Target electromagnetic scattering characteristic prediction method in multi-task Gaussian process

Abstract

The invention discloses a target electromagnetic scattering characteristic prediction method in a multi-task Gaussian process, and the method comprises the following steps: deducing the relation between the frequency and a far region scattering field in a physical optical method, introducing a single-input single-output Gaussian process regression model, and combining a mean value function in Gaussian process regression and uncertainty analysis to obtain a target electromagnetic scattering characteristic prediction model; constructing a frequency domain scattering characteristic proxy model; through kernel function selection and hyper-parameter selection, electromagnetic scattering characteristic rapid prediction about frequency is realized; then, the relation between the frequency and a far-region scattering field in a physical optical method is introduced into a single-input multi-output Gaussian process regression model, and a frequency domain scattering characteristic agent model based on the multi-task Gaussian process is constructed; compared with a single-task Gaussian process regression model, the multi-task Gaussian process frequency domain scattering characteristic agent model provided by the invention can greatly shorten the calculation time, and the scattering characteristic of the metal target can be quickly and efficiently calculated through the method.

Description

Technical Field

[0001] The present invention belongs to the technical field of rapid prediction of target electromagnetic scattering characteristics, and in particular to a method for predicting target electromagnetic scattering characteristics based on a multi-task Gaussian process. Background Art

[0002] A target's radar cross-section (RCS) is a crucial feature for radar target recognition. In real-world environments, obtaining target RCS characteristics typically requires extensive simulation calculations or actual measurements. Currently, the most widely used simulation methods include full-wave simulation and high-frequency methods. However, with increasing electromagnetic wave frequencies, RCS simulation calculations for complex, multi-scale targets face high time costs and significant memory consumption. Therefore, research on efficient and accurate methods for obtaining target scattering characteristics is a key area of research in the field of target characterization. Rapid prediction of target scattering characteristics based on surrogate models is a key research direction. Surrogate models are mathematical tools used to simulate and predict complex systems based on probabilistic statistics and machine learning. They are approximate models of the original system, describing the relationship between input variables and predicted results through mathematical relationships or algorithms. Specifically, in radar target feature extraction and recognition, surrogate model technology can reconstruct complete angular, frequency, and dimensional responses even when data is missing, significantly improving the performance and application range of radar systems. Numerous surrogate model methods exist, primarily including statistical learning surrogate models and neural network surrogate models. Statistical learning surrogate models include methods such as Gaussian process regression and support vector regression. These methods have strong mathematical interpretability and perform well with small samples, but lack certain physical interpretability. Neural network surrogate models include methods such as fully connected neural networks, convolutional neural networks, and graph neural networks. These methods have strong nonlinear fitting capabilities, but require massive amounts of training data, their generalization capabilities are limited by the distribution of training data, and they also lack certain physical meaning. Among these methods, Gaussian process regression has the advantages of small sample requirements, theoretical optimal solutions, fast computational speed with small sample data, and the ability to obtain the variance of prediction results, making it one of the most representative surrogate models. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for predicting target electromagnetic scattering characteristics based on a multi-task Gaussian process.

[0004] The technical solution to achieve the purpose of the present invention is: a method for predicting target electromagnetic scattering characteristics using a multi-task Gaussian process, comprising the following steps:

[0005] Step 1: Based on the physical optics method and Gaussian process regression model, the relationship between the frequency and the far-field scattering field in the physical optics method is established to achieve rapid prediction of the target's electromagnetic scattering characteristics;

[0006] Step 2: Introduce a single-input, single-output Gaussian process regression model. Combined with the mean function in Gaussian process regression, including the prior mean function and the posterior mean function, and uncertainty analysis, a frequency-domain scattering characteristic proxy model with physical interpretability is constructed. The RCS data of a single target in a known frequency band is used as input data. Through kernel function selection and hyperparameter selection, the GPR model is trained to predict the RCS data of the target in an unknown frequency band, thereby achieving rapid prediction of the electromagnetic scattering characteristics with respect to frequency.

[0007] Step 3. Introduce the relationship between frequency and far-field scattering in the physical optics method into the single-input multi-output Gaussian process regression model to construct a frequency-domain scattering characteristic proxy model based on a multi-task Gaussian process; use the RCS data of multiple targets in known frequency bands as input data at the same time, train the GPR model through kernel function selection and hyperparameter selection, and predict the RCS data of multiple targets in unknown frequency bands, thereby realizing the rapid prediction of the electromagnetic scattering characteristics of multiple targets with respect to frequency.

