An adaptive sample enhancement method based on IDW-Kriging

Abstract

This invention discloses an adaptive sample augmentation method based on IDW-Kriging, comprising: configuring initial parameters; preprocessing sample data in the original sample set to reduce the degree of variation in the sample data, obtaining a preprocessed sample set; obtaining initial IDW interpolation curves and initial Kriging interpolation curves based on the preprocessed sample data; comparing the initial IDW interpolation curves and the initial Kriging interpolation curves to automatically find potential extreme points greater than the interpolation threshold; resimulating at the potential extreme points to obtain new sample data to supplement the sample set, and preprocessing the new sample data; re-obtaining IDW interpolation curves and Kriging interpolation curves based on the updated sample set; determining whether the maximum number of iterations has been reached; if not, executing the above steps; if so, saving the Kriging interpolation curve as the augmented sample set. This invention can adaptively find potential extreme points and achieve adaptive sample construction without adjusting the numerical simulation density based on empirical judgment.

Description

Technical Field

[0001] This invention belongs to the field of big data model building technology, specifically relating to an adaptive sample augmentation method based on IDW-Kriging. Background Technology

[0002] In the process of product design, manufacturing, and operation and maintenance, it is necessary to simulate and predict the mechanical, thermal, and electromagnetic properties of products to achieve optimal product design and operational control. Traditional methods achieve this goal through empirical formulas and numerical simulation. Empirical formulas are only applicable to calculations of typical structures, and the calculation error is relatively large for simulations of complex product performance. Although numerical simulations provide accurate calculation results, they are time-consuming, computationally expensive, and lack timely feedback, making it difficult to meet the high real-time simulation application requirements of intelligent manufacturing and intelligent operation and maintenance scenarios, such as virtual-real integration and autonomous decision-making.

[0003] Proxy models are a method that has emerged in recent years for real-time prediction of product performance based on large amounts of sample data. Kriging interpolation is the most widely used proxy model method. Kriging interpolation can obtain computational results comparable to numerical simulation in a computation time equivalent to empirical formulas, and therefore it is widely used in geographic data computation, structural fatigue life estimation, fluid dynamics optimization, and aircraft radar cross-section (RCS) simulation.

[0004] However, in practice, it has been found that the Kriging interpolation method suffers from a large prediction error near extreme points when processing datasets with large gradients. When Kriging is applied to simulated samples, the error between the predicted and actual simulated values ​​is small at locations with small gradients, but at minimum or maximum values, the error can reach 5 dBsm, which is unacceptable in engineering applications. Therefore, the Kriging interpolation method cannot be directly applied for simulation prediction in processing such datasets.

[0005] While increasing the number of samples can alleviate this problem, uniformly increasing sample density is unlikely to cover all extreme points, resulting in a very limited improvement in overall simulation accuracy. Furthermore, the amount of sample data cannot increase indefinitely; a significant increase in sample data would lead to a sharp increase in the memory and time consumed by Kriging interpolation calculations. Summary of the Invention

[0006] To address the aforementioned problems in the existing technology, this invention provides an adaptive sample augmentation method based on IDW-Kriging. The technical problem to be solved by this invention is achieved through the following technical solution:

[0007] This invention provides an adaptive sample augmentation method based on IDW-Kriging, comprising:

[0008] S1: Configure initial parameters, including the kriging parameter θ. kriging IDW interpolation smoothing parameter α, sampling interval, difference threshold and maximum number of iterations for IDW interpolation method and Kriging interpolation method;

[0009] S2: Preprocess the sample data in the original sample set to reduce the degree of variation in the sample data and obtain the preprocessed sample set;

[0010] S3: Obtain the initial IDW interpolation curve and the initial Kriging interpolation curve based on the preprocessed sample set;

[0011] S4: Compare the initial IDW interpolation curve with the initial Kriging interpolation curve to obtain potential extreme points greater than the difference threshold;

[0012] S5: Re-simulate at the potential extreme point to obtain new sample data to supplement the sample set, and preprocess the new sample data to obtain an updated sample set;

[0013] S6: Obtain the IDW interpolation curve and Kriging interpolation curve again based on the updated sample set;

[0014] S7: Determine whether the maximum number of iterations has been reached. If not, proceed to step S4. If yes, save the Kriging interpolation curve as the enhanced sample set.

[0015] In one embodiment of the present invention, S2 includes:

[0016] S2.1: Obtain multiple raw sample data points using actual measurement or simulation methods to form a raw sample set. The raw sample data in the raw sample set includes angular domain sample points or frequency domain sample points, wherein the input data for the angular domain sample points is the incident angle. The output data is the RCS value. The input data for the frequency domain sample points is the frequency, and the output data is the RCS value.

[0017] S2.2: Perform data translation on the RCS values ​​of the angular domain sample points or frequency domain sample points in the original sample data, and define a new RCS value σ′(before)=σ(before)+1, so that after the data translation, the RCS value σ′(before) of each angular domain sample point or frequency domain sample point is in the interval [1,+∞], where σ(before) is the RCS value of the angular domain sample point or frequency domain sample point in the original sample data;

[0018] S2.3: Use the log function to transform σ′(before), the transformation formula is:

[0019] σ′(after)=10·lg(σ′(before)),

[0020] Where σ′(after) represents the RCS value of each angular domain sample point or frequency domain sample point after data transformation, that is, the RCS value after preprocessing.

