amount of researches are developed on the basis of the assumption that the gaussian process is stationary. However, smoothness is strict and limiting. In the actual antenna design problem, the assumption of the non-stationary process is often needed, so that the non-stationary proxy model is built to reflect the characteristics of the actual antenna problem.
In the aspect of scientific research, the antenna design problem is a nonlinear and time-consuming optimization problem of electromagnetic simulation, and the mapping relation from the antenna structural parameters to the electromagnetic field radiation distribution belongs to a non-stationary random process in the probability sense, so that the research has the latest scientific research result application value. The method utilizes the non-stationary Gaussian process model to mine the electromagnetic distribution mechanism inside the antenna, establishes a corresponding non-stationary Gaussian process proxy model, replaces expensive electromagnetic simulation with the proxy model to make appropriate prediction, and combines an evolution algorithm to perform rapid global optimization. Finally, the difficulty of time-consuming electromagnetic simulation in antenna design is overcome by combining a non-stationary Gaussian model with an evolution algorithm (model-assisted evolution algorithm). Therefore, the evolution algorithm assisted by the non-stationary model has important research value and significance for the research of antenna design.
Disclosure of Invention
The invention aims to solve the technical problem that the actual problem of an antenna cannot be reflected due to the fact that research is conducted on the basis of a stable proxy model in the prior art, and provides an antenna electromagnetic optimization method and system based on a non-stable Gaussian process model.
The technical scheme adopted for solving the technical problems is as follows: an antenna electromagnetic optimization method based on a non-stationary Gaussian process model is constructed, which comprises the following steps:
s1, when antenna design is carried out, firstly, constructing a population, and carrying out initialization setting on the scale of the population, wherein each individual in the population respectively represents a training sample point, and each training sample point respectively represents an antenna; after the population is evaluated by adopting electromagnetic simulation, a target value corresponding to each individual is obtained, 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 to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space;
s3, in the process of training a non-stationary Gaussian process model, performing global optimization on parameters to be solved in the model by adopting a differential evolution algorithm; the parameters to be solved comprise a weight coefficient theta representing the change of the training sample point x;
s4, selecting a potential sample point from the random population through an expected lifting strategy according to the random population obtained after differential evolution to perform electromagnetic simulation; and adding the potential sample points into a training set, updating a non-stationary Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.
The invention discloses an antenna electromagnetic optimization system based on a non-stationary Gaussian process model, which comprises the following modules:
the system comprises a population construction module, a population analysis module and a population analysis module, wherein the population construction module is used for firstly constructing a population and carrying out initialization setting on the scale of the population when carrying out antenna design, each individual in the population respectively represents a training sample point, and each training sample point respectively represents an antenna; after the population is evaluated by adopting electromagnetic simulation, a target value corresponding to each individual is obtained, 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 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 to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space;
the differential evolution module is used for carrying out global optimization on parameters to be solved in the model by adopting a differential evolution algorithm in the process of training the non-stationary Gaussian process model; the parameters to be solved comprise 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 through an expected lifting strategy according to the random population obtained after differential evolution to perform electromagnetic simulation; and adding the potential sample points into a training set, updating a non-stationary Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.
The antenna electromagnetic optimization method and system based on the non-stationary Gaussian process model have the following beneficial effects:
1. the online electromagnetic optimization method and the online electromagnetic optimization system are more flexible and have generality through theoretical analysis;
2. through theoretical analysis, the line electromagnetic optimization method and system can more accurately approximate the electromagnetic simulation (expensive optimization function) of the antenna;
3. the electromagnetic optimization method and the system provide an evolution algorithm framework assisted by a non-stationary Gaussian process model, solve the selection of a proxy model in data-driven evolution optimization, and bring a certain reference significance for solving the problem of data-driven optimization in the future;
4. the online electromagnetic optimization method and the online electromagnetic optimization system can effectively solve the problem of antenna design optimization, further reduce electromagnetic simulation calculation cost, improve antenna design optimization efficiency (increase calculation efficiency), promote rapid optimization and contribute to increase production speed.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a simulation diagram of an implementation of a stationary and non-stationary process;
FIG. 2 is a flow diagram of an antenna electromagnetic optimization method based on a non-stationary Gaussian process model embodying the present disclosure;
FIG. 3 is a schematic diagram of a non-stationary Gaussian model training flow;
FIG. 4 is a data driven-non stationary Gaussian model assisted evolution algorithm pseudocode.