[0008] A computer device comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the steps of the above method are implemented when the processor executes the program.

[0009] A computer-readable storage medium stores a computer program, which implements the steps of the above method when executed by a processor.

[0010] A computer program product comprises a computer program, which implements the steps of the above method when executed by a processor.

[0011] Compared with the prior art, the advantages of the present invention are as follows: (1) Compared with the traditional method in which the linear term and the fluctuation term are predicted using the least squares method and the Gaussian process respectively, the present invention makes use of the role of the prior mean function to simplify the prediction of the linear term and the fluctuation term to only require one method, which is the Gaussian process regression. This avoids the conversion operation between different methods and helps to improve the computational efficiency while ensuring the accuracy; (2) The mean function of the Gaussian process is combined with the RCS calculation formula in the physical optics method, which gives the Gaussian process a physical meaning and has stronger physical interpretability; (3) The multi-task Gaussian process regression is used, which can significantly shorten the computation time while having physical interpretability compared with the single-task Gaussian process. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 Schematic diagram of the pyramid model and its dimensioning.

[0013] Figure 2 Schematic diagram of the cylindrical model and its dimensioning.

[0014] Figure 3(a) to Figure 3(b)Comparison of RCS prediction results for the pyramid target. Figure 3(a) shows the constant mean, and Figure 3(b) shows the linear mean.

[0015] Figure 4(a) to Figure 4(b) Comparison of the RCS prediction results for cylindrical targets. Figure 4(a) shows the constant mean, and Figure 4(b) shows the linear mean.

[0016] Figure 5(a) to Figure 5(b) The prediction results of multi-task Gaussian process regression are shown in Figure 5(a) for task one and Figure 5(b) for task two. DETAILED DESCRIPTION

[0017] The present invention proposes a method for predicting target electromagnetic scattering characteristics based on a multi-task Gaussian process. The steps are as follows: First, a rapid prediction study of target electromagnetic scattering characteristics based on physical optics and Gaussian process regression (GPR) is carried out. The relationship between frequency and far-field scattering field in physical optics (PO) is derived, and a single-input single-output (SISO) Gaussian process regression model is introduced. Combined with the mean function in Gaussian process regression, including the prior mean function and the posterior mean function, as well as uncertainty analysis, a frequency-domain scattering characteristic proxy model with physical interpretability is constructed. Through kernel function selection and hyperparameter selection, rapid prediction of electromagnetic scattering characteristics with respect to frequency is achieved. Then, the relationship between frequency and far-field scattering field in physical optics (PO) is introduced into a single-input multiple-output (SIMO) Gaussian process regression model to construct a frequency-domain scattering characteristic proxy model based on a multi-task Gaussian process. Finally, the effectiveness of the physically interpretable frequency-domain scattering characteristic proxy model is demonstrated through the swept-frequency radar cross-section (RCS) curves of cylindrical and pyramidal models. Furthermore, compared to the single-task Gaussian process regression model, the proposed multi-task Gaussian process frequency-domain scattering characteristic proxy model significantly reduces computation time. This method enables the rapid and efficient calculation of the scattering characteristics of metal targets.

[0018] The present invention focuses on two aspects: on the one hand, by combining the physical optics method to give a certain physical meaning to the Gaussian process regression, and using its prior mean function and posterior mean function formula, a frequency domain scattering characteristic proxy model is constructed. Through the design of the mean function, the complex model that requires the use of least squares method and Gaussian process to achieve prediction is simplified to a model that only requires Gaussian process regression, so that the target scattering characteristics can be quickly predicted, reducing the conversion operations between different methods, while ensuring accuracy, helping to improve computational efficiency. On the other hand, the present invention proposes a single-input and multi-output Gaussian process regression model for the prediction of target scattering characteristics. By constructing a multi-task target scattering characteristic data set, it can achieve the prediction of multiple tasks at the same time. Compared with the traditional single-task Gaussian process regression model, when faced with multiple data samples, this method can greatly improve the prediction efficiency while ensuring accuracy.

[0019] The present invention is further described in detail below with reference to the accompanying drawings.