[0021] In one embodiment of the present invention, S3 includes:

[0022] S3.1: Based on the preprocessed sample set, the RCS values ​​of multiple prediction points are obtained using the IDW interpolation method, and an IDW interpolation curve is formed based on the multiple prediction points. The preprocessed sample set includes preprocessed angular domain sample points or preprocessed frequency domain sample points, wherein the preprocessed angular domain sample points include the incident angle. and the corresponding preprocessed RCS value; the preprocessed frequency domain sample points include the frequency and the corresponding preprocessed RCS value;

[0023] S3.2: Based on the preprocessed sample set, obtain the initial Kriging interpolation curve using the Kriging interpolation method.

[0024] In one embodiment of the present invention, S3.1 includes:

[0025] Based on the preprocessed sample set, the RCS values ​​of multiple prediction points are obtained using the IDW interpolation method. The calculation formula is as follows:

[0026]

[0027] Where Z represents the predicted point at the preset angle, and d Zi This represents the distance between the predicted point Z and the known sample point i in the preprocessed sample set, where α is a set parameter, and w Zi The distance weight between the predicted point Z and the known sample point i is represented by N, where N represents the number of known sample points in the preprocessed sample set, and RCS is the distance weight between the predicted point Z and the known sample point i. i Given the RCS value of sample point i, This represents the predicted value of the RCS of the predicted point Z;

[0028] An IDW interpolation curve is formed based on the multiple prediction points.

[0029] In one embodiment of the present invention, S4 includes:

[0030] The actual difference between the RCS values ​​of the same prediction point on the initial IDW interpolation curve and the initial Kriging interpolation curve is calculated sequentially. It is then determined whether the difference is greater than a preset threshold. If so, the current prediction point is considered a potential extreme point, and all potential extreme points on the initial IDW interpolation curve and the initial Kriging interpolation curve are obtained.

[0031] In one embodiment of the present invention, S5 includes:

[0032] At the potential extreme point, a numerical method is used to re-simulate, and the corresponding RCS value is obtained by simulating the input angle or frequency at the potential extreme point, thereby forming new sample data and supplementing it to the current sample set;

[0033] The new sample data is then subjected to data shifting and log function processing.

[0034] In one embodiment of the present invention, S7 includes:

[0035] Determine if the current iteration count has reached the maximum iteration count. If not, proceed to step S4 to continue searching for potential extreme points on the current IDW interpolation curve and the current Kriging interpolation curve. If yes, obtain the current Kriging interpolation curve.

[0036] The sample data on the current Kriging interpolation curve is transformed using the following formula:

[0037]

[0038] Where σ′(after) is the RCS value of the interpolation point on the current Kriging interpolation curve, and σ(after) is the RCS value of the interpolation point after restoration on the current Kriging interpolation curve.

[0039] Another aspect of the present invention provides a storage medium storing a computer program for performing the steps of the adaptive sample augmentation method based on IDW-Kriging as described in any of the above embodiments.

[0040] Another aspect of the present invention provides an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor, when calling the computer program in the memory, implements the steps of the adaptive sample augmentation method based on IDW-Kriging as described in any of the above embodiments.

[0041] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0042] 1. This invention proposes an adaptive sample augmentation method based on IDW-Kriging. First, initial IDW and Kriging interpolation curves are obtained from an existing sample dataset using IDW and Kriging interpolation methods. The location of extreme points is predicted by comparing the differences between the two interpolation methods at the same position. Then, sample data is added at the predicted extreme point locations. Finally, the overall prediction accuracy of the Kriging interpolation method is significantly improved with a small amount of additional sample data. This method can adaptively find the required simulated sample points. For models of varying complexity, it can adaptively control the amount of data, unlike traditional surrogate models which require empirical judgment of simulation value density. This approach has broader application prospects and overcomes the limitation of traditional interpolation methods that require empirical judgment to adjust the numerical simulation density.

[0043] 2. The adaptive sample augmentation method of this invention performs data preprocessing before constructing the surrogate model to improve its accuracy, ultimately reducing the amount of data required for simulation and achieving the goal of constructing a high-quality RCS surrogate model. The effectiveness of the IDW-Kriging-based adaptive sample augmentation method of this invention was verified using a surrogate model of the RCS of an aircraft. The results show that the prediction accuracy of the surrogate model after adaptive sample augmentation is improved by more than 70% compared to the traditional Kriging model.

[0044] 3. The method of this invention can better capture small but important local features in curves or 3D surface models. Compared with traditional uniform one-time sampling, the method of this invention more accurately reconstructs the relationship between the input and output of the target. The error analysis graph of the surrogate model constructed by this invention has significantly fewer spikes than that of uniform sampling, and the maximum value of the spikes is smaller, meaning the error is smaller.

[0045] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0046] Figure 1 This is a flowchart of an adaptive sample augmentation method based on IDW-Kriging provided in an embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram of the processing procedure of an IDW-Kriging adaptive sample augmentation method provided in an embodiment of the present invention;

[0048] Figure 3 This is a schematic diagram of the IDW interpolation curves when the parameter α takes different values, as provided in the embodiments of the present invention.

