Rcs prediction method based on gaussian process regression and two-dimensional proxy model

Abstract

The application relates to a target RCS prediction method based on a Gaussian process regression and a two-dimensional agent model, which comprises the following steps: a three-dimensional rectangular block is established, and parameters are set; a data set is acquired; a composite covariance function is constructed, and a Newton gradient method is used to optimize and solve hyperparameters; a two-dimensional agent model is constructed according to the optimized hyperparameters, a covariance vector and a covariance matrix of the agent model are calculated, and a prediction result is obtained; and the prediction result is evaluated according to evaluation indexes. The two-dimensional agent model has good performance; through training of training data, a corresponding relationship between input and output is found, and a test set is used to realize a prediction function; overall, time consumed is obviously less than that of a traditional electromagnetic scattering algorithm, and calculation cost is saved; the method is not only suitable for a radar signal analysis field, but also suitable for remote sensing research, geophysical science and the like, and analysis cost is low.

Description

Technical Field

[0001] This invention relates to the fields of machine learning and signal processing technology, and in particular to a target RCS prediction method based on Gaussian process regression and a two-dimensional surrogate model. Background Technology

[0002] In real-world environments, the radar cross section (RCS) of an object is affected by uncertainties such as object size, coating thickness, object shape, and material properties. Accurately calculating the RCS under these conditions is challenging. Current research primarily employs stochastic analysis methods for uncertainty analysis. Traditional stochastic analysis methods include Monte Carlo methods, perturbation methods, stochastic combination methods, and stochastic Galerkin methods. These methods all rely on repeated full-wave simulations, with the computational scale proportional to the stochastic dimension. This results in a significant consumption of computational resources for multidimensional stochastic problems. Therefore, the ability to accurately and quickly predict the uncertainty of an object's RCS while conserving computational resources has become a key focus in recent years.

[0003] Machine learning and other methods have offered new insights into this problem. Deep learning has been employed to analyze large datasets and identify the relationship between input and output. While deep learning is a powerful nonlinear model capable of handling complex relationships, it also suffers from drawbacks such as excessive data requirements and a tendency to overfit. Furthermore, because the models primarily analyze large datasets, the underlying mathematical theory is not always sufficiently robust. Supervised learning methods have also been used, offering high predictive power and broad applicability, but they perform poorly in handling nonlinearities and optimizing parameter settings. Moreover, most of these studies focus on analyzing a single uncertainty factor, neglecting multidimensional analysis and consuming significant time. Summary of the Invention

[0004] To address the problem that existing technologies consume significant computational resources and time during calculations, the present invention aims to provide a target RCS prediction method based on Gaussian process regression and a two-dimensional surrogate model that can simultaneously analyze multidimensional uncertainties, save computational resources, reduce computation time, and achieve rapid prediction while ensuring accuracy.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a target RCS prediction method based on Gaussian process regression and a two-dimensional surrogate model, the method comprising the following sequential steps:

[0006] (1) Establish a three-dimensional rectangular block and set parameters, including incident wave frequency, relative permittivity and indeterminate shape;

[0007] (2) Obtain the dataset. The dataset is a two-dimensional dataset. The two-dimensional dataset is divided to obtain the training set and the test set of the two-dimensional dataset.

[0008] (3) Construct a composite covariance function based on Gaussian process regression, and use Newton's gradient method to optimize the hyperparameters of the composite covariance function to obtain the optimized hyperparameters;

[0009] (4) Construct a two-dimensional surrogate model based on the optimized hyperparameters, calculate the covariance vector and covariance matrix of the two-dimensional surrogate model, and obtain the prediction results;

[0010] (5) Evaluate the prediction results based on the evaluation indicators.

[0011] The step (1) specifically refers to: creating a three-dimensional rectangular block in the FEKO simulation software. The length, width, and height of the three-dimensional rectangular block are all 1m. The indefinite shape is the height of the three-dimensional rectangular block. The incident wave frequency is set to 300MHz.