[0020] The present invention is a method for predicting target electromagnetic scattering characteristics using a multi-task Gaussian process, comprising the following steps:

[0021] Step 1: Conduct a study on the rapid prediction of target electromagnetic scattering characteristics based on the physical optics method and the Gaussian process regression model, and establish the relationship between the frequency and the far-field scattering field in the physical optics method;

[0022] For metal targets, the physical optics method assumes that the induced current J' i It only exists in the illuminated area, that is, the bright area, and the induced current is zero in the dark area. i , and the backscattered field of the bright area is obtained; then, based on the calculation formula of the scattered field, the expression σ(k) of the target's single-station RCS with respect to wavenumber k is derived. Combined with the conversion relationship between wavenumber and frequency, the calculation formula σ(f) of RCS with respect to frequency is obtained; the linear term and exponential term in σ(f) are approximated respectively, and the approximate expression of the single-station RCS of the metal target with respect to frequency based on the physical optics method is obtained. The details are as follows:

[0023] For metal targets, based on the induced current J' i , the backscattered field in the bright area can be expressed as:

[0024]

[0025] in is the surface source S′ i is the unit outward normal vector, k is the incident wave vector, k is the wave number, is the unit incident wave vector, and the conversion relationship between the three is: k=|k|, is the unit vector of the incident electric field, η is the wave impedance. r is the distance from the origin O to the observation point r o The distance, r' is the source point; N is the number of surface elements, S' i represents the i-th bin.

[0026] Combining the vector operation principle, we can get:

[0027]

[0028] Based on the calculation formula of the scattered field, the expression of the target's single-station RCS with respect to k is derived:

[0029]

[0030] in k = 2πf / c, where f is the frequency and c is the speed of light in a vacuum. Therefore, the wave number term is the frequency term, and σ(k) can be further derived as:

[0031]

[0032] Through the expression of σ(f), we find that It can be approximately regarded as a cosine function, thus converted to

[0033]

[0034] where γ i is the unknown coefficient, Ψ i (f) is a function of frequency. It is a linear function of frequency and can be replaced by αf. Further adding the intercept term β, it is converted to αf+β, and σ(f) can be approximately expressed as:

[0035]

[0036] Step 2: Introduce a single-input, single-output Gaussian process regression model. Combined with the mean functions (both prior and posterior mean functions) in Gaussian process regression and uncertainty analysis, construct a physically interpretable proxy model of frequency-domain scattering characteristics. Using the RCS data of a single target in a known frequency band as input, the GPR model is trained through kernel function selection and hyperparameter selection to predict the target's RCS data in an unknown frequency band, thereby enabling rapid prediction of electromagnetic scattering characteristics with respect to frequency.

[0037] Construct sample set D = {(f i ,σ i )|=1,2,...,n}, n is the number of samples; where f i represents the frequency of the incident wave, σ iis the RCS value of the corresponding frequency point. Usually, it is assumed that the function to be predicted satisfies the Gaussian distribution, that is,

[0038]

[0039] Where GP(·) represents Gaussian process regression, m(f) represents the mean function, and K is the covariance matrix whose elements are expressed by the covariance function k(f i , f j ) calculated, K ij =k(f i , f j ). The covariance function is often used to represent the correlation between data. I is the identity matrix, and f and f′ are both sample data.

[0040] Gaussian process regression is based on the concept of maximum likelihood estimation. Its essence is to find a set of hyperparameters that maximizes the probability of observing sample data under these hyperparameters. Combined with the probability density function p(y|x,θ) of the Gaussian distribution:

[0041]

[0042] Where θ is a hyperparameter, y is the sample output y={σ i |i=1,2,...,n}, x is the sample input x={f i |i=1,2,...,n} Is the length of the training sample. Taking the logarithm of the probability density function gives the marginal likelihood function L(θ), and further taking the negative number, we get the negative logarithmic marginal likelihood function -L(θ).

[0043]

[0044] According to the idea of maximum likelihood estimation, the maximum value of the logarithmic marginal likelihood function is obtained, that is, the negative logarithmic marginal likelihood function is minimized, and finally the optimal hyperparameter value θ is obtained. * .

[0045]

[0046] The commonly used hyperparameter optimization method in Gaussian process regression is the gradient descent method.