[0049] Figure 4The spatial correlation function R and distance d are related to different values ​​of the Kriging interpolation parameter θ. k Relationship diagram;

[0050] Figure 5 This is a process diagram illustrating the iterative increase of sample points and the correction of the surrogate model curve as proposed in an embodiment of the present invention;

[0051] Figure 6 It is a corner domain proxy model obtained by using the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging of this invention;

[0052] Figure 7 This is a schematic diagram of the error analysis of the corner domain proxy model obtained by the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging of this invention;

[0053] Figure 8 This is a scatter plot of the fitted samples from the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging of this invention;

[0054] Figure 9 This is a comparison of the frequency domain proxy models obtained in the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging of this invention;

[0055] Figure 10 This is a corner-domain error analysis diagram comparing the proxy model constructed using the traditional Kriging method and the proxy model constructed using the method of this invention;

[0056] Figure 11 It is a scatter plot of the corner domain fitting of the surrogate model constructed using the traditional Kriging method and the method of this invention;

[0057] Figure 12 This is a comparison chart of frequency domain error analysis between the surrogate models constructed using the traditional Kriging method and the method of this invention. Detailed Implementation

[0058] To further illustrate the technical means and effects adopted by the present invention to achieve the intended purpose, the following describes in detail an adaptive sample enhancement method based on IDW-Kriging proposed according to the present invention, in conjunction with the accompanying drawings and specific embodiments.

[0059] The foregoing and other technical contents, features, and effects of the present invention will be clearly presented in the following detailed description of specific embodiments in conjunction with the accompanying drawings. Through the description of the specific embodiments, a more in-depth and concrete understanding can be gained of the technical means and effects adopted by the present invention to achieve its intended purpose. However, the accompanying drawings are for reference and illustration only and are not intended to limit the technical solutions of the present invention.

[0060] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations are intended to cover non-exclusive inclusion, such that an article or apparatus comprising a list of elements includes not only those elements but also other elements not expressly listed. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the article or apparatus that includes said element.

[0061] Example 1

[0062] This embodiment addresses the problem of large prediction errors near extreme points when performing Kriging interpolation on large gradient data sample sets. It proposes an adaptive sample augmentation method based on IDW (Inverse Distance Weight) and Kriging to improve the accuracy of Kriging interpolated surface models.

[0063] Please see Figure 1 and Figure 2 , Figure 1 This is a flowchart of an adaptive sample augmentation method based on IDW-Kriging provided in an embodiment of the present invention; Figure 2 This is a schematic diagram illustrating the processing of an IDW-Kriging adaptive sample augmentation method provided in an embodiment of the present invention. The basic idea of ​​this adaptive sample augmentation method is as follows: by performing IDW interpolation and Kriging interpolation on the original sample set respectively, and comparing the difference between the two interpolation methods at the same location, the extreme value position of the surrogate model surface is automatically detected; then, sample data is added to the existing data sample set at this extreme value position; by continuously iterating the above operations, adaptive augmentation of the data samples can be achieved, significantly reducing prediction errors.

[0064] The specific steps of this adaptive sample augmentation method include:

[0065] S1: Configure initial parameters, including the kriging parameter θ. krigingThe IDW interpolation smoothing parameter α, sampling interval, difference threshold between IDW interpolation method and Kriging interpolation method, and maximum number of iterations are specified.

[0066] In this embodiment, the kriging parameter θ kriging The IDW interpolation smoothing parameter α is set to 5, the sampling interval is set to 1 degree, and the difference threshold between the IDW and Kriging interpolation methods is set to 1. This means that if the difference between the IDW and Kriging interpolation methods at the same position is greater than 1, an additional interpolation point is added. The maximum number of iterations is set to 4. This embodiment uses RCS (Radar Cross Section) as an example for simulation, including frequency domain simulation and angular domain simulation. These correspond to the construction of a surrogate model with relatively gradual changes and the construction of a surrogate model with drastic changes, respectively.

[0067] S2: Preprocess the sample data in the original sample set to reduce the degree of variation in the sample data and obtain a preprocessed sample set. In this embodiment, the original sample set is composed of the original sample points (i.e., sample data) used for training the surrogate model obtained by numerical simulation. The Log function is used to process and shift the data to reduce the impact of large gradient data on the surrogate model.

[0068] This step specifically includes:

[0069] S2.1: Obtain multiple original sample data based on actual measurement or simulation methods to form an original sample set.

[0070] Typically, the sample data in the original sample set should be in the form of xy, i.e., the sample data corresponding to the input and output, or (x,y)-z, i.e., the sample data corresponding to one output for multiple inputs. In this embodiment, the original sample set is formed by numerical simulation using Feko simulation software to obtain angular domain sample points or frequency domain sample points. That is, the original sample data in the original sample set includes angular domain sample points or frequency domain sample points, wherein the angular domain sample points are the incident angles. The format is a one-to-one correspondence between (pitch angle, azimuth angle) and RCS value, and the frequency domain sample points are in a one-to-one correspondence format between frequency and RCS value. That is, if this sample set is used to train the angular domain proxy model, then the input to the angular domain proxy model is... The output is the RCS value corresponding to that angle; if the sample set is used to train the frequency domain proxy model, then the input of the frequency domain proxy model is the frequency, and the output is the RCS value at that frequency.

[0071] S2.2: Perform data shifting on the original sample data before interpolation.

[0072] This step requires data preprocessing of the original sample data in the original sample set to reduce the degree of variation in the sample data. Since the object values ​​to be constructed in the proxy model may vary greatly, a log function is used for processing. The formula for this transformation is:

[0073] σ(after) = 10·lgσ(before)

[0074] Where σ(before) represents the original sample data in the original sample set, and σ(after) represents the value required to build the surrogate model after transformation.