[0012] In step (2), obtaining the dataset specifically refers to obtaining the dataset through the FEKO simulation software. For two-dimensional data, the input variable for the FEKO simulation software is the relative permittivity. and the height of the 3D rectangular block ,Right now , , i∈[1,n], The wavelength of the incident wave is 1. For the uncertainty factor, the output of the FEKO simulation software is the single-station RCS: , , To form a two-dimensional dataset.

[0013] Step (3) specifically refers to the following equation representing the Gaussian process regression:

[0014]

[0015] In the formula, Let be the mean function of the Gaussian process regression. It is a composite covariance function. Represent a Gaussian process;

[0016] The expression is as follows:

[0017]

[0018] In the formula, It is the squared exponential covariance function. It is a periodic covariance function;

[0019] The expression is:

[0020]

[0021] In the formula, , All are hyperparameters;

[0022] The expression is:

[0023]

[0024] In the formula, , All are hyperparameters;

[0025] The expressions for the four hyperparameters are shown below:

[0026]

[0027] In the formula, A vector of hyperparameters;

[0028] The hyperparameter is chosen as the log-marginal likelihood function, expressed as follows:

[0029]

[0030] In the formula, Let logarithmic marginal likelihood function include hyperparameters. Let be the determinant of the covariance matrix. The RCS output corresponding to the 3D rectangular block. The number of samples;

[0031] Using an optimizer based on Newton's gradient descent, the marginal likelihood partial derivatives of each hyperparameter are derived as follows:

[0032]

[0033] Newton's gradient method is shown in the following equation:

[0034]

[0035] in, For the number of iterations, For learning rate, Simultaneously, initial values ​​are set for four hyperparameters, and the number of iterations is determined. Set the learning rate to 30. Set to 0.0000001. Represents the loss function about The gradient of the loss function represents the rate of change of the loss function at the current weight.

[0036] Step (4) specifically refers to: in Gaussian process regression, when the input variables of the test set appear... When the test set output is:

[0037]

[0038] in, The RCS output corresponding to the 3D rectangular block. Let covariance vector be the vector. Let be the covariance matrix. The above formula is the formula for a two-dimensional surrogate model, and the prediction result is: ;

[0039] and The formulas are as follows:

[0040]

[0041]

[0042] In the formula, Input variables representing the test set Input variables of the training set covariance, The input variables of the training set With itself The covariance.

[0043] Step (5) specifically refers to: introducing two indicators, relative error and coefficient of determination, to measure the system performance. The expression for the relative error is:

[0044]

[0045] The expression for the coefficient of determination is:

[0046]

[0047] in, It is the RCS output by the test set. It is the predicted RCS output by the two-dimensional surrogate model. It is the mean of the RCS output of the test set; The value range is [0,1]. The larger the value, the better the prediction effect.

[0048] As can be seen from the above technical solution, the beneficial effects of the present invention are as follows: First, the present invention considers the influence of both relative permittivity and indeterminate shape on RCS, achieving dimensionality enhancement compared to ordinary one-dimensional surrogate model research; Second, the two-dimensional surrogate model of the present invention has good performance, with a relative error of 0.04868 and a coefficient of determination of 99.716%. When the coefficient of determination is close to 1, it indicates that the two-dimensional surrogate model fits the data very well and can explain most of the data variability; Third, the present invention finds the correspondence between input and output by training on training data, and then realizes the prediction function through the test set. Compared with traditional electromagnetic scattering algorithms, such as MoM or MLFMM, the present invention consumes significantly less time overall than traditional electromagnetic scattering algorithms, saving computational costs; Fourth, the present invention is not only applicable to the field of radar signal analysis, but also applicable to remote sensing research, geophysical science, and other fields, with lower analysis costs. Attached Figure Description

[0049] Figure 1 This is a flowchart of the method of the present invention;

[0050] Figure 2 This is a three-dimensional rectangular block model diagram of the present invention;

[0051] Figure 3 This is the FEKO sample step sampling diagram of the present invention;

[0052] Figure 4 This is a three-dimensional result diagram of the sample of the present invention;

[0053] Figure 5 This is a one-dimensional small-sample relative permittivity prediction diagram of the present invention;

[0054] Figure 6 This is a one-dimensional large-sample relative permittivity prediction diagram of the present invention;

[0055] Figure 7 This is a diagram showing the three-dimensional standard deviation of the two-dimensional predicted RCS and the actual RCS of this invention.