FIG. 5 is an algorithm result comparison of the disclosed algorithm and other model assisted evolution optimization;
fig. 6 is a graph comparing the performance of the algorithm of the disclosed invention and other model-assisted evolutionary optimization at d=2-dimensional test problem sets;
fig. 7 is a graph comparing the performance of the algorithm of the disclosed invention and other model assisted evolutionary optimization at d=5-dimensional test problem sets;
fig. 8 is a graph comparing the performance of the algorithm of the disclosed invention and other model assisted evolutionary optimization at d=10-dimensional test problem sets;
fig. 9 is an initial geometry of an antenna design;
FIG. 10 is a graph of the gain of an algorithmically optimized optimal antenna;
FIG. 11 is a graph of a standing wave of an algorithmically optimized optimal antenna;
FIG. 12 is a block diagram of an antenna electromagnetic optimization system based on a non-stationary Gaussian process model in accordance with the present disclosure.
Detailed Description
For a clearer understanding of technical features, objects and effects of the present invention, a detailed description of embodiments of the present invention will be made with reference to the accompanying drawings.
Since the antenna design optimization problem is of great interest, and data-driven proxy model-assisted evolutionary optimization yields satisfactory results in solving the actual antenna design problem. In the evolution optimization assisted by the on-line data driven proxy model, firstly, the most critical scientific problem is how to select a reasonable proxy model and improve the accuracy of the proxy model, if the established proxy model can not describe the original function, the misleading search is converged to an erroneous area rather than an area of optimal solution of the original problem; since the establishment of accurate proxy models is important for solving the antenna design optimization problem, a great deal of research effort has been developed in succession to establish more accurate proxy models.
In these extensive studies. The Gaussian process agent model is widely applied, and the predicted fitness value has higher precision and simultaneously provides the confidence coefficient of the predicted fitness value; in addition, the performance of the gaussian process model is better than other proxy models (polynomials, radial basis functions, artificial neural networks, support vector machines) in solving optimization problems with dimensions below 15 dimensions. However, in the current large amount of researches, the assumption is made that the gaussian process is a stationary process, i.e., the large amount of researches are developed on the basis of the assumption that the gaussian process is stationary. However, smoothness is strict and limiting. In the practical problem of antenna design, however, a non-stationary process is often required, please refer to fig. 1, which is a simulation diagram of a stationary and a non-stationary process, in which in fig. 1, a solid line represents a non-stationary process implementation, and a dotted line represents a stationary process implementation, where the abscissa and ordinate represent different indexes in different application scenarios, for example, in an application scenario of a transmission signal sequence: the abscissa represents time and the ordinate represents intensity; it is further clear from the figure that the characteristics of most of the antenna design problems actually solved are non-stationary characteristics, so that the non-stationary proxy model is built to reflect the characteristics of the actual antenna design problems.
Referring to fig. 2, a flow chart of an antenna electromagnetic optimization method based on a non-stationary gaussian process model according to the present disclosure is shown, which includes the following steps:
s1, constructing a population when designing an antenna, and initializing the scale of the population, wherein each individual in the population represents a training sample point, and the training sample points are single antennas; after the population is evaluated by adopting electromagnetic simulation, a target value corresponding to each individual is obtained, wherein the target value is a function value corresponding to an antenna optimization problem target established under a given antenna structure; wherein:
initializing the scale of the population, specifically initializing the scale of the population by using a Latin square sampling method;
in the present embodiment, the size of the initial population is defined as: 11×d-1; d is greater than or equal to 1, and d is the dimension of the solution problem in the optimization process.
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 to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space;
the present step is also a key point in the whole scheme, and the following explanation can be specifically made:
first, for the model establishment:
the key point is the construction of a non-stationary proxy model, and the form of the established non-stationary Gaussian process model (NGP) is as follows:
wherein f (x) is a regression function, and beta is a weight coefficient of the regression function f (x) to be solved; p is the total number of predefined regression functions; z (x) to N (0, sigma) 2 z ) Is a stationary term, wherein the mean value of the stationary term is 0, and the variance term to be solved is sigma 2 z ；
The correlation measure between any two training sample points x and x' is determined by a gaussian kernel function, and the mathematical form of the gaussian kernel function is defined as:
wherein θ is a weight coefficient to be solved representing the change of the sample point x; the value range of i is [1, d ], d represents the dimension of x.