[0075] Due to the characteristics of the log function, when the original sample data σ(before) is in the interval [0, 1], it will cause σ(after) to change drastically. Therefore, in this embodiment, the RCS values ​​of the angular domain sample points or frequency domain sample points in the original sample data are preprocessed before interpolation. A new sample data σ′(before) = σ(before) + 1 is defined, so that the RCS value σ′(before) of each angular domain sample point or frequency domain sample point after data translation is in the interval [1, +∞], where σ(before) is the RCS value of the angular domain sample point or frequency domain sample point in the original sample data.

[0076] S2.3: Use the log function to transform the new sample data σ′(before). The transformation formula is as follows:

[0077] σ′(after)=10·lg(σ′(before))

[0078] Where σ′(after) represents the output data of each angular domain sample point or frequency domain sample point after data transformation, that is, the preprocessed RCS value, which together with the corresponding input data constitutes the preprocessed sample data.

[0079] S3: Obtain the initial IDW interpolation curve and the initial Kriging interpolation curve based on the preprocessed sample set.

[0080] Step S3 in this embodiment specifically includes:

[0081] S3.1: Based on the preprocessed sample set, the RCS values ​​of multiple prediction points are obtained using the IDW interpolation method, and an IDW interpolation curve is formed based on the multiple prediction points.

[0082] The preprocessed sample set includes preprocessed angular domain sample points or preprocessed frequency domain sample points, wherein the preprocessed angular domain sample points include the incident angle. The preprocessed frequency domain sample points include the frequency and the corresponding preprocessed RCS value (after steps S2.2 and S2.3).

[0083] The IDW (Inverse Distance Weight) interpolation method calculates the value of a predicted point based on the distance between the predicted point and the sample point. When calculating the attribute value of a point in space, the IDW interpolation method allocates a larger weight to its surrounding neighboring values, and the weight value decreases as the distance increases.

[0084] Suppose we have sample points A and B with coordinates (x, y, y) respectively. a ,y a ) and (x b ,y b If the distance between sample points A and B is defined as follows:

[0085]

[0086] Where, d ab Let Z be the distance between two sample points A and B. For any predicted point Z at different angles, the RCS value is calculated using the following formula:

[0087]

[0088] Where, d Zi This represents the distance between the predicted point Z and the known sample point i in the preprocessed sample set. In this embodiment, the known sample point is either a preprocessed angular domain sample point or a preprocessed frequency domain sample point. α is a set parameter, and w Zi This represents the distance weight value w between the predicted point Z and the known sample point i. The greater the distance between the predicted point Z and the known sample point i, the higher the distance weight value w. Zi The smaller N is, the more likely N is to represent the number of known sample points i in the preprocessed sample set. RCS i Let i be the RCS value of the known sample point i. This represents the predicted RCS value of the predicted point Z, when α approaches 0. It approaches the average of the RCS of all known sample points when α approaches infinity. The value of the interpolation point is approximated by the known sample point closest to the predicted point. By adjusting the value of α, interpolation point curves with different degrees of smoothness can be obtained. Here, the known sample point i is the sample point in the original sample set after preprocessing in step S2.

[0089] Subsequently, the IDW interpolation curve can be generated based on the multiple prediction points.

[0090] Please see Figure 3, Figure 3 This is a schematic diagram of the IDW interpolation curves when the parameter α takes different values, as provided in this embodiment of the invention. It can be seen that the smaller the parameter α, the flatter the fitted IDW interpolation curve (corresponding to the red curve in the figure); the larger the parameter α, the steeper the fitted IDW interpolation curve. Specifically, Figure 3 (a) shows the interpolation result when parameter α = 0.5. Since the parameter α is very small, the correlation between the predicted points (corresponding to the interpolation points in the figure) and the known sample points (corresponding to the real data in the figure) is not greatly affected by distance, and the interpolation curve tends to fluctuate around the mean of the known sample points. When the value of α is 0, the interpolation curve degenerates into the mean of the sample points. Figure 3 (d) is the interpolation result when parameter α = 4. Since the parameter α is large, the correlation between the predicted point and the known sample point will weaken rapidly as the distance increases. The predicted point is more affected by the neighboring points. The interpolation curve is closer to the real data formed by the known points. The interpolation curve will approximately pass through the interpolation point.

[0091] S3.2: Based on the preprocessed sample set, obtain the initial Kriging interpolation curve using the Kriging interpolation method.

[0092] The Kriging interpolation method uses data from known points in space to predict the values ​​of unknown points through a weighted summation. However, its weights are not simply calculated based on the reciprocal of the distance, but rather a set of optimal coefficients is calculated. This method is an existing interpolation method based on optimal, unbiased linear prediction, relying on the spatial autocorrelation of the data for prediction. The expression for the Kriging model is:

[0093] G(x)=g T (x)β+z(x)

[0094] Where g(x) = [g1(x), g2(x), ..., g p (x)] T Let x be a polynomial matrix, and β = [β1, β2, ..., β]. p ] T denoted as , where is the regression coefficient and p is the number of polynomials. T (x)β is a parameterized polynomial model, which can be in constant form, linear form, or second-order polynomial form. z(x) is a stochastic process with a mean of 0.

[0095] Since g(x) has little impact on the fitting results of the Kriging interpolation method, a simple constant-type polynomial is used in this embodiment, with the condition p = 1 and g(x) = 1, i.e., g T (x)β=β.