[0047]

[0048] The left side of the arrow represents the hyperparameters updated in each iteration, and the right side of the arrow represents the hyperparameters obtained in the previous iteration. ris the learning rate. Usually, different learning rates are set due to different sample data and kernel functions. By selecting an appropriate kernel function, setting an appropriate learning rate and number of iterations, Gaussian process regression can obtain the theoretical optimal solution for data prediction. When calculating the partial derivative of the likelihood function with respect to the hyperparameters in the gradient descent method, the partial derivative is usually calculated for each hyperparameter separately, θ={θ1,θ2,...,θ n}, the calculation formula is as follows:

[0049]

[0050] tr(·) represents the trace of the matrix. Through the above derivation, the Gaussian process regression model can be trained based on the sample data, that is, the hyperparameters can be optimized.

[0051] After obtaining the optimal hyperparameters, the sample output y and the test sample input x * The corresponding output y * The joint distribution of is:

[0052]

[0053] K x，x The covariance matrix constructed for the training sample input x, K x，x* Input x for training samples and x for test samples * The constructed covariance matrix, K x*，x Input x for the test sample * The covariance matrix constructed with the training sample input x, K x*，x* Input x for the test sample * The covariance matrix constructed, x * ={f i* |i=1,2,...,m},y * ={σ i* |i=1,2,...,m}, m is the length of the test sample. The final output of the test sample satisfies,

[0054]

[0055] Input x for the test sample * The covariance vector constructed with the training sample input x. Usually, m 后验 (x ＊ ) is called the posterior mean function, m 先验 (x) is called the prior mean function. When the prior mean function m 先验 When (x) is a constant zero, the posterior mean function can be converted to the common form:

[0056]

[0057] Variance of the test sample It can be used to measure the uncertainty of the prediction results.

[0058]

[0059] Input x for the test sample * The covariance vector constructed between Input x for the test sample * The covariance vector constructed with the training sample input x, Input x for training samples and x for test samples * The covariance vector constructed. According to the mean and variance of the test sample, the confidence interval is calculated. The upper and lower bounds of the confidence interval are calculated as follows:

[0060]

[0061] The RCS calculation formula derived by physical optics is further converted into

[0062]

[0063] Where h(f) is called the global trend term. Previous work has used least squares to optimize parameters such as α and β. ξ(f) is called the local fluctuation term. GPR is used to capture data volatility and make predictions. When training the GPR model, previous researchers set the prior mean function to zero and used ξ(f) to predict the fluctuation characteristics of the RCS data, ignoring the predictive power of the prior mean function itself.

[0064] Therefore, the RCS calculation formula σ(f) derived by the physical optics method is combined with the posterior mean function m in GPR. 后验 (f) Combine and use the prior mean function m 先验 (f) h(f) describes the overall trend term of RCS data. This paper proposes a Gaussian process regression model with physical interpretability.

[0065] Considering the simplified form of h(f), m 先验 (f) can also be characterized by a linear mean function.

[0066] m 先验 (f) = 20lg(|αf+β|) (52)

[0067] The α and β parameters can be optimized together with other hyperparameters in the kernel function. The final constructed posterior mean function is

[0068]

[0069] where f* is the test frequency, f is the training frequency, is the test frequency f * The covariance vector constructed with the training frequency f, K f，f is the covariance matrix constructed between the training frequency points.

[0070] Furthermore, the SM (Spectral Mixture) kernel function, as a frequency-domain mixed Gaussian kernel function, is good at capturing complex multi-scale periodic and non-stationary patterns. Its mathematical expression is:

[0071]

[0072] Where Q is the number of mixed components. The larger the value, the more complex the data characteristics the model can capture. However, the increase in the Q value will increase the calculation time and memory consumption, and also bring the risk of overfitting. Therefore, the appropriate Q value should be selected according to the complexity of the data. q Represents the weight of each component. The larger the weight, the more significant the contribution of the frequency component to the final covariance function. q represents the mean of each component, represents the variance of each component, and f and f′ are both frequency variables. The selection of initial values for the kernel function's hyperparameters directly impacts the final optimization results. Therefore, the present invention randomly initializes the hyperparameters multiple times and utilizes the uncertainty of the prediction results to select appropriate initial values. The optimal hyperparameters for the model are determined by setting the learning rate and the number of iterations, combined with gradient descent.