[0056] Figure 8 This is a comparison chart of the predicted RCS and the actual RCS after the re-application of this invention;

[0057] Figure 9 This is a graph showing the relative error and coefficient of determination of the present invention;

[0058] Figure 10 This is a comparison chart of the calculation time of the present invention and the calculation time of the method of moments. Detailed Implementation

[0059] like Figure 1As shown, a target RCS prediction method based on Gaussian process regression and a two-dimensional surrogate model is presented. This method includes the following sequential steps:

[0060] (1) Establish a three-dimensional rectangular block and set parameters, including incident wave frequency, relative permittivity and indeterminate shape;

[0061] (2) Obtain the dataset. The dataset is a two-dimensional dataset. The two-dimensional dataset is divided to obtain the training set and the test set of the two-dimensional dataset.

[0062] (3) Construct a composite covariance function based on Gaussian process regression, and use Newton's gradient method to optimize the hyperparameters of the composite covariance function to obtain the optimized hyperparameters;

[0063] (4) Construct a two-dimensional surrogate model based on the optimized hyperparameters, calculate the covariance vector and covariance matrix of the two-dimensional surrogate model, and obtain the prediction results;

[0064] (5) Evaluate the prediction results based on the evaluation indicators.

[0065] like Figure 2 As shown, step (1) specifically refers to: creating a three-dimensional rectangular block in the FEKO simulation software. The length, width, and height of the three-dimensional rectangular block are all 1m, and the indeterminate shape is the height of the three-dimensional rectangular block. The incident wave frequency is set to 300MHz.

[0066] In step (2), obtaining the dataset specifically refers to obtaining the dataset through the FEKO simulation software. For two-dimensional data, the input variable for the FEKO simulation software is the relative permittivity. and the height of the 3D rectangular block ,Right now , , i∈[1,n], The wavelength of the incident wave is 1. For the uncertainty factor, the output of the FEKO simulation software is the single-station RCS: , , To form a two-dimensional dataset.

[0067] Step (3) specifically refers to the following equation representing the Gaussian process regression:

[0068]

[0069] In the formula, Let be the mean function of the Gaussian process regression. It is a composite covariance function. Represent a Gaussian process;

[0070] The expression is as follows:

[0071]

[0072] In the formula, It is the squared exponential covariance function. It is a periodic covariance function;

[0073] The expression is:

[0074]

[0075] In the formula, , All are hyperparameters;

[0076] The expression is:

[0077]

[0078] In the formula, , All are hyperparameters;

[0079] The expressions for the four hyperparameters are shown below:

[0080]

[0081] In the formula, A vector of hyperparameters;

[0082] The hyperparameter is chosen as the log-marginal likelihood function, expressed as follows:

[0083]

[0084] In the formula, Let logarithmic marginal likelihood function include hyperparameters. Let be the determinant of the covariance matrix. The RCS output corresponding to the 3D rectangular block. The number of samples;

[0085] Using an optimizer based on Newton's gradient descent, the marginal likelihood partial derivatives of each hyperparameter are derived as follows:

[0086]

[0087] Newton's gradient method is shown in the following equation:

[0088]

[0089] in, For the number of iterations, For learning rate, Simultaneously, initial values ​​are set for four hyperparameters, and the number of iterations is determined. Set the learning rate to 30. Set to 0.0000001. Represents the loss function about The gradient of the loss function represents the rate of change of the loss function at the current weight.

