Radar performance equivalent calculation method under non-iid test conditions based on physical constraints

Abstract

The application provides a radar performance equivalent calculation method based on physical constraints under non-identically distributed test conditions, and belongs to the technical field of radar performance evaluation. The method comprises the following steps: obtaining radar data, calculating the radar action distance according to the radar data; standardizing the radar data and the radar action distance; constructing a PC-BPNN network model; constructing a loss function of the PC-BPNN, and adding a gradient monotonicity physical regularization constraint to the loss function to punish the gradient in the model output that violates the change rule of the radar action distance; based on the loss function of the PC-BPNN, using a training set to train the PC-BPNN network model to obtain a trained PC-BPNN network model; and based on the trained PC-BPNN network model, using a test set that is not independent and identically distributed with the training sample to perform equivalent calculation on the radar performance. Through the physical constraint and the gradient monotonicity regularization, the radar detection performance calculation precision and the physical consistency under the non-independent and identically distributed test conditions are improved.

Description

Technical Field

[0001] This invention belongs to the field of radar performance evaluation technology, and in particular relates to a method for equivalent calculation of radar performance under non-uniformly distributed test conditions based on physical constraints. Background Technology

[0002] Radar detection performance evaluation typically relies on field tests, simulation tests, or semi-physical tests to obtain performance data, and then analyzes indicators such as range, detection probability, false alarm probability, and measurement accuracy. Field tests can realistically reflect the operating state of a radar system and are an important basis for equipment performance evaluation. However, actual field tests often face problems such as complex test conditions, huge sample space, scarce test targets, high test costs, and long cycles. Due to limitations in test resources and safety conditions, field tests can usually only cover a limited number of target types, environmental conditions, and parameter combinations, making it difficult to fully reflect the performance variation patterns of the radar system under all operating conditions. Therefore, based on existing test or simulation data, equivalent extrapolation of radar detection performance under conditions where no tests have been conducted is an important means to overcome the limitations of field tests, reduce test costs, and improve evaluation coverage.

[0003] Existing methods for estimating radar detection performance mainly include analytical estimation methods based on physical mechanisms and data-driven machine learning methods.

[0004] Analytical estimation methods based on physical mechanisms are among the earliest methods used in radar performance evaluation. These methods are grounded in radar range equations, electromagnetic wave propagation theory, target scattering characteristics, and signal detection theory. By establishing explicit relationships between performance indicators and influencing factors, they achieve performance conversion under different experimental conditions. Their advantages include clear physical meaning, a clear derivation process, and strong interpretability of results, making them suitable for performance estimation tasks with well-defined parameters and relatively ideal scenarios. However, actual radar detection processes are influenced by multiple factors, including the radar system, environmental propagation, target characteristics, and signal processing mechanisms. Complex nonlinear coupling relationships also exist between these factors, making it difficult to accurately establish explicit relationships between parameters and detection performance. Furthermore, some parameters are difficult to obtain precisely. When there are deviations between the actual scenario and the analytical assumptions, analytical estimation methods are prone to large errors, making it difficult to meet the high-precision performance evaluation requirements under complex engineering conditions.

[0005] In recent years, the development of machine learning technology has provided new ideas for radar detection performance estimation. Data-driven radar performance estimation methods transform radar detection performance evaluation into a classification or regression problem in machine learning. By learning the nonlinear mapping relationship between input parameters and performance indicators through existing experimental or simulation data, performance prediction under unknown conditions can be achieved. This method avoids the dependence on fully explicit modeling of the physical process, and its prediction accuracy is superior to traditional analytical estimation methods. However, this method has the following problems: data-driven methods heavily rely on the quality and quantity of training data. Considering the cost of radar target detection, it is difficult to establish large-scale experimental datasets for radar detection performance. Furthermore, in real field tests and equipment evaluations, training samples often only cover a limited range of parameters, while the samples to be estimated may come from untested ranges or unknown scenarios, leading to non-independent and identically distributed conditions between training and test samples. Conventional machine learning models are prone to problems such as decreased prediction accuracy, extrapolation results that do not conform to physical laws, and insufficient interpretability.

[0006] In summary, existing analytical estimation methods and data-driven methods cannot meet the requirements for radar detection performance estimation under the condition that the test samples are not independent and identically distributed. Summary of the Invention

[0007] The purpose of this invention is to provide an equivalent estimation method for radar performance under non-identical distribution test conditions based on physical constraints. By adding physical prior features calculated by analytical formulas to the features and adding gradient monotonicity regularization to the loss function, the physical consistency and estimation accuracy of the estimation are improved. This solves the technical problems in the existing analytical estimation methods, such as the inconsistency between idealized assumptions and actual scenarios, decreased prediction accuracy under non-independent and identically distributed conditions of training and test samples, extrapolation results that do not conform to physical laws, and insufficient interpretability.