In the existingN training sample points x 1 ,…,x N ∈R d And the corresponding target value is y=y 1 ,…,y N In the case of a non-stationary gaussian process, a model is built, but in this model, the hyper-parameters in some models need to be determined first. The super-parameters comprise a weight coefficient beta of a regression function f (x), a weight coefficient theta representing the change of a sample point x, and a variance term sigma to be solved 2 z However, these estimates of the super-parameters are obtained by maximizing likelihood functions, which have a log form of:
wherein C is a 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 a regression function matrix in n×p dimensions, G (x) is a regression function vector in p×1 dimensions;
the above super parameter calculation mode is:
1. estimation of parameter β:
under the present embodiment, β is estimated by least square method to obtain an estimated valueThe calculation formula is as follows:
2. variance term sigma 2 z Is estimated by (a):
the estimated value obtained by the calculation is calculatedThe second term parameter sigma is calculated by taking into the following calculation formula (5) 2 z The calculation formula is as follows:
3. estimation of the weight coefficient θ representing the change of the sample point x:
the calculation is carried outSum sigma 2 z Substituting the maximum likelihood function defined above and the formula (3) to obtain a likelihood function for estimating θ:
since the maximum likelihood function is related only 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 theta, solving the parameter theta by adopting a differential evolution algorithm based on a formula (7);
the above is the hyper-parameter calculation process to be solved, and after the hyper-parameters are obtained, the non-stationary Gaussian process model can be determined; then the prediction for any untested sample point x can be obtained by best linear unbiased estimation, for example:
when there is no information prior distribution, the mean value of the untested sample points x is:
the corresponding variances are:
here h is:
h＝g(x)-G T C -1 r； (10)
wherein g (x) is a regression function vector of dimension p 1,is the coefficient matrix of the p 1-dimensional regression function, and r is the covariance matrix of N1 consisting of the non-test points x and the training data.
In the present embodiment, a Radial Basis Function (RBF) is used as the regression function, which is used in the algorithm and has the following form:
wherein c is the clustering center of the N training sample points,the form of the defined regression function is represented, II 3 The third power of the distance between x and c is represented. Currently, the number of RBFs is set to 2d+1 according to the Kelmogorov theorem; the cluster center c is determined by a Kmeans clustering method.
S3, after a non-stationary Gaussian process model is built based on the step S2, training data are input into the non-stationary Gaussian process model to carry out model training, and in the training process, a differential evolution algorithm is adopted to carry out global optimization on parameters to be solved in the model (refer to FIG. 4, which is a data driving-evolution algorithm pseudo code assisted by the non-stationary Gaussian model); wherein:
the parameters to be solved comprise a weight coefficient theta representing the change of the training sample point x;
the model is input as follows: in step S1, taking the population estimated by the expensive original function (namely electromagnetic simulation) as a training set, and selecting training data from the training set; generating a dataset based on the individuals obtained after the evaluation of each individual in the population;
the output of the model is: and the optimal solution in the data set is the optimal antenna and the structure corresponding to the optimal antenna.
And (3) evolving a population by adopting a differential evolution algorithm, namely, adopting a non-stationary Gaussian process model to replace an expensive evaluation function to reevaluate individuals in the population in the training process, so as to obtain a random population.
S4, selecting a potential sample point from the random population obtained after differential evolution through an expected lifting strategy to perform electromagnetic simulation (namely, expensive evaluation); the potential sample points are added to the training set and the non-stationary gaussian process model is updated until the number of simulations (i.e., the number of expensive evaluations) is exhausted. (the current training process may refer to fig. 4, which is a data driven-non stationary gaussian model assisted evolution algorithm pseudocode) wherein:
after re-evaluating individuals in the population by adopting a non-stationary Gaussian process model instead of an expensive evaluation function, selecting the most potential solution probability point (namely, the solution with the best target value of a training sample point) in the population, evaluating by using the expensive evaluation function (namely, electromagnetic simulation), adding the solution into a training data set, and updating the data set; after the expensive number of evaluations (i.e., the number of simulations) has run out, the optimal solution in the current dataset is output.