[0096] z(x) can approximate the region of the fitted surface. z(x) follows a normal distribution, and its mean, variance, and covariance can be expressed as:

[0097] E[z(x)]=0

[0098] Var[z(x)]=σ 2

[0099] Cov[z(x i ), z(x j )]=σ 2 R(x i x j , θ), i, j=1,2,…,m

[0100] Where σ is the standard deviation of the random error, R(x) i x j ,θ) represents any two sample points x i and x j The spatial correlation function determines the fitting accuracy of the Kriging model, where m represents the total number of sample points. Similar to the IDW interpolation method, the sample points x... i and x j The closer the distances, the greater their correlation. θ represents the objective to be optimized. When x i =x j When their RCS are equal, then R(x) i x j ,θ)=1; when x i -x j As x approaches infinity, R(x) i x j Spatial correlation function R(x) → 0. i x j The expression for θ is:

[0101]

[0102] in, The subscript k indicates the k-th component, d k =x ik -x jk d k Represents a known sample point x i The k-th component and the known sample point x j The distance between the k-th components, where n represents the number of components.

[0103] The RCS value of the interpolation point (predicted point) predicted by the Kriging interpolation method can be represented by a linear combination of the RCS values ​​of known sample points in the known sample set:

[0104]

[0105] Where Y = [Y1, Y2, ..., Y] m ] T This represents the RCS value of a known sample point, where c is the coefficient matrix, derived from the formula G(x) = g T We can obtain: (x)β+z(x)

[0106] Y = Fβ + Z

[0107] Where F = [g T (x1), g T (x2), ..., g T (x m )] T {x1, x2, ..., x} m Let} be the known sample points, Z = [z(x1), z(x2), ..., z(x3)]. m )] T .

[0108] The prediction error of the Kriging model is:

[0109]

[0110] To ensure that the mean of the prediction error of the Kriging model is 0, the following conditions must be met: Right now:

[0111] c T Fg T (x)=0

[0112] The mean squared error of the Kriging model's prediction error is:

[0113]

[0114] Among them, r=[R(x, x1, θ), R(x, x2, θ),…, R(x, x m The correlation between predicted sample points and known sample points is calculated using the correlation function [θ), where R is the correlation matrix:

[0115]

[0116] To minimize the root mean square error of the Kriging model's prediction results, the mean prediction error c needs to be minimized while satisfying the constraints. T Fg T Find the optimal coefficient matrix c when (x) = 0. This can be achieved using the Lagrange multiplier method.

[0117] L(c, λ) = σ 2 (1+c TRc-2c T r)-λ T [c T Fg T (x)]

[0118] Where λ is a Lagrange multiplier. Taking the partial derivative with respect to c and simplifying it to matrix form, we get:

[0119]

[0120] Solving the above equation yields:

[0121]

[0122]

[0123] The predicted response value can be obtained from the above formula.

[0124]

[0125] By generalized least squares, we get:

[0126]

[0127]

[0128] For a constant g(x), the above expression can be simplified to:

[0129]

[0130]

[0131] Here, 1 represents an n-dimensional unit vector.

[0132] The expression for the Kriging model prediction is:

[0133]

[0134] In order to calculate and σ 2 The θ value in matrix R can be calculated using maximum likelihood estimation:

[0135]

[0136] In the Kriging model, the mean can be... Variance σ 2The expression is converted into a function containing the variable θ, and a Kriging model is constructed by solving the optimization problem in the above equation. This invention uses the DACE toolbox provided by Matlab to solve for θ. This toolbox requires the upper and lower boundary values ​​of θ and the initial value, and calculates the value of θ through maximum likelihood estimation.

[0137] Subsequently, β and z(x) are obtained through θ, and G(x) is obtained, which is the initial Kriging interpolation curve obtained by the Kriging interpolation method.

[0138] Please see Figure 4 , Figure 4 The spatial correlation function R and distance d are related to different values ​​of the Kriging interpolation parameter θ. k The relationship diagram shows that the smaller the Kriging interpolation parameter θ, the stronger the correlation; conversely, the larger the parameter θ, the stronger the spatial correlation function R and the distance d. k The weaker the correlation.

[0139] S4: Compare the initial IDW interpolation curve with the initial Kriging interpolation curve to obtain potential extreme points greater than the difference threshold.

[0140] Specifically, the actual difference between the RCS values ​​of the same prediction point on the initial IDW interpolation curve and the initial Kriging interpolation curve is calculated sequentially. It is then determined whether the difference exceeds a preset threshold. If it does, the current prediction point is considered a potential extreme point; otherwise, the next prediction point is sequentially evaluated. This process is repeated to obtain all potential extreme points on the initial IDW interpolation curve and the initial Kriging interpolation curve. Here, "same prediction point" refers to the incident angle on the initial IDW interpolation curve and the initial Kriging interpolation curve. Points that are equal (angular domain sample points) or have the same frequency (frequency domain sample points).

[0141] For example, for angular domain sample points, both the initial IDW interpolation curve and the initial Kriging interpolation curve include a preset number of prediction points, and the x-coordinates of the prediction points on the two curves (corresponding to the incident angles) are... While the corresponding values ​​are the same, the ordinates (corresponding to RCS values) may differ. The actual difference between the ordinates of the predicted points at the same x-axis on the initial IDW interpolation curve and the initial Kriging interpolation curve is calculated sequentially. It is then determined whether this actual difference exceeds a preset difference threshold. If so, the current predicted point is considered a potential extreme point. This method is applied to all potential extreme points on the curve. Please refer to [link to relevant documentation]. Figure 5 , Figure 5 This is a flowchart illustrating the process of iteratively increasing sample points and correcting the surrogate model curve, as proposed in an embodiment of the present invention. Figure 5(a) It can be seen that in the first iteration, a large number of potential extreme points are obtained by comparing the maximum difference between the two interpolation methods.

