CN111625923A - 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; performing electromagnetic simulation on the population to obtain a target value corresponding to each individual; then, the evaluated population is used as a training set, and training data are selected from the training set; in the process of the non-stationary Gaussian process model training, a differential evolution algorithm is adopted to carry out global optimization on the parameters to be solved in the model; according to a 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 the potential sample points into a training set, and updating the non-stable 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 the electromagnetic optimization simulation efficiency of an antenna based on a non-stationary Gaussian model.
Background
The antenna is used as energy receiving, transmitting and converting equipment and widely applied to the fields of communication, radar, electronic countermeasure and the like, and can realize the functions of high-safety radar, electronic warfare, wireless communication and the like. The research of antennas has therefore been widely recognized. 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 problems. The optimization algorithms generally involved are a traditional optimization method and an artificial intelligence optimization method.
The traditional optimization method (newton method, conjugate function method, gradient descent method, etc.) usually uses derivative information of the correlation function, and the derivative information is determined by limits and only reflects the local characteristics of the function, so the traditional optimization method is difficult to obtain a global optimal solution or cannot be used.
In order to obtain a global optimal solution, currently, the problem of antenna design is mostly considered to be solved through an artificial intelligence Evolution Algorithm (EA); the evolution algorithm is to iterate the whole population (called population) according to the idea of evolution, and the quality of the whole population is continuously improved by applying some evolution operators (cross, mutation and selection) to the population; in the method, 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, conductibility and the like of the target 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 with nonlinearity, multiple modes, large scale, high constraint and great uncertainty.
However, existing studies are all based on the assumption that evolutionary algorithms are easy, computationally inexpensive to perform in performing the evaluation of objectives and constraints, and that there are explicit objective, constraint function expressions. However, in real-life, the problems are not so simple, and in actual antenna design, the evaluation of the adaptive values of the antenna optimization problems comes from expensive electromagnetic simulation experiments, which consumes a large amount of calculation cost, however, the time consumed in the optimization process is more unacceptable when hundreds of optimizations are involved each time.
Establishing an accurate proxy model is important for solving the optimization problem of antenna design, so a great deal of research work has been successively developed to establish a more accurate proxy model. In these numerous studies. The Gaussian process proxy model is widely applied, and has higher accuracy and provides confidence 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 the optimization problem with dimensions below 15. However, in the current many studies, the gaussian process is assumed to be a stationary process, i.e. many studies are conducted on the basis of the stationary gaussian process. However, the smoothness is strict and limited. In the actual antenna design problem, the assumption of a non-stationary process is often required, so that the characteristic of the antenna actual problem can be better reflected by establishing a non-stationary proxy model.
From the scientific research perspective, the antenna design problem is a nonlinear optimization problem which is very time-consuming for electromagnetic simulation, and the mapping relation from the antenna structure parameters to the electromagnetic field radiation distribution belongs to a non-stable random process in the probability sense, so the research has the latest scientific research result application value. The method applies a non-stationary Gaussian process model to mine an internal electromagnetic distribution mechanism of 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 applying a non-stationary Gaussian model and combining 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 antenna electromagnetic optimization method and system based on the non-stationary Gaussian process model are provided aiming at the defect that the actual problem of the antenna cannot be reflected due to the fact that research is carried out on the basis of the stationary proxy model in the prior art.
The technical scheme adopted by the invention for solving the technical problems is as follows: an antenna electromagnetic optimization method based on a non-stationary Gaussian process model is constructed, and the method comprises 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,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdA 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.
The invention discloses an antenna electromagnetic optimization system based on a non-stationary Gaussian process model, which comprises 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 N training sample points x1,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdA 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.
The implementation of the antenna electromagnetic optimization method and the system based on the non-stationary Gaussian process model has the following beneficial effects:
1. through theoretical analysis, the line electromagnetic optimization method and system are more flexible and general;
2. through theoretical analysis, the line electromagnetic optimization method and the line electromagnetic optimization system can more accurately approximate antenna electromagnetic simulation (expensive optimization function);
3. the electromagnetic optimization method and the system provide an evolution algorithm framework assisted by a non-stationary Gaussian process model, solve the problem of selection of a proxy model in data-driven evolution optimization, and bring certain reference significance for solving the problem of data-driven optimization in the future;
4. the line electromagnetic optimization method and the line electromagnetic optimization system can effectively solve the antenna design optimization problem, further reduce electromagnetic simulation calculation cost, improve antenna design optimization efficiency (accelerate calculation efficiency), promote rapid optimization and contribute to accelerating production speed.