[0090] Step (4) specifically refers to: in Gaussian process regression, when the input variables of the test set appear... When the test set output is:

[0091]

[0092] in, The RCS output corresponding to the 3D rectangular block. Let covariance vector be the vector. Let be the covariance matrix. The above formula is the formula for a two-dimensional surrogate model, and the prediction result is: ;

[0093] and The formulas are as follows:

[0094]

[0095]

[0096] In the formula, Input variables representing the test set Input variables of the training set covariance, The input variables of the training set With itself The covariance, for detailed calculation, can be found in the above section on the composite covariance function. The calculation formula.

[0097] like Figure 9 As shown, step (5) specifically refers to: introducing two indicators, relative error and coefficient of determination, to measure the system performance. The expression for the relative error is:

[0098]

[0099] The expression for the coefficient of determination is:

[0100]

[0101] in, It is the RCS output by the test set. It is the predicted RCS output by the two-dimensional surrogate model. It is the mean of the RCS output of the test set; The value range is [0,1]. The larger the value, the better the prediction effect.

[0102] like Figure 4 As shown, step sampling in the FEKO simulation software, where the relative permittivity... The sampling range is [2,5], and the sampling interval is 0.1; the uncertainty coefficient is... The sampling range is [0.1, 1], and the sampling interval is 0.1; a total of 310 datasets were obtained. For ease of subsequent research, the dataset was divided into sections based on relative permittivity. The dataset with an even number of elements serves as the training set, with a training set size of 160, and the relative permittivity... The dataset with an odd number of elements is the test set, and the number of test sets is 150.

[0103] Although the main research object of this invention is the two-dimensional proxy model, it is still necessary to analyze and preprocess the results of the one-dimensional model, such as... Figure 5 As shown, the training set has 19 elements, and the test set has 30 elements; Figure 6 As shown, the sample set was expanded based on the original data, with 46 training samples and 62 test samples. It can be found that expanding the sample dataset has an impact on the model's prediction performance.

[0104] exist Figure 7 One problematic point was found where the error reached 5.78 dBSm, while the errors in other areas were generally only around 0.578 dBSm. (dBSm is decibels per square meter.) Figure 4 Analysis can reveal the problem, and find that... In the range of [3.5, 5.0] and Within the range of [0.3, 0.7], the object's RCS experienced a sharp drop. Figure 4 During the sampling process, the sampling interval may be too sparse, resulting in insufficient information collected for the area, thus leading to significant prediction errors. To address this issue, resampling is performed. In [4,5] and 176 data points within the range [0.1, 1] were sampled as the training set, and 90 data points were randomly sampled as the test set; the optimizer, iterator, and learning rate settings are as previously shown. Figure 8As shown, by comparing the output results of the FEKO simulation software (actual RCS) with the output results of Gaussian process regression (GPR) (predicted RCS), it can be found that the error has been greatly improved.

[0105] like Figure 9 As shown, the relative error and coefficient of determination of the two-dimensional surrogate model are displayed. It can be found that the relative error is around 0.048 and the coefficient of determination reaches 0.99716, indicating that the two-dimensional surrogate model has good performance.

[0106] like Figure 10 The figure shows a comparison of the time consumption of the present invention and the calculation using the method of moments (MoM) in the FEKO simulation software. It can be seen that in terms of overall time consumption, the present invention is superior to the method of moments, improving efficiency while ensuring accuracy.

[0107] In summary, the two-dimensional surrogate model of this invention exhibits excellent performance. The relative error based on this invention is 0.04868, and the coefficient of determination reaches 99.716%. When the coefficient of determination is close to 1, it indicates that the two-dimensional surrogate model fits the data very well and can explain most of the data variability. By training on training data, the correspondence between input and output is found, and then the prediction function is achieved through a test set. Compared with traditional electromagnetic scattering algorithms, such as MoM or MLFMM, the overall time consumed by this invention is significantly less than that of traditional electromagnetic scattering algorithms, saving computational costs. It is not only applicable to the field of radar signal analysis but also to remote sensing research, geophysical science, and other fields, with lower analysis costs.