[0073] Step 3: Introduce the relationship between frequency and far-field scattering from the physical optics method into a single-input, multi-output Gaussian process regression model to construct a frequency-domain scattering characteristic proxy model based on a multi-task Gaussian process. The RCS data of multiple targets in known frequency bands are simultaneously used as input. Through kernel function selection and hyperparameter selection, the GPR model is trained to predict the RCS data of multiple targets in unknown frequency bands, thereby achieving rapid prediction of the electromagnetic scattering characteristics of multiple targets with respect to frequency.

[0074] For a single input variable x, the D-dimensional output Y=[y1,...,y d ,...,y D ], y d ={σ di |i=1,2,...,n}, assuming that each dimension outputs y d is generated by an independent Gaussian process, then

[0075]

[0076] The prior mean function m d =[m d (f1),...,md (f n )], covariance matrix K d By kernel function k d (f i ,f j ). The joint prior distribution of all outputs is

[0077]

[0078] Where M=[m1,...,m d ,...,m D ], block diagonal covariance matrix for

[0079]

[0080] For the test sample x * , each dimension outputs y d Independent, its posterior mean and variance can be decomposed into independent calculations of each dimension.

[0081]

[0082] in Input x for the test sample in the dth dimension * The covariance vector constructed with the training sample input x, K d|x，x is the covariance matrix constructed between the training sample inputs x in the dth dimension.

[0083] The mean function of the multi-task Gaussian process can be derived as,

[0084]

[0085] is the test frequency f in the dth dimension * The covariance vector constructed with the training frequency f, K d|f,f is the covariance matrix constructed between the training frequency points f in the dth dimension. d is α=[a1,...,α d ,...,α D ] elements, β d is β=[β1,...,β d ,...,β D ] elements, the mean functions of different tasks can be calculated independently, and the variance of different tasks It can be deduced that,

[0086]

[0087] is the test sample frequency f* The covariance vector constructed between is the test frequency f * The covariance vector constructed with the training frequency f, The training frequency f and the test frequency f * The constructed covariance vector.

[0088] The effectiveness of the proposed method is determined by the relative root mean square error (RMSE) calculation formula.

[0089]

[0090] where y i Represents the prediction results, Represents the true value, and N represents the number of predicted data.

[0091] Example 1

[0092] This embodiment performs RCS prediction verification of the single-task Gaussian process regression model in the frequency domain for the metal prism model and the metal cylinder model. th Gen Inter(R) Core(TM) i5-13500HX CPU @ 2.5GHz, 32GB of memory. Based on the PyCharm platform, Python version is 3.10.15.

[0093] (1) Metal prism model and its dimensions as follows Figure 1 As shown, the incident wave frequency range is 8-12GHz, the incident wave angle is θ=0°, The model sampling data is shown in Table 1. For the pyramid model, the linear mean function and the constant mean function are used respectively, the learning rate is set to 0.1, the number of iterations is 500, and the SM kernel function is used. The two methods set the same initial values of the hyperparameters to achieve the purpose of controlling the variables. The prediction results are shown in Figure 3(a) to Figure 3(b) Figure 3(a) shows the prediction results using the constant mean, and Figure 3(b) shows the prediction results using the linear mean. The RMSE using the linear mean function is 0.03, while the RMSE using the constant mean is 3.20, as shown in Table 2. This shows that using the linear mean function can significantly improve prediction accuracy.

[0094] (2) Metal cylindrical model and its dimensions as follows Figure 2 As shown, the incident wave frequency range is 4-12GHz, the incident wave angle is θ=40°, The model sampling data is shown in Table 1. For the cylindrical model, the linear mean function and the constant mean function are used respectively, the learning rate is set to 0.1, the number of iterations is 500, and the SM kernel function is used. The two methods set the same initial values of the hyperparameters to achieve the purpose of controlling the variables. The prediction results are shown in Figure 4(a) to Figure 4(b) Figure 4(a) shows the prediction results using the constant mean, and Figure 4(b) shows the prediction results using the linear mean. The RMSE using the linear mean function is 0.45, while the RMSE using the constant mean is 5.38, as shown in Table 2. This shows that using the linear mean function can significantly improve prediction accuracy.

[0095] Furthermore, the prediction variance of Gaussian process regression can reflect the uncertainty of the prediction results. For example, the 95% confidence intervals in Figures 3 and 4 show that when the prediction result is relatively accurate, the confidence interval is relatively narrow, and when the prediction result error is large, the confidence interval is relatively wide. Compared with other prediction methods, Gaussian process regression can provide a reference for the accuracy of prediction results when the true value is unknown.