Please refer to fig. 3, which is a schematic diagram of a non-stationary gaussian model training flow, and includes the following steps when model training is performed:
firstly, acquiring an initial sample point and a data set, judging whether a stopping criterion is reached (the stopping criterion is judging whether the simulation times are exhausted), and outputting the best solution in the data set, namely a structure corresponding to an optimal antenna, when the stopping criterion is reached; otherwise, executing the next step;
secondly, selecting a training sample, taking a non-stationary Gaussian process model as a proxy model, and training the proxy model;
secondly, adopting a differential evolution algorithm to evolve population, and taking a proxy model as an expensive evaluation function;
and finally, according to the random population obtained after differential evolution, selecting a potential sample point (a potential point) from the population through an expected lifting strategy for electromagnetic simulation, updating the data set, stopping training until a stopping criterion is reached, and outputting the best solution in the data set.
The disclosed algorithm mainly solves the problem of high electromagnetic simulation calculation cost in antenna design. The invention provides a non-stationary Gaussian assisted evolution algorithm which solves the problem that electromagnetic simulation calculation in antenna design is high in cost, and the algorithm is different from other model assisted evolution optimization algorithms in model establishment. In the current research field, the Gaussian agent model has better effect than other approximation technologies in solving the problem that the dimension is lower than 15 dimensions. In the present invention, the proposed data-driven-NGP model assisted evolution algorithm (DD-NGP-MAEA) is compared with the steady gaussian process assisted evolution algorithm (DD-SGP-MAEA), with the goal of minimizing in this experiment. The CEC2014 expensive optimization problem is taken as a test problem set, and the test problem set is used for testing the performance of an algorithm. The data-driven-NGP model assisted evolution algorithm (DD-NGP-MAEA) and the steady gaussian process assisted evolution algorithm were tested and compared on d=2, 5,10, respectively, to obtain the following conclusion (see fig. 5):
1. algorithm performance comparison at d=2 test problem set:
as shown in fig. 6, the two algorithms DD-NGP-MAEA and DD-SGP-MAEA are independently run 25 times on the test set problem to obtain mean ± variance, best value, worst value results; and Friedman Rank and Wlicoxon Rank sum test statistical tests were performed on these results.
By comparison, it is apparent that the algorithm proposed in this embodiment is superior to the existing stationary gaussian model assisted evolution algorithm. The method is characterized in that:
a. average value:
the average value of DD-NGP-MAEA is smaller than DD-SGP-MAEA;
b. statistical test comparison:
friedman Rank value of DD-NGP-MAEA is smaller than DD-SGP-MAEA.
Through Wlicoxon Rank sum test, DD-NGP-MAEA and DD-SGP-MAEA have a significant difference at the significance index α=0.05.
2. Algorithm performance comparison of test problem set at d=5:
as shown in fig. 7, the two algorithms DD-NGP-MAEA and DD-SGP-MAEA are independently run 25 times on the test set problem to obtain mean ± variance, best value, worst value results; and Friedman Rank and Wlicoxon Rank sum test statistical tests were performed on these results.
By comparison, it is apparent that the algorithm proposed in this embodiment is superior to the existing stationary gaussian model assisted evolution algorithm. The method is characterized by two aspects:
a. average value:
the average value of DD-NGP-MAEA is mostly smaller than DD-SGP-MAEA, and F5 is divided, because the F7 function becomes narrower along with the increase of the dimension, the valley of fitness landscape of the function becomes narrower, and great challenges are brought to all algorithms;
b. statistical test comparison:
friedman Rank value of DD-NGP-MAEA is smaller than DD-SGP-MAEA.
Through Wlicoxon Rank sum test, DD-NGP-MAEA and DD-SGP-MAEA have a significant difference at the significance index α=0.1.
3. Algorithm performance comparison of the test problem set at d=10:
as shown in fig. 8, the two algorithms DD-NGP-MAEA and DD-SGP-MAEA are independently run 25 times on the test set problem to obtain mean ± variance, best value, worst value results; and Friedman Rank and Wlicoxon Rank sum test statistical tests were performed on these results.
By comparison, it is apparent that the algorithm proposed in this embodiment has better performance than the existing stationary gaussian model assisted evolution algorithm in the d=10-dimensional test problem. The method is characterized by two aspects:
a. average value:
the average value of DD-NGP-MAEA is smaller than DD-SGP-MAEA;
b. statistical test comparison:
friedman Rank value of DD-NGP-MAEA is smaller than DD-SGP-MAEA.