Claims

1. A target RCS prediction method based on Gaussian process regression and a two-dimensional surrogate model, characterized in that: The method includes the following steps in sequence: (1) Establish a three-dimensional rectangular block and set parameters, including incident wave frequency, relative permittivity and indeterminate shape; (2) Obtain the dataset. The dataset is a two-dimensional dataset. The two-dimensional dataset is divided to obtain the training set and the test set of the two-dimensional dataset. (3) Construct a composite covariance function based on Gaussian process regression, and use Newton's gradient method to optimize the hyperparameters of the composite covariance function to obtain the optimized hyperparameters; (4) Construct a two-dimensional surrogate model based on the optimized hyperparameters, calculate the covariance vector and covariance matrix of the two-dimensional surrogate model, and obtain the prediction results; (5) Evaluate the prediction results based on the evaluation indicators; The step (1) specifically refers to: creating a three-dimensional rectangular block in the FEKO simulation software. The length, width, and height of the three-dimensional rectangular block are all 1m. The indefinite shape is the height of the three-dimensional rectangular block. The incident wave frequency is set to 300MHz. In step (2), obtaining the dataset specifically refers to obtaining the dataset through the FEKO simulation software. For two-dimensional data, the input variable for the FEKO simulation software is the relative permittivity. and the height of the 3D rectangular block ,Right now , , i∈[1,n], The wavelength of the incident wave is... For the uncertainty factor, the output of the FEKO simulation software is the single-station RCS: , , To form a two-dimensional dataset.

2. The target RCS prediction method based on Gaussian process regression and two-dimensional surrogate model according to claim 1, characterized in that: Step (3) specifically refers to the following equation representing the Gaussian process regression: ， In the formula, Let be the mean function of the Gaussian process regression. It is a composite covariance function. Represent a Gaussian process; The expression is as follows: ， In the formula, It is the squared exponential covariance function. It is a periodic covariance function; The expression is: ， In the formula, , All are hyperparameters; The expression is: ， In the formula, , All are hyperparameters; The expressions for the four hyperparameters are shown below: ， In the formula, A vector of hyperparameters; The hyperparameter is chosen as the log-marginal likelihood function, expressed as follows: ， In the formula, Let logarithmic marginal likelihood function include hyperparameters. Let be the determinant of the covariance matrix. The RCS output corresponding to the 3D rectangular block. The number of samples; Using an optimizer based on Newton's gradient descent, the marginal likelihood partial derivatives of each hyperparameter are derived as follows: ， Newton's gradient method is shown in the following equation: ， in, For the number of iterations, For learning rate, Simultaneously, initial values ​​are set for four hyperparameters, and the number of iterations is determined. Set the learning rate to 30. Set to 0.0000001. Represents the loss function about The gradient of the loss function represents the rate of change of the loss function at the current weight.

3. The target RCS prediction method based on Gaussian process regression and two-dimensional surrogate model according to claim 1, characterized in that: Step (4) specifically refers to: in Gaussian process regression, when the input variables of the test set appear... When the test set output is: ， in, The RCS output corresponding to the 3D rectangular block. Let covariance vector be the vector. Let be the covariance matrix. The above formula is the formula for a two-dimensional surrogate model, and the prediction result is: ; and The formulas are as follows: ， ， In the formula, Input variables representing the test set Input variables of the training set covariance, The input variables of the training set With itself The covariance.

4. The target RCS prediction method based on Gaussian process regression and two-dimensional surrogate model according to claim 1, characterized in that: Step (5) specifically refers to: introducing two indicators, relative error and coefficient of determination, to measure the system performance. The expression for the relative error is: ， The expression for the coefficient of determination is: ， in, It is the RCS output by the test set. It is the predicted RCS output by the two-dimensional surrogate model. It is the mean of the RCS output of the test set; The value range is [0,1]. The larger the value, the better the prediction effect.

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