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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 stationary and non-stationary processes;
FIG. 2 is a schematic flow chart of a method for implementing the antenna electromagnetic optimization method based on the non-stationary Gaussian process model disclosed by the invention;
FIG. 3 is a schematic diagram of a non-stationary Gaussian model training process;
FIG. 4 is data driven-non-stationary Gaussian model assisted evolutionary algorithm pseudo-code.
FIG. 5 is a comparison of algorithm results for evolutionary optimization with the assistance of the disclosed algorithm and other models;
FIG. 6 is a graph comparing the performance of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization algorithms in a d-2 dimensional test problem set;
FIG. 7 is a graph comparing the performance of the disclosed algorithm and other model-assisted evolutionary optimization algorithms in a d-5 dimensional test problem set;
FIG. 8 is a graph comparing the performance of the algorithm disclosed in the present invention and other model-assisted evolutionary optimization algorithms in a d-10 dimensional test problem set;
FIG. 9 is an initial geometry of an antenna design;
FIG. 10 is a graph of gain curves for an optimal antenna optimized by the algorithm;
FIG. 11 is a graph of the standing wave of the optimized antenna optimized by the algorithm;
FIG. 12 is a structural diagram of an antenna electromagnetic optimization system based on a non-stationary Gaussian process model according to the present invention.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
As antenna design optimization problems have gained wide attention, and data-driven proxy model assisted evolution optimization has yielded satisfactory results in solving practical antenna design problems. In the evolution optimization assisted by the on-line data-driven agent model, firstly, the most critical scientific problem is how to select a reasonable agent model and improve the accuracy of the agent model, if the established agent model cannot depict the original function, the misleading search is caused to converge to an error region instead of the region of the optimal solution of the original problem; since establishing an accurate proxy model is important for solving the antenna design optimization problem, a great deal of research work has been successively developed to establish a more accurate proxy model.
In these numerous studies. The Gaussian process proxy model is widely applied, and has higher accuracy and provides confidence 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 the optimization problem with dimensions below 15. However, in the current many studies, the gaussian process is assumed to be a stationary process, i.e. many studies are conducted on the basis of the stationary gaussian process. However, the smoothness is strict and limited. In practical problems of antenna design, an assumption of non-stationary process is often required, please refer to fig. 1, which is a simulation diagram for implementing stationary and non-stationary processes, in fig. 1, a solid line represents the implementation of the non-stationary process, and a dotted line represents the implementation of a stationary process, wherein horizontal and vertical coordinates represent different indexes in different application scenarios, for example, in an application scenario of transmitting a signal sequence: the abscissa represents time, and the ordinate represents intensity; it can be further clarified from the figure that the characteristics of most of the antenna design problems actually solved are non-stationary characteristics, so that the establishment of the non-stationary proxy model can better reflect the characteristics of the actual antenna design problems.
Please refer to fig. 2, which is a schematic flowchart of an antenna electromagnetic optimization method based on a non-stationary gaussian process model according to the present invention, including the following steps:
s1, constructing a population during antenna design, and initializing the scale of the population, wherein each individual in the population represents a training sample point, and the training sample point is a single 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; wherein:
initializing the scale of the population, specifically initializing the scale of the population by adopting a Latin square sampling method;
under the current embodiment, the size of the initial population is defined as: 11 xd-1; d is more than or equal to 1, and d is the dimension for solving the 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 for model training; wherein the training data comprises N training sample points x1,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdA real number space;
the current step is also a key point in the whole scheme, and the following explanation can be specifically made:
firstly, aiming at the establishment of a model:
the key point at present 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, 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.