[0142] S5: Re-simulate at potential extreme points to obtain new sample data to supplement the sample set, and preprocess the new sample data to obtain an updated sample set.

[0143] Specifically, at potential extreme points, numerical methods are used to re-simulate and obtain new sample data. That is, by using the input angle or frequency (value on the horizontal axis) at the potential extreme point, numerical methods are used to simulate and obtain the corresponding RCS value (value on the vertical axis), which is then added to the current sample set as new sample data. Subsequently, the new sample data undergoes data shifting and log function processing, the specific process of which is the same as step S2 and will not be repeated here. In this embodiment, Figure 5 The blue circled sample points represent the RCS values ​​corresponding to the incident angles of the potential extreme points obtained from the simulation. These newly added sample points will be added to the existing sample dataset for the next simulation.

[0144] It should be noted that before proceeding to the next step of interpolation curve fitting, the new sample data needs to be preprocessed, which is similar to the data shift and log function processing in step S2, and then used together with the original preprocessed sample set in the sample set for subsequent interpolation curve fitting.

[0145] S6: Re-obtain the IDW interpolation curve and Kriging interpolation curve based on the updated sample set.

[0146] Specifically, the current IDW interpolation curve and Kriging interpolation curve are recalculated based on the sample data in the updated sample set after data shifting and log function processing. In this embodiment, Figure 5 This is the four-iteration process of the adaptive sample augmentation method based on IDW-Kriging of the present invention. The gray curve is the result obtained by Kriging interpolation, and the red curve is the result obtained by IDW interpolation.

[0147] S7: Determine if the current iteration count has reached the maximum iteration count. If not, proceed to step S4. If yes, save the Kriging interpolation curve as the enhanced sample set.

[0148] Specifically, determine whether the current iteration count has reached the preset maximum iteration count. If not, proceed to step S4, that is, continue to search for potential extreme points on the current IDW interpolation curve and the current Kriging interpolation curve. If so, obtain the current Kriging interpolation curve.

[0149] Since the RCS values ​​of the sample data in the sample set were shifted and processed using the log function before the interpolation, this step restores the RCS values ​​of the sample data. Specifically, the sample data on the current Kriging interpolation curve is transformed using the following formula:

[0150]

[0151] Where σ′(after) is the RCS value of the interpolation point on the current Kriging interpolation curve, and σ(after) is the RCS value of the interpolation point after restoration on the current Kriging interpolation curve, the transformed sample data has the same format as the original sample data, i.e., the incident angle. The data is in the format of value or frequency-RCS value, and all the converted sample data together form the final surrogate model.

[0152] The following simulation experiments further illustrate the effectiveness of the adaptive sample augmentation method based on IDW-Kriging of this invention.

[0153] This example provides a construction instance of an aircraft corner domain proxy model, which contains a large number of drastically changing peak points. Please refer to... Figure 6 and Figure 7 , Figure 6 The corner domain proxy model is obtained by using the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging of this invention. It can be seen that the number of spikes in the corner domain proxy model after four iterations is significantly less than the number of spikes before the iteration, and the proxy model effect is significantly improved. Figure 7 This diagram illustrates the error analysis of the angular domain proxy model obtained during the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging proposed in this invention. The method proposed in this invention accurately reconstructs the target's RCS as a function of the incident angle by adaptively iterating to find potential peak points. Please refer to [link to relevant documentation]. Figure 8 , Figure 8 The scatter plots fitted using the IDW-Kriging-based adaptive sample augmentation method of this invention in the first and fourth iterations show a significant improvement in the fitted scatter plot performance after iterations. In the first iteration, due to the limited number of sample points, many predicted values ​​from the surrogate model deviated considerably from the true values. By adaptively sampling to locate potential RCS narrow peaks, the fitted scatter plot performance was significantly improved after four iterations, with the vast majority of RCS values ​​approaching or equaling the true values. This method effectively reduces the amount of data in the surrogate model that deviates from the actual simulation data and is suitable for constructing surrogate models for objects with drastic numerical changes.

[0154] This invention also provides an example of constructing a frequency domain proxy model, in which the curve changes more smoothly. Please see [link to example]. Figure 9 , Figure 9 This paper compares the frequency domain surrogate models obtained in the first and fourth iterations of the adaptive sample augmentation method based on IDW-Kriging proposed in this invention. The method proposed in this invention can correctly find narrow peaks and supplement sampling points, making the frequency domain surrogate model more accurate, and still showing significant improvement in the construction of relatively smooth surrogate models. The method of this invention is suitable for constructing surrogate models with smooth numerical changes.

[0155] This invention also provides an error comparison between the traditional Kriging method and the method of this invention in constructing drastically changing proxy models. Please see below. Figure 10 , Figure 10 This is a corner-domain error analysis diagram comparing the surrogate model constructed using the traditional Kriging method and the surrogate model constructed using the method of this invention. Figure 10 (a) is the error analysis diagram of the Kriging surrogate model obtained by uniformly sampling with the same amount of data. Figure 10 (b) is the error analysis diagram of the surrogate model obtained using the method of this invention. The uniform sampling method (Kriging method) can basically reconstruct the trend of the target's RCS with angle variation, but it performs poorly at locations with large extreme value changes. The method proposed in this invention better captures these small but important local features and more accurately reconstructs the target's RCS variation with the incident angle. The surrogate model obtained using the method of this invention has significantly fewer spikes than the uniform sampling method, and the maximum value of the spikes is smaller, meaning the error is smaller. It is evident that, with the same amount of data, the error of the surrogate model obtained using the method of this invention is significantly reduced.