Through Wlicoxon Rank sum test, DD-NGP-MAEA and DD-SGP-MAEA have a significant difference at the significance index α=0.05.
4. Taking an elliptical slot microstrip patch antenna as an example, as shown in fig. 9, the initial geometry of the antenna design is compared by experiments, and the algorithm provided by the invention has excellent performance when the antenna design is performed, and the advantages are embodied in the gain of the antenna and the standing wave ratio of the antenna:
firstly, when the algorithm of the present invention is used for designing the antenna, the gain of the antenna is greater than 0 (please refer to fig. 10):
antenna gain, which is one of the most important parameters in selecting a base station antenna, is a measure of the ability of an antenna to transmit and receive signals in a particular direction. In general, gain improvement relies mainly on reducing the vertical radiation-facing lobe width while maintaining omnidirectional radiation performance in the horizontal plane. Antenna gain is of paramount importance to the operational quality of a mobile communication system, as it determines the signal level at the cell edge. Increasing the gain may increase the coverage of the network in a certain direction or increase the gain margin in a certain range. Any cellular system is a bi-directional process, increasing the gain of the antenna can reduce the bi-directional system gain budget margin.
Secondly, when the algorithm provided by the invention is used for designing the antenna, the standing wave ratio of the antenna is less than 2.0 (please refer to fig. 11):
the standing wave ratio of the common antenna is smaller than 2.0, which is a good index, and many finished antennas all require the standing wave ratio to be smaller than 2.0, and some even to be 2.5.
To sum up:
a. the calculation cost of an antenna designed by the DD-NGP-MAEA algorithm is reduced to more than 10 times;
b. the DD-NGP-MAEA algorithm designs the antenna with excellent performance (particularly in the gain of the antenna and the standing wave ratio of the antenna).
Please refer to fig. 12, which is a block diagram of an antenna electromagnetic optimization system based on a non-stationary gaussian process model, the system includes a population building module L1, a model training module L2, a differential evolution module L3 and an electromagnetic simulation module L4, wherein each of the modules functions as follows:
functional role of population building block L1:
constructing a population, and initializing the scale of the population, wherein each individual in the population represents a training sample point, and each sample point represents an antenna; and after the population is evaluated, a target value corresponding to each individual is obtained, wherein the target value is a function value corresponding to an antenna optimization problem target established under a given antenna structure.
Functional role of model training module L2:
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 to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space.
Referring to fig. 12, the model training module further includes β value estimation sub-modules L21 and σ 2 z A value calculation sub-module L22, a maximum likelihood function construction sub-module L23, a maximum likelihood function simplification sub-module L24, a theta value calculation sub-module L25 and a non-stationary Gaussian process model construction module L26, wherein the functions of each module are as follows:
the beta value estimation submodule L21 is used for estimating beta by a least square method to obtain an estimated valueThe calculation formula is as follows:
wherein C is a 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 a regression function matrix in n×p dimensions, G (x) is a regression function vector in p×1 dimensions;
σ 2 z the value calculation submodule L22 is used for calculating the estimated value obtained by the beta value estimation moduleThe second term parameter sigma is calculated by taking the following calculation formula 2 z The calculation formula is as follows:
the maximum likelihood function construction submodule L23 is used for constructing a maximum likelihood function:
wherein det (C) is a determinant of covariance matrix C;
the maximize likelihood function simplifying sub-module L24 is used for the followingSum sigma 2 z Substituting the maximum likelihood function into the maximum likelihood function constructed by the maximum likelihood function constructing submodule, and simplifying the maximum likelihood function to obtain a likelihood function for evaluating theta:
-N logσ 2 z -log(det(C))；
the theta value calculation sub-module L25 is used for solving and obtaining theta by adopting a differential evolution algorithm based on a likelihood function for evaluating theta;
the non-stationary Gaussian process model building module L26 is used for estimating the obtained parameter betaσ 2 z And the parameter theta is brought into the non-stationary Gaussian process model, so that the construction of the non-stationary Gaussian process model is completed.
The differential evolution module L3 functions:
in the process of training a non-stationary Gaussian process model, global optimization is carried out on parameters to be solved in the model by adopting a differential evolution algorithm; the parameters to be solved comprise a weight coefficient theta representing the change of the training sample point x.