At the existing N training sample points x1,…,xN∈RdAnd the corresponding target value is y ═ y1,…,yNIn the case of (2) a model of the non-stationary gaussian process is established, but in this model it is first necessary to determine some hyper-parameters in the model, including the weight coefficients β of the regression function f (x), the weight coefficients θ representing the change of the sample point x, and the variance term to be solved as σ2 zHowever, the estimation of these hyper-parameters is obtained by maximizing a likelihood function, the log form of which is:
where C is a covariance matrix of N x N, N is the number of input sample points, and y is (y)1,…,yN)TIs an N-dimensional column vector; g ═ G (G)j(xi) 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;
the calculation mode of the hyper-parameter is as follows:
1. estimation of the parameter β:
in this embodiment, β is estimated by the least square method to obtain an estimated value
The calculation formula is as follows:
2. variance term σ2 zEstimation of (2):
the estimated value obtained by the calculation is used
Substituting the following calculation formula (5), calculating the second term parameter sigma2 zThe calculation formula is as follows:
3. estimation of the weighting factor θ representing the change of the sample point x:
obtained by the above calculation
And σ2 zSubstituting the maximum likelihood function defined above and equation (3) to obtain a likelihood function for estimating θ:
since the maximum likelihood function is only related to the parameters in the covariance matrix, the above equation (6) can be abbreviated as follows:
-N logσ2 z-log(det(C))； (7)
finally, when the parameter theta is solved, based on a formula (7), solving by adopting a differential evolution algorithm to obtain theta;
the above is the calculation process of the hyper-parameters 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 an optimal linear unbiased estimation, such as:
when there is no prior distribution of information, the mean of the untested sample points x is:
the corresponding variance is:
here h is:
h＝g(x)-GTC-1r； (10)
wherein g (x) is a regression function vector of dimension p x 1,
is a coefficient matrix of a regression function in dimensions p x 1, and r is a covariance matrix of N x 1 consisting of untested points x and training data.
In the present embodiment, a Radial Basis Function (RBF) is used as the regression function, and the form of the regression function used in the above algorithm is:
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. Currently, the number of RBFs is set to be 2d +1 according to the Kolmogoroff theorem; the above-mentioned cluster center c is determined by means of Kmeans clustering.
S3, after constructing the non-stationary Gaussian process model based on the step S2, inputting training data to the non-stationary Gaussian process model for model training, and in the training process, adopting a differential evolution algorithm to carry out global optimization on parameters to be solved in the model (refer to FIG. 4, which is an evolution algorithm pseudo code assisted by a data-driven non-stationary Gaussian model); wherein:
the parameter to be solved comprises a weight coefficient theta representing the change of the training sample point x;
the inputs to the model are: in step S1, selecting training data from a training set using a population evaluated by an expensive primitive function (i.e., electromagnetic simulation) as the training set; generating a data set based on the individuals obtained after evaluating each individual in the population;
the output of the model is: and (4) optimal solutions in the data set, namely optimal antennas and structures corresponding to the optimal antennas.
And (3) evolving a population by adopting a differential evolution algorithm, namely, in the training process, adopting a non-stationary Gaussian process model to replace an expensive evaluation function to re-evaluate individuals in the population to obtain a random population.
S4, selecting a potential sample point from the population for electromagnetic simulation (namely expensive evaluation) through an expected lifting strategy according to the random population obtained after differential evolution; the potential sample points are added into a training set, and a non-stationary Gaussian process model is updated until the simulation times (namely the expensive evaluation times) are exhausted. (the current training process can refer to fig. 4, which is data-driven-non-stationary gaussian model assisted evolutionary algorithm pseudo-code) where:
after adopting a non-stationary Gaussian process model to replace an expensive evaluation function to re-evaluate individuals in a population, selecting a most potential solution conditioning point (namely the best solution of a training sample point target value) 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; until the expensive evaluation times (i.e., simulation times) are exhausted, the optimal solution in the current data set is output.