[0156] Figure 11 These are scatter plots of corner-domain fitting for the surrogate models constructed using the traditional Kriging method and the method of this invention. Both are scatter plots of fitting 65341 data points with 44820 samples. Figure 11 (a) is the fitted scatter plot of the traditional uniform sampling method. Figure 11 (b) is the fitted scatter plot obtained by the IDW-Kriging adaptive method proposed in this invention. The adaptive sampling method of this invention can find the location of narrow peaks in the RCS data. As can be clearly seen in the fitted scatter plot, this method effectively reduces the amount of data deviating from the true simulation data values. With the same amount of data, the surrogate model constructed by this invention is far superior to the uniformly sampled Kriging surrogate model.

[0157] This invention provides an error comparison between the traditional Kriging method for constructing proxy models and the method of this invention for constructing gently changing proxy models. Please refer to [link to relevant documentation]. Figure 12, Figure 12 This is a comparison of the frequency domain error analysis of the surrogate models constructed using the traditional Kriging method and the method of this invention. It can be seen that, with the same amount of data, the method of this invention effectively identifies two narrow peaks, making the model more accurate. Ordinary uniform sampling methods, however, cannot capture these details.

[0158] Furthermore, Table 1 presents the root mean square error (RMSE) analysis of the surrogate models obtained by different methods in the angular domain. Here, sample points refer to the sample points before curve fitting, and the ratio of data volume before and after fitting refers to the ratio of the number of sample points before curve fitting to the number of predicted points after fitting. Under the initial uniformly sampled sample set, the RMSE of the obtained surrogate model is 1.1404. After four iterations using the IDW-Kriging-1 method (similar to the method of this invention but excluding the data preprocessing step S2), the RMSE of the obtained surrogate model is 0.2352, and the RMSE of the obtained surrogate model decreases by 79.66%. This indicates that the method of increasing sample points through adaptive iteration effectively improves the accuracy of the results. Further, after adding the data preprocessing step (IDW-Kriging-2, i.e., the method of this invention), the RMSE is 0.1685, and the RMSE decreases again by 28.36%.

[0159] Table 1. Corner Domain Error Analysis

[0160]

[0161] Table 2 presents the root mean square error (RMSE) analysis of the surrogate models obtained by different methods in the frequency domain. With the initial uniformly sampled sample set, the RMSE of the obtained surrogate model is 0.4560. After four iterations using the IDW-Kriging-1 method (similar to the method of this invention but excluding the data preprocessing step S2), the RMSE of the obtained surrogate model is 0.1275, representing a 72.04% decrease in RMSE. Furthermore, after adding the data preprocessing step (IDW-Kriging-2, i.e., the method of this invention), the RMSE is 0.951, and the RMSE decreases by another 25.41%, indicating that data preprocessing can also improve the accuracy of the surrogate model.

[0162] Table 2. Frequency Domain Error Analysis

[0163]

[0164] Table 3 shows the root mean square error (RMSE) analysis of the surrogate models obtained by the traditional Kriging method and the IDW-Kriging method in the angular domain with the same number of samples. Compared with the traditional Kriging uniform sampling, the RMSE of the surrogate model obtained by the IDW-Kriging method (IDW-Kriging-1, similar to the method of this invention but excluding the data preprocessing process in step S2) decreased by 70.09%. This indicates that the IDW-Kriging method in this paper effectively improves the accuracy of the results. Compared with the traditional Kriging interpolation method, it can find highly representative samples to avoid redundant simulations and reduce the computational cost required for simulation. Comparing Kriging-1, IDW-Kriging-1, and IDW-Kriging-2, it can be seen that the IDW-Kriging method of this invention uses less data to achieve a similar RMSE. Furthermore, after adding the data preprocessing process (IDW-Kriging-2, i.e., the method of this invention), the RMSE decreases again.

[0165] Table 1. Corner Domain Error Analysis

[0166]

[0167] Table 4 shows the root mean square error (RMSE) analysis of the surrogate models obtained by the traditional Kriging method and the IDW-Kriging method in the frequency domain with the same number of samples. The RMSE of the surrogate model obtained by the IDW-Kriging method (IDW-Kriging-1, similar to the method of this invention but without the data preprocessing process in step S2) decreased by 65.34% compared to the traditional uniform sampling (IDW-Kriging-1). Furthermore, after adding the data preprocessing process (IDW-Kriging-2, i.e., the method of this invention), the RMSE decreased again.

[0168] Table 4. Frequency Domain Error Analysis

[0169]

[0170] This invention, based on the IDW-Kriging adaptive sample augmentation method, compares the interpolation results of IDW and Kriging interpolation models. Based on a preset difference threshold, it automatically detects the extreme value positions of the Kriging interpolation surface and determines the locations of the sample points to be supplemented, solving the problem of uniform sampling ignoring peak points. Through simulation of the sample points at these locations, the sample set data points are expanded, interpolation is recalculated, and new IDW and Kriging interpolation results are obtained. Iterative adaptive searching for potential extreme points and sample augmentation significantly improves the prediction accuracy of the Kriging model. After reaching the maximum number of iterations, the Kriging interpolation model is saved as a surrogate model, greatly reducing the number of simulations and time required, while maintaining higher accuracy than the traditional Kriging model.