[0008] To solve the above-mentioned technical problems, the specific technical solution of the present invention is as follows:

[0009] A method for equivalent estimation of radar performance under non-uniformly distributed test conditions based on physical constraints, the method comprising the following steps:

[0010] Step S1: Acquire radar data and calculate the radar range based on the radar data; standardize the radar data and radar range to obtain standardized radar features and standardized radar range; use the standardized radar range as a physical prior feature and together with the standardized radar features to construct the training set, validation set, and test set;

[0011] Step S2: Construct a PC-BPNN network model, which includes an input layer, a hidden layer, and an output layer; add radar detection performance index features to the input features, that is, calculate the standardized radar range as a physical prior feature, fuse it with the standardized radar features to obtain an 11-dimensional fusion feature, and input the fusion feature as the input feature into the PC-BPNN network model so that the model can obtain physical trend information guidance during the data fitting process;

[0012] Step S3: Construct the loss function of PC-BPNN, and add gradient monotonicity physical regularization constraint to the loss function to penalize gradients in the model output that violate the radar range variation law;

[0013] Step S4: Based on the loss function of PC-BPNN, train the PC-BPNN network model using the training set to obtain the trained PC-BPNN network model.

[0014] Step S5: Based on the trained PC-BPNN network model, use a test set that is not independent and identically distributed with the training samples to perform an equivalent calculation of the radar performance.

[0015] Furthermore, the formula for calculating radar range is as follows:

[0016]

[0017] in, Indicates the radar's effective range; This represents the peak power of the radar transmitter. Antenna gain; For radar operating wavelength, The speed of light in a vacuum. The operating frequency of the radar; For target RCS; J / K is the Boltzmann constant; Standard room temperature is typically 290K. This is the receiver's equivalent noise bandwidth; The receiver noise figure; This represents the total loss of the radar system. The minimum output signal-to-noise ratio required to satisfy a given detection probability and false alarm probability.

[0018] Furthermore, standardization is achieved through the following formula:

[0019]

[0020] in, Indicates the first Radar characteristics after standardization of radar data; Indicates the first Radar data; Indicates the first The mean of the radar dataset; Indicates the first The variance of the radar dataset.

[0021] Furthermore, the loss function of PC-BPNN is expressed as follows:

[0022]

[0023] in, Represents the loss function of PC-BPNN; This represents the loss in the basic mean squared error fitting. Represents the physical monotonicity regularity term; This represents the physical constraint weight coefficient.

[0024] Furthermore, the basic mean squared error fitting loss adopts the mean squared error loss to ensure the model's fitting accuracy to the training data, as shown below:

[0025]

[0026] in, Indicates the number of samples; This represents the true value, referring to the calculated standardized radar range. This represents the predicted value, which is... Indicates the first The predicted range of radar for each sample.

[0027] Furthermore, the physical monotonicity regularization term is expressed as follows:

[0028]

[0029] in, Indicates the dimension of the input features; Indicates the number of samples; Indicates the first Monotonicity constraint symbol for dimensional features; Indicates the first The first sample 3D input features; Indicates the first Predicted radar range for each sample; Indicates model output pairs The gradient.

[0030] Further, step S4 includes the following steps:

[0031] Step S41: Initialize the PC-BPNN network structure, learning rate, number of iterations, and regularization parameters;

[0032] Step S42: Randomly initialize the weight matrix and bias vector of each layer of the network, and set the convergence threshold and the maximum number of iterations;

[0033] Step S43: Forward Propagation: Input the input features of the training set into the network, calculate the output of each neuron in the network layer according to the calculation formula, and finally obtain the model prediction value. ;

[0034] Step S44: Loss Calculation: Calculate the basic mean squared error fitting loss and the physical monotonicity regularization term to obtain the model training loss;

[0035] Step S45: Backpropagation: Based on the chain rule, calculate the gradient of the loss function with respect to the weights and biases of each layer from the output layer to the input layer.

[0036] Step S46: Parameter Update: Use the gradient descent optimizer to update the weights and biases of each layer along the negative gradient direction of the loss function;

[0037] Step S47: Determine if the stopping condition is met: The stopping condition is that the loss value is lower than the convergence threshold, or the maximum number of iterations is reached; if not met, return to step 3 to continue iterating;

[0038] Step S48: The iteration ends, save the trained network weights and standardized parameters, and output the trained PC-BPNN network model.

[0039] Compared with the prior art, the present invention has the following beneficial technical effects:

[0040] 1) This invention proposes a radar detection performance estimation method based on a data-driven model constrained by physical information. It fully combines the high precision of data-driven methods with the strong extrapolation ability of physical information, improving the estimation accuracy, physical consistency, and interpretability of the estimation model. To further improve prediction accuracy, a data-driven model constrained by physical information is introduced into the radar detection performance estimation task. The Physical-Constrained Back Propagation Neural Network (PC-BPNN) adds radar analytical formulas to the input features to estimate the radar range as physical prior features, providing physical supervision information gain to the model and effectively improving its extrapolation performance. Simultaneously, gradient monotonicity regularization between each input feature and the output is added to the loss function, penalizing gradients that violate physical laws. This ensures that the model maintains a mapping relationship between input and output that conforms to radar physical principles across the entire feature space, further strengthening the model's ability to learn core physical laws. Compared to conventional data-driven methods, the PC-BPNN model significantly improves prediction performance, especially under the condition that the training and test samples are not independently and identically distributed, with a significantly improved coefficient of determination (R-squared, R...). 2 The performance improved by 20%, and the root mean square error (RMSE) decreased by 40%.