Electromagnetic simulation module L4 functions:
according to the random population obtained after differential evolution, selecting a potential sample point from the population through an expected lifting strategy to perform electromagnetic simulation (namely, expensive evaluation); the potential sample points are added to the training set and the non-stationary gaussian process model is updated until the electromagnetic simulation times (the expensive evaluation times) are exhausted.
Based on the analysis results, the antenna electromagnetic optimization method and system based on the non-stationary Gaussian process model disclosed by the invention have the following two obvious advantages:
1. the antenna electromagnetic optimization method and system are more flexible and have generality;
2. the antenna electromagnetic optimization method and system can more accurately approximate the expensive optimization function.
In addition, the antenna electromagnetic optimization method and the system also provide a general evolution algorithm framework assisted by a non-stationary Gaussian process model, solve the difficulty of high calculation cost in antenna design, solve the difficulty of agent model selection in data driving optimization, namely the selection of agent models, and bring a certain reference significance for solving the problem of data driving optimization in the future.
In addition, the antenna electromagnetic optimization method and the system can effectively solve the problem of time consumption of electromagnetic simulation of antenna design, further reduce the calculation cost of electromagnetic simulation, improve the antenna design efficiency (quicken the calculation efficiency), promote quick optimization and contribute to quickening the production speed.
The embodiments of the present invention have been described above with reference to the accompanying drawings, but the present invention is not limited to the above-described embodiments, which are merely illustrative and not restrictive, and many forms may be made by those having ordinary skill in the art without departing from the spirit of the present invention and the scope of the claims, which are to be protected by the present invention.
Claims (5)
1. An antenna electromagnetic optimization method based on a non-stationary Gaussian process model is characterized by comprising the following steps of:
s1, when designing an antenna structure, firstly constructing a population, and initializing the scale of the population, wherein each individual in the population respectively represents a training sample point, and each training sample point respectively represents an antenna structure; after the population is evaluated by adopting electromagnetic simulation, a target value corresponding to each individual is obtained, wherein the target value is a target function value corresponding to an antenna optimization problem 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 to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space; specifically, the mathematical form of the constructed non-stationary gaussian process model is as follows:
wherein g (x) is a regression function, and beta is a weight coefficient of the regression function g (x) to be solved; p is the total number of predefined regression functions; z (x) to N (0, sigma) 2 Z ) Is a stationary term, wherein the mean value of the stationary term is 0, and the variance term to be solved is sigma 2 Z ；
The correlation measure between any two training sample points x and x' is determined by a gaussian kernel function whose mathematical form is defined as:
wherein θ is a weight coefficient to be solved for representing the activity degree of the sample point x; the value range of i is [1, d ], d represents the dimension of x;
by solving for beta 1 ,…,β p 、θ 1 ,…,θ d Sum sigma 2 z To construct a non-stationary gaussian process model, comprising the steps of:
s21, estimating beta by a least square method to obtain an estimated valueThe calculation formula is as follows:
wherein C is a covariance matrix of N, N is the number of input sample points, y= (t) 1 ,…,y N ) T Is an N-dimensional column vector; g= (G) j (x i ) I=1, …, N, j=1, …, p, G is a regression function matrix in n×p dimensions, G (x) is a regression function vector in p×1 dimensions;
s22, calculating the estimated value obtained in the step S21The second term parameter sigma is calculated by taking the following calculation formula 2 z The calculation formula is as follows:
s23, constructing a maximum likelihood function:
wherein det (C) is a determinant of covariance matrix C;
s24, willSum sigma 2 Z Substituting the maximum likelihood function constructed in the step S23, and simplifying the maximum likelihood function to obtain a likelihood function about the evaluation theta:
-Nlogσ 2 Z -log(det(C)).
s25, based on likelihood functions about the evaluation of theta, the theta can be obtained by solving through a differential evolution algorithm;
s26, estimating the parameter beta obtained in the step S21Sigma obtained based on step S22 2 z And carrying the parameter theta obtained in the step S25 into a non-stationary Gaussian process model to complete the construction of the non-stationary Gaussian process model;
s3, in the process of training a non-stationary Gaussian process model, performing global optimization on parameters to be solved in the model by adopting a differential evolution algorithm; the parameters to be solved comprise a weight coefficient theta for representing the activity degree of the training sample point x;
s4, selecting a potential sample point from the random population through an expected lifting strategy according to the random population obtained after differential evolution to perform electromagnetic simulation; and adding the potential sample points into a training set, updating a non-stationary Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.