Please refer to fig. 3, which is a schematic diagram of a non-stationary gaussian model training process, and the method includes the following steps:
firstly, acquiring an initial sample point and a data set, judging whether a stopping criterion is reached (the stopping criterion is whether simulation times are exhausted), and outputting the best solution in the data set, namely the structure corresponding to the optimal antenna when the stopping criterion is reached; otherwise, executing the next step;
secondly, selecting a training sample, taking a non-stable Gaussian process model as a proxy model, and training the proxy model;
secondly, evolving the population by adopting a differential evolution algorithm, and taking the agent model as an expensive evaluation function;
and finally, according to the random population obtained after differential evolution, selecting a potential sample point (simulating point) from the population through an expected lifting strategy to perform 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 that electromagnetic simulation calculation in antenna design is high in cost. The invention provides a non-stationary Gaussian-assisted evolution algorithm, which solves the problem that the electromagnetic simulation calculation in antenna design is high in cost, and the algorithm is different from the algorithm of evolution optimization assisted by other models and is the establishment of the models. In the current research field, the Gaussian proxy model has better effect than other approximation techniques in solving the problem that the dimension is lower than 15 dimensions. In the invention, the proposed data-driven-NGP model-assisted evolution algorithm (DD-NGP-MAEA) is compared with the evolution algorithm (DD-SGP-MAEA) assisted by a stable Gaussian process, and the aim in the experiment is minimization. The CEC2014 expensive optimization problem is used as a test problem set, and the test problem set is used for testing the performance of the algorithm. The following conclusions can be obtained by testing and comparing the above-mentioned data-driven-NGP model-assisted evolution algorithm (DD-NGP-MAEA) and the evolution algorithm assisted by the stationary gaussian process on d 2,5, and 10, respectively (see fig. 5):
1. performance of the algorithm was compared on d 2 test question set:
as shown in fig. 6, it shows the results of mean ± variance, best value, and worst value obtained by independently running two algorithms DD-NGP-MAEA and DD-SGP-MAEA 25 times on the test set problem; and statistical tests were performed on Friedmanrank and Wlicon Rank test for these results.
By comparison, it is obvious that the algorithm provided in the embodiment is superior to the existing evolution algorithm assisted by a smooth gaussian model. The method is embodied from two aspects:
a. average value:
the average values of DD-NGP-MAEA are all smaller than DD-SGP-MAEA;
b. statistical test comparison:
the Friedman Rank value of the DD-NGP-MAEA is less than that of the DD-SGP-MAEA.
The significance indexes alpha of DD-NGP-MAEA and DD-SGP-MAEA are 0.05 of significance difference through Wlicon Rank sum test.
2. Performance of the algorithm was compared at d-5 for the test problem set:
FIG. 7 shows the mean + -variance, best value, and worst result of the DD-NGP-MAEA and DD-SGP-MAEA algorithms running 25 times independently on the test set problem; and statistical tests were performed on Friedmanrank and Wlicon Rank test for these results.
By comparison, it is obvious that the algorithm provided in the embodiment is superior to the existing evolution algorithm assisted by a smooth gaussian model. The method is embodied from two aspects:
a. average value:
the average value of DD-NGP-MAEA is mostly smaller than that of DD-SGP-MAEA except F5, because the valley of the fit landscapes of the function becomes narrower as the dimension of the F7 function increases, which brings great challenges to all algorithms;
b. statistical test comparison:
the Friedman Rank value of the DD-NGP-MAEA is less than that of the DD-SGP-MAEA.
By Wlicon Rank sum test, the DD-NGP-MAEA and the DD-SGP-MAEA have significance difference at the significance index alpha of 0.1.
3. Performance of the algorithm was compared at d 10 for the test question set:
as shown in fig. 8, it shows the results of mean ± variance, best value, and worst value obtained by independently running two algorithms DD-NGP-MAEA and DD-SGP-MAEA 25 times on the test set problem; and statistical tests were performed on Friedmanrank and Wlicon Rank test for these results.
By comparison, it is obvious that the performance of the algorithm proposed in the present embodiment is superior to that of the existing evolution algorithm assisted by a smooth gaussian model on the d-10-dimensional test problem. The method is embodied from two aspects:
a. average value:
the average values of DD-NGP-MAEA are all smaller than DD-SGP-MAEA;
b. statistical test comparison:
the Friedman Rank value of the DD-NGP-MAEA is less than that of the DD-SGP-MAEA.
The significance indexes alpha of DD-NGP-MAEA and DD-SGP-MAEA are 0.05 of significance difference through Wlicon Rank sum test.