[0171] Another embodiment of the present invention provides a storage medium storing a computer program for executing the steps of the IDW-Kriging-based adaptive sample augmentation method described in the above embodiments. A further aspect of the present invention provides an electronic device including a memory and a processor. The memory stores a computer program, and the processor, when invoking the computer program in the memory, implements the steps of the IDW-Kriging-based adaptive sample augmentation method as described in the above embodiments. Specifically, the integrated modules implemented as software functional modules can be stored in a computer-readable storage medium. These software functional modules, stored in a storage medium, include several instructions to cause an electronic device (which may be a personal computer, server, or network device, etc.) or processor to execute some steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as a USB flash drive, a portable hard drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk, or an optical disk.

[0172] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. An adaptive sample augmentation method based on IDW-Kriging, characterized in that, include: S1: Configure initial parameters, including kriging parameters. IDW interpolation smoothing parameters Sampling interval, difference threshold and maximum number of iterations for IDW interpolation and Kriging interpolation methods; S2: Preprocess the sample data in the original sample set to reduce the degree of variation in the sample data and obtain the preprocessed sample set; S3: Obtain the initial IDW interpolation curve and the initial Kriging interpolation curve based on the preprocessed sample set; S4: Compare the initial IDW interpolation curve with the initial Kriging interpolation curve to obtain potential extreme points greater than the difference threshold; S5: Re-simulate at the potential extreme point to obtain new sample data to supplement the sample set, and preprocess the new sample data to obtain an updated sample set; S6: Obtain the IDW interpolation curve and Kriging interpolation curve again based on the updated sample set; S7: Determine whether the maximum number of iterations has been reached. If not, proceed to step S4. If yes, save the Kriging interpolation curve as the enhanced sample set. S2 includes: S2.1: Obtain multiple raw sample data points using actual measurement or simulation methods to form a raw sample set. The raw sample data in the raw sample set includes angular domain sample points or frequency domain sample points, wherein the input data for the angular domain sample points is the incident angle. The output data is the RCS value. The input data of the frequency domain sample points is the frequency, and the output data is the RCS value. S2.2: Perform data translation on the RCS values ​​of the angular domain sample points or frequency domain sample points in the original sample data to define new RCS values. This makes the RCS value of each angular domain sample point or frequency domain sample point after data translation... All are within the range ,in, The RCS value of the angular domain sample point or frequency domain sample point in the original sample data; S2.3: Use the log function to... The data transformation is performed using the following formula: ， in, This represents the RCS value of each angular domain sample point or frequency domain sample point after data transformation, i.e., the RCS value after preprocessing.

2. The adaptive sample augmentation method based on IDW-Kriging according to claim 1, characterized in that, S3 includes: S3.1: Based on the preprocessed sample set, the RCS values ​​of multiple prediction points are obtained using the IDW interpolation method, and an IDW interpolation curve is formed based on the multiple prediction points. The preprocessed sample set includes preprocessed angular domain sample points or preprocessed frequency domain sample points, wherein the preprocessed angular domain sample points include the incident angle. and the corresponding preprocessed RCS value; the preprocessed frequency domain sample points include the frequency and the corresponding preprocessed RCS value; S3.2: Based on the preprocessed sample set, obtain the initial Kriging interpolation curve using the Kriging interpolation method.

3. The adaptive sample augmentation method based on IDW-Kriging according to claim 2, characterized in that, S3.1 includes: Based on the preprocessed sample set, the RCS values ​​of multiple prediction points are obtained using the IDW interpolation method. The calculation formula is as follows: in, Z This indicates the predicted point at the preset angle. Indicates the prediction point Z Compared with known sample points in the preprocessed sample set i The distance between them The parameters are set. Indicates the prediction point Z With known sample points i Distance weight values ​​between N This indicates the number of known sample points in the preprocessed sample set. RCS i For known sample points i RCS value, Indicates the prediction point Z The predicted value of RCS; An IDW interpolation curve is formed based on the multiple prediction points.

4. The adaptive sample augmentation method based on IDW-Kriging according to claim 1, characterized in that, S4 includes: The actual difference between the RCS values ​​of the same prediction point on the initial IDW interpolation curve and the initial Kriging interpolation curve is calculated sequentially. It is then determined whether the difference is greater than a preset threshold. If so, the current prediction point is considered a potential extreme point, and all potential extreme points on the initial IDW interpolation curve and the initial Kriging interpolation curve are obtained.

5. The adaptive sample augmentation method based on IDW-Kriging according to claim 2, characterized in that, S5 includes: At the potential extreme point, a numerical method is used to re-simulate, and the corresponding RCS value is obtained by simulating the input angle or frequency at the potential extreme point, thereby forming new sample data and supplementing it to the current sample set; The new sample data is then subjected to data shifting and log function processing.

6. The adaptive sample augmentation method based on IDW-Kriging according to claim 1, characterized in that, S7 includes: Determine if the current iteration count has reached the maximum iteration count. If not, proceed to step S4 to continue searching for potential extreme points on the current IDW interpolation curve and the current Kriging interpolation curve. If yes, obtain the current Kriging interpolation curve. The sample data on the current Kriging interpolation curve is transformed using the following formula: ， in, This represents the RCS value of the interpolation point on the current Kriging interpolation curve. The RCS value of the current Kriging interpolation curve after the interpolation point is restored.

7. A storage medium storing a computer program for performing the steps of the IDW-Kriging-based adaptive sample augmentation method as described in any one of claims 1 to 6.

8. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, and the processor, when invoking the computer program in the memory, implements the steps of the adaptive sample augmentation method based on IDW-Kriging as described in any one of claims 1 to 6.

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