2. The method for electromagnetic optimization of an antenna based on a non-stationary gaussian process model according to claim 1, wherein in step S1, the scale of the population is initialized by a latin square sampling method;
the size of the initial population is: 11×d-1; d is greater than or equal to 1, d represents the dimension of the antenna optimization problem in the optimization process.
3. The method of antenna electromagnetic optimization based on a non-stationary gaussian process model according to claim 1, characterized in that the radial basis function is defined mathematically as a regression function, which is defined as:
wherein c is the clustering center of the N training sample points,the form of the defined regression function is represented, II 3 A cubic kernel function representing the distance between x and c.
4. An antenna electromagnetic optimization system based on a non-stationary gaussian process model is characterized by comprising the following modules:
the system comprises a population construction module, a population detection module and a population detection module, wherein the population construction module is used for firstly constructing a population and carrying out initialization setting on the scale of the population when carrying out antenna design, each individual in the population respectively represents a training sample point, and each training sample point respectively represents an antenna structure; after the population is evaluated by adopting electromagnetic simulation, a target value corresponding to each individual is obtained, wherein the target value is a target function value corresponding to an antenna optimization problem established under a given antenna structure;
the model training module adopts the estimated population as a training set, and training data are selected from the training set; inputting the training data into a non-stationary Gaussian process model to perform model training; wherein the training data comprises N training sample points x 1 ,…,x N ∈R d And a target value y corresponding to each sample point 1 ,…,y N The method comprises the steps of carrying out a first treatment on the surface of the R is a real number, d is a dimension, R d Is real space;
in the model training module, the mathematical form of the constructed non-stationary Gaussian process model is as follows:
wherein g (x) is a regression function, and β is the regression to be solvedThe weight coefficient of the normalization function g (x); p is the total number of predefined regression functions; z (x) to N (0, sigma) 2 Z ) Is a stationary term, wherein the mean value of the stationary term is 0, and the variance term to be solved is sigma 2 Z ；
The correlation measure between any two training sample points x and x' is determined by a gaussian kernel function whose mathematical form is defined as:
wherein θ is a weight coefficient to be solved for representing the activity degree of the sample point x; the value range of i is [1, d ], d represents the dimension of x;
by solving for beta 1 ,…,β p 、θ 1 ,…,θ d Sum sigma 2 z To construct a non-stationary gaussian process model, in a model training module, further comprising the following sub-modules:
a beta value estimation sub-module for estimating beta by least square method to obtain an estimated valueThe calculation formula is as follows:
wherein C is a 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 a regression function matrix in n×p dimensions, G (x) is a regression function vector in p×1 dimensions;
σ 2 z the value calculation submodule calculates an estimated value obtained by calculation of the beta value estimation submoduleThe second term parameter sigma is calculated by taking the following calculation formula 2 z The calculation formula is as follows:
a maximum likelihood function construction sub-module for constructing a maximum likelihood function:
wherein det (C) is a determinant of covariance matrix C;
maximizing likelihood function simplifies the sub-module, toSum sigma 2 Z Substituting the maximum likelihood function into the maximum likelihood function constructed by the maximum likelihood function constructing submodule, and simplifying the maximum likelihood function to obtain a likelihood function about estimating theta:
-Nlogσ 2 Z -log(det(C))
a theta value calculation submodule, which is used for solving a likelihood function for evaluating theta by adopting a differential evolution algorithm to obtain theta;
a non-stationary Gaussian process model building module for estimating the obtained parameter betaσ 2 z and the parameter theta are brought into a non-stationary Gaussian process model, and the construction of the non-stationary Gaussian process model is completed;
the differential evolution module is used for carrying out global optimization on parameters to be solved in the model by adopting a differential evolution algorithm in the process of training the non-stationary Gaussian process model; the parameters to be solved comprise 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 through an expected lifting strategy according to the random population obtained after differential evolution to perform electromagnetic simulation; and adding the potential sample points into a training set, updating a non-stationary Gaussian process model until the simulation times are exhausted, and outputting an optimal antenna and an antenna structure corresponding to the antenna.
5. The non-stationary gaussian process model based antenna electromagnetic optimization system according to claim 4, characterized in that the radial basis function is defined mathematically as a regression function, which is defined as:
wherein c is the clustering center of the N training sample points,the form of the defined regression function is represented, II 3 The third power of the distance between x and c is represented.
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