4. Taking design of an elliptical slot microstrip patch antenna as an example, as an initial geometric structure of the antenna design shown in fig. 9, through experimental comparison, the algorithm provided by the invention has excellent performance when the antenna is designed, and the advantages are embodied in the gain of the antenna and the standing-wave ratio of the antenna:
first, when the antenna design is performed by the algorithm proposed by the present invention, the gains of the antennas are all greater than 0 (please refer to fig. 10):
antenna gain is a measure of the ability of an antenna to transmit and receive signals in a particular direction, and is one of the most important parameters for selecting a base station antenna. Generally, gain improvement relies primarily on reducing the lobe width of the vertically oriented radiation, while maintaining omnidirectional radiation performance in the horizontal plane. The antenna gain is extremely important to the operating quality of the mobile communication system because 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 within a certain range. Any cellular system is a bi-directional process, and increasing the gain of the antenna can reduce the bi-directional system gain budget margin at the same time.
Secondly, when the antenna is designed, the standing-wave ratios of the antennas are all less than 2.0 (please refer to fig. 11):
the standing-wave ratio of the general antenna is less than 2.0, which is a better index, and many finished antennas require the standing-wave ratio to be less than 2.0, and some antennas even reach 2.5.
In summary, the following steps:
a. the calculation 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 (particularly 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 according to 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, wherein each of the above modules functions as:
the functional role of the population building module L1 is as follows:
constructing a population, and carrying out initialization setting on 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, obtaining 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.
Functional role of the model training module L2:
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 N training sample points x1,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdIs a real space.
Referring to fig. 12, the model training module further includes β value estimation submodules L21 and σ2 zThe system comprises a value calculating operator module L22, a maximum likelihood function construction submodule L23, a maximum likelihood function simplification submodule L24, a theta value calculating operator module L25 and a non-stationary Gaussian process model building module L26, wherein the functions of each module are as follows:
the β value estimation submodule L21 is used for estimating β by a least square method to obtain an 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,…,yN)TIs an N-dimensional column vector; g ═ G (G)j(xi) 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 zthe value calculating operator module L22 is used for calculating the estimated value obtained by the β value estimating module
Substituting the following calculation formula to calculate a second term parameter sigma2 zThe calculation formula is as follows:
the maximum likelihood function construction submodule L23 is used to construct the maximum likelihood function:
where det (C) is the determinant of the covariance matrix C;
the maximum likelihood function reduction submodule L24 is used for converting
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 L25 is used for solving to obtain theta by adopting a differential evolution algorithm based on a likelihood function for evaluating the theta;
the non-stationary Gaussian process model building module L26 is used for obtaining 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.
The function of the differential evolution module L3 is:
in the process of model training of the non-stationary Gaussian process, global optimization is carried out on 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.
The electromagnetic simulation module L4 has the following functions:
according to a 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 into a training set, and a non-stationary Gaussian process model is updated until the electromagnetic simulation times (expensive evaluation times) are exhausted.
Based on the analysis result, the antenna electromagnetic optimization method and the antenna electromagnetic optimization system based on the non-stationary Gaussian process model 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 the system can more accurately approximate an expensive optimization function.
In addition, the antenna electromagnetic optimization method and the antenna electromagnetic optimization system also provide a general non-stationary Gaussian process model-assisted evolution algorithm framework, solve the high calculation cost difficulty encountered in antenna design, solve the proxy model selection difficulty in data-driven optimization, namely the proxy model selection, and bring certain reference significance for solving the data-driven optimization problem in the future.
In addition, the antenna electromagnetic optimization method and the antenna electromagnetic optimization system can effectively solve the problem of time consumption of antenna design electromagnetic simulation, further reduce electromagnetic simulation calculation cost, improve antenna design efficiency (accelerate calculation efficiency), promote rapid optimization and contribute to accelerating production speed.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.
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,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdA 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,…,βp、θ1,…,θdAnd 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,…,yN)TIs an N-dimensional column vector; g ═ G (G)j(xi) 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,…,xN∈RdAnd a target value y corresponding to each sample point1,…,yN(ii) a R is a real number, d is a dimension, RdA 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,…,βp、θ1,…,θdAnd 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,…,yN)TIs an N-dimensional column vector; g ═ G (G)j(xi) 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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