[0096] Table 1 Sampling data of different models

[0097]

[0098] Table 1 shows the sampling information of the pyramid model and cylindrical model, including the frequency sampling interval, number of training samples, and number of prediction samples.

[0099] Table 2 Comparison of single-task Gaussian process prediction results and time consumption

[0100]

[0101] Table 2 shows the RMSE and time consumption of the prism model and the cylindrical model when using the constant mean and the linear mean, respectively. It shows that in terms of prediction accuracy, the linear mean function is better than the constant mean function, and the two are comparable in computational efficiency.

[0102] Example 2

[0103] This example validates frequency-domain RCS prediction using a multi-task Gaussian process regression model for metal pyramid and cylinder models. This example was implemented on a 12th Gen Intel(R) Core(TM) i5-12400F CPU @ 2.5GHz with 32GB of RAM. It was run on Visual Studio Code, using Python version 3.10.15.

[0104] In order to demonstrate the efficiency improvement of multi-task Gaussian process regression compared to single-task Gaussian process regression, this embodiment still uses the model and data in Example 1. The difference is that in Example 1, two single-task Gaussian process regression models are used to realize the RCS prediction of the prism and cylinder models respectively, while in this embodiment, one multi-task Gaussian process regression is used to predict the two models simultaneously. The prediction results are shown in Figure 2. Figure 5(a) to Figure 5(b) As shown, Figure 5(a) is the prediction result of the pyramid model of task 1, and Figure 5(b) is the prediction result of the cylindrical model of task 2. The curve fitting between the true value and the prediction result shows that the multi-task Gaussian process can achieve the effect of simultaneous prediction of multiple tasks. The specific error and time consumption are shown in Table 3. The RMSE of the pyramid and cylindrical RCS predicted using the multi-task Gaussian process are 0.10 and 0.60 respectively, and the time consumption is 5.50 seconds. In Example 1, the sum of the time consumption of the two single tasks is 8.99 seconds, which shows that the multi-task Gaussian process regression can significantly improve the prediction efficiency while ensuring the prediction accuracy.

[0105] Table 3 Comparison of multi-task Gaussian process prediction results and time consumption

[0106]

[0107] The above embodiments are illustrations of specific implementation methods of the present invention, rather than limitations of the present invention. Technicians in the relevant technical fields can make various changes and modifications to obtain corresponding equivalent technical solutions without departing from the spirit and scope of the present invention. Therefore, all equivalent technical solutions should be included in the patent protection scope of the present invention.

Claims

1. A method for predicting target electromagnetic scattering characteristics based on a multi-task Gaussian process, characterized in that: Here are the steps: Step 1: Based on the physical optics method and Gaussian process regression model, the relationship between the frequency and the far-field scattering field in the physical optics method is established to achieve rapid prediction of the target's electromagnetic scattering characteristics; Step 2: Introduce a single-input, single-output Gaussian process regression model. Combined with the mean function in Gaussian process regression, including the prior mean function and the posterior mean function, and uncertainty analysis, a frequency-domain scattering characteristic proxy model with physical interpretability is constructed. The RCS data of a single target in a known frequency band is used as input data. Through kernel function selection and hyperparameter selection, the GPR model is trained to predict the RCS data of the target in an unknown frequency band, thereby achieving rapid prediction of the electromagnetic scattering characteristics with respect to frequency. Step 3. Introduce the relationship between frequency and far-field scattering in the physical optics method into the single-input multi-output Gaussian process regression model to construct a frequency-domain scattering characteristic proxy model based on a multi-task Gaussian process; use the RCS data of multiple targets in known frequency bands as input data at the same time, train the GPR model through kernel function selection and hyperparameter selection, and predict the RCS data of multiple targets in unknown frequency bands, thereby realizing the rapid prediction of the electromagnetic scattering characteristics of multiple targets with respect to frequency.

2. The target electromagnetic scattering characteristics prediction method of the multi-task Gaussian process according to claim 1 is characterized in that: Step 1 establishes the relationship between frequency and far-field scattering in the physical optics method: For metal targets, the physical optics method assumes that the induced current J' i It only exists in the illuminated area, that is, the bright area, and the induced current is zero in the dark area; Based on the induced current J' i , the backscattered field of the bright area is obtained; then, based on the calculation formula of the scattered field, the expression σ(k) of the target's single-station RCS with respect to wavenumber k is derived. Combined with the conversion relationship between wavenumber and frequency, the calculation formula σ(f) of RCS with respect to frequency is obtained; the linear term and exponential term in σ(f) are approximated respectively, and the approximate expression of the single-station RCS of the metal target with respect to frequency based on the physical optics method is obtained; the details are as follows: For metal targets, based on the induced current J' i , the backscattered field in the bright area is expressed as: in is the surface source S′ i is the unit outward normal vector, k is the incident wave vector, k is the wave number, is the unit incident wave vector, and the conversion relationship between the three is: k=|k|, is the unit vector of the incident electric field, η is the wave impedance; r is the distance from the origin O to the observation point r o The distance, r' is the source point; N is the number of surface elements, S' i represents the i-th facet; Combining the vector operation principle, we can get: Based on the calculation formula of the scattered field, the expression of the target's single-station RCS with respect to k is derived: in k = 2πf / c, where f is the frequency and c is the speed of light in a vacuum. Therefore, the wave number term is the frequency term, and σ(k) is derived as: Through the expression of σ(f), we find that Approximately regarded as a cosine function, thus converted to where γ i is the unknown coefficient, Ψ i (f) is a function of frequency; It is a linear function of frequency, replaced by αf; add the intercept term β, convert it to af+β, and σ(f) is approximately expressed as:

3. The target electromagnetic scattering characteristics prediction method of the multi-task Gaussian process according to claim 1 is characterized in that: Step 2 constructs a physically interpretable proxy model of frequency-domain scattering characteristics: introduces a single-input, single-output Gaussian process regression model; analyzes the mean function in Gaussian process regression, including the prior mean function and the posterior mean function, as well as the uncertainty; establishes a physically interpretable single-task Gaussian process regression model; selects an appropriate kernel function and performs hyperparameter optimization to train the constructed proxy model; the details are as follows: Construct sample set D = {(f i ,σ i )|i=1,2,...,n}, n is the number of samples; where f i represents the frequency of the incident wave, σ i is the RCS value of the corresponding frequency point; assuming that the function to be predicted satisfies the Gaussian distribution, that is, Where GP(·) represents Gaussian process regression, m(f) represents the mean function, and K is the covariance matrix whose elements are expressed by the covariance function k(f i , f j ) calculated, K ij =k(f i ,f j ); the covariance function is used to represent the correlation between data; I is the identity matrix, f and f′ are sample data; Gaussian process regression is based on the idea of maximum likelihood estimation. Its essence is to find a set of hyperparameters that maximizes the probability of observing sample data under this set of hyperparameters; combined with the probability density function p(y|x,θ) of the Gaussian distribution: Where θ is a hyperparameter, y is the sample output y={σ i |i=1,2,...,n}, x is the sample input x={f i |i=1,2,...,n}, n is the length of the training sample; Taking the logarithm of the probability density function gives the marginal likelihood function L(θ), and taking the negative logarithm gives the negative logarithmic marginal likelihood function -L(θ): According to the idea of maximum likelihood estimation, the maximum value of the logarithmic marginal likelihood function is obtained, that is, the negative logarithmic marginal likelihood function is minimized, and finally the optimal hyperparameter value θ is obtained. * ; The hyperparameter optimization method in Gaussian process regression is gradient descent method. The left side of the arrow is the hyperparameter after each iteration update, and the right side of the arrow is the hyperparameter obtained in the previous iteration; r It is the learning rate. Due to the different sample data and kernel functions, the set learning rate is also different; By selecting the kernel function, setting the learning rate and the number of iterations, Gaussian process regression can obtain the theoretical optimal solution for data prediction. When calculating the partial derivative of the likelihood function with respect to the hyperparameters in the gradient descent method, the partial derivative is calculated for each hyperparameter separately, θ={θ1,θ2,...,θ n The calculation formula is as follows: tr(·) represents the trace of the matrix. Through the above derivation, the Gaussian process regression model is trained according to the sample data, that is, the hyperparameters are optimized. After obtaining the optimal hyperparameters, the sample output y and the test sample input x * The corresponding output y * The joint distribution of is: K x，x The covariance matrix constructed for the training sample input x, Input x for training samples and x for test samples * The constructed covariance matrix is Input x for the test sample * The covariance matrix constructed with the training sample input x, Input x for the test sample * The covariance matrix constructed is m is the length of the test sample; the output of the final test sample satisfies, Input x for the test sample * The covariance vector constructed with the training sample input x; m 后验 (x * ) is called the posterior mean function, m 先验 (x) is called the prior mean function; when the prior mean function m 先验 When (x) is a constant zero, the posterior mean function is transformed into the following form: Variance of the test sample Used to measure the uncertainty of the forecast results: Input x for the test sample * The covariance vector constructed between Input x for the test sample * The covariance vector constructed with the training sample input x, Input x for training samples and x for test samples * The covariance vector constructed is used to calculate the confidence interval based on the mean and variance of the test sample. The upper and lower bounds of the confidence interval are calculated as follows: The RCS calculation formula derived by physical optics is further converted into Where h(f) is the overall trend term, ξ(f) is the local fluctuation term, and GPR is used to capture the volatility of the data and make predictions; the RCS calculation formula σ(f) derived from the physical optics method is combined with the posterior mean function m in GPR. 后验 (f) Combine and use the prior mean function m 先验 (f) h(f) describes the overall trend of RCS data and proposes a Gaussian process regression model with physical interpretability; Considering the simplified form of h(f), m 先验 (f) Characterized by a linear mean function: m 先验 (f)=20lg(|αf+β|) (22) The α and β parameters are optimized together with other hyperparameters in the kernel function; the final constructed posterior mean function is where f * is the test frequency, f is the training frequency, is the test frequency f * The covariance vector constructed with the training frequency f, K f，f is the covariance matrix constructed between the training frequency points; As a frequency-domain mixed Gaussian kernel function, the SM kernel function is good at capturing complex multi-scale periodic and non-stationary patterns. Its mathematical expression is: Where Q is the number of mixture components, w q Represents the weight of each component, μ q represents the mean of each component, represents the variance of each component, and f and f′ are both frequency variables; By randomly initializing the hyperparameters multiple times and using the uncertainty of the prediction results to select the initial values, the optimal hyperparameters of the model are solved by setting the learning rate and the number of iterations and combining the gradient descent method.

4. The target electromagnetic scattering characteristics prediction method of the multi-task Gaussian process according to claim 1 is characterized in that: In step 3, a frequency-domain scattering characteristic proxy model based on a multi-task Gaussian process is constructed. The relationship between frequency and far-field scattering field in the physical optics method is introduced into a single-input multi-output Gaussian process regression model. The RCS data of multiple targets in known frequency bands are simultaneously used as input data. Through kernel function selection and hyperparameter optimization, the GPR model is trained to predict the RCS data of multiple targets in unknown frequency bands, thereby realizing the rapid prediction of the electromagnetic scattering characteristics of multiple targets with respect to frequency. The details are as follows: For a single input variable x, the D-dimensional output Y=[y1,...,y d ,...,y D ], y d ={σ di |i=1,2,...,n}, assuming that each dimension outputs y d is generated by an independent Gaussian process, then The prior mean function m d =[m d (f1),...,m d (f n )], covariance matrix K d By kernel function k d (f i ,f j ) is composed; the joint prior distribution of all outputs is Where M=[m1,...,m d ,...,m D ], block diagonal covariance matrix for For the test sample x * , each dimension outputs y d Independent, its posterior mean and variance are decomposed into independent calculations of each dimension; in Input x for the test sample in the dth dimension * The covariance vector constructed with the training sample input x, K d|x，x is the covariance matrix constructed between the training sample inputs x in the dth dimension; The mean function of the multi-task Gaussian process is derived as: is the test frequency f in the dth dimension * The covariance vector constructed with the training frequency f, K d|f，f is the covariance matrix constructed between the training frequency points f in the dth dimension; where α d is α=[α1,...,α d ,...,α D ] elements, β d is β=[β1,...,β d ,...,β D ] elements, the mean functions of different tasks are calculated independently, and the variances of different tasks The derivation is: is the test sample frequency f * The covariance vector constructed between is the test frequency f * The covariance vector constructed with the training frequency f, The training frequency f and the test frequency f * The constructed covariance vector.

5. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 4 are implemented.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method according to any one of claims 1 to 4 are implemented.

7. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 4 are implemented.

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