[0041] 2) This invention, through a radar detection performance estimation method based on the PC-BPNN model, avoids establishing an explicit relationship between detection performance indicators and parameters, transforming radar detection performance evaluation into a regression problem in machine learning. By adding radar analytical formulas to the input features to estimate the radar range as a physical prior feature, it provides physical supervision information gain to the model, effectively improving the model's extrapolation performance. At the same time, by adding gradient monotonicity regularization of each input feature and output to the loss function, it penalizes gradients that violate physical laws, ensuring that the model maintains a mapping relationship between input and output that conforms to radar physical principles throughout the entire feature space, further strengthening the model's ability to learn core physical laws.

[0042] 3) This invention adopts the PC-BPNN model, a data-driven method based on physical information constraints, which combines the strong extrapolation of analytical estimation methods with the high precision of data-driven methods. It also incorporates gradient monotonicity regularization to improve the estimation accuracy and physical consistency of radar detection performance under non-independent and identically distributed test conditions. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0044] Figure 1 This is a schematic diagram of the radar performance equivalent estimation method based on physical constraints under non-uniformly distributed test conditions according to the present invention.

[0045] Figure 2 This is a schematic diagram of the PC-BPNN network model architecture of the present invention.

[0046] Figure 3 This is a schematic diagram showing the comparison between the actual and predicted values ​​of the PC-BPNN network model of this invention.

[0047] Figure 4 This is a scatter plot showing the comparison between the actual and predicted values ​​of the PC-BPNN network model of this invention. Detailed Implementation

[0048] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] Radar detection performance is the core combat capability that a radar system comprehensively demonstrates in the entire process of target detection, tracking and identification. Its evaluation index system mainly includes multiple dimensions such as detection capability, measurement accuracy and resolution capability, which directly determine the actual working efficiency and engineering application value of the radar system.

[0050] The core evaluation metrics for radar detection performance mainly include maximum radar range (Rmax), signal-to-noise ratio (SNR), and detection probability (P). d ), False Alarm Probability (P) faThe core evaluation indicators for radar detection performance include measurement accuracy and radar resolution. Among these, radar range comprehensively reflects the combined effects of factors such as transmit power, antenna gain, target RCS, propagation loss, receiver noise, and detection threshold, and is the most direct engineering representation of radar detection capability. This invention selects the maximum radar range as the core output indicator for data-driven equivalent calculation.

[0051] This invention proposes an equivalent estimation method for radar performance under non-uniformly distributed test conditions based on physical constraints, such as... Figure 1 As shown, the method includes the following steps:

[0052] Step S1: Acquire radar data and calculate the radar range based on the radar data; standardize the radar data and radar range to obtain standardized radar features and standardized radar range; use the standardized radar range as a physical prior feature and together with the standardized radar features to construct the training set, validation set, and test set.

[0053] Specifically, the radar data includes the transmitter's peak power. Antenna gain Operating frequency Noise figure Receiver bandwidth Mean RCS (Radar Cross Section), RCS fluctuation type, radial velocity Meteorological clutter type, clutter intensity; a total of 10 dimensions.

[0054] The formula for calculating radar range is as follows:

[0055]

[0056] in, Indicates the radar's effective range; This represents the peak power of the radar transmitter. Antenna gain; For radar operating wavelength, The speed of light in a vacuum. The operating frequency of the radar; For target RCS; J / K is the Boltzmann constant; Standard room temperature is typically 290K. This is the receiver's equivalent noise bandwidth; The receiver noise figure; This represents the total loss of the radar system. The minimum output signal-to-noise ratio required to satisfy a given detection probability and false alarm probability.

[0057] The dataset contains significantly different numerical features with varying dimensions and ranges. Directly inputting these features into the model can lead to large-value features dominating training and neglecting the physical impact of smaller-value features. To address this issue, this invention employs Z-score normalization to standardize radar data and radar range, converting them into a standard normal distribution with a mean of 0 and a standard deviation of 1. The standardization process is achieved using the following formula:

[0058]

[0059] in, Indicates the first Radar characteristics after standardization of radar data; Indicates the first Radar data; Indicates the first The mean of the radar dataset; Indicates the first The variance of the radar dataset.

[0060] After standardization processing, the radar data yields 10 dimensions of standardized radar features, which, together with the standardized radar range, are used as input features for subsequent models.

[0061] A dataset was constructed using standardized radar features and standardized radar ranges. The training set employed typical discrete fixed parameter combinations from radar engineering, simulating fixed operating conditions achievable in field experiments. A random sampling method was used to generate 500 valid training samples, providing the model with engineering-representative training data. The parameter value ranges of the radar data before standardization are shown in Table 1.

[0062] Table 1. Parameter table of radar data before training set standardization

[0063]

[0064] The validation set maintains discrete values ​​for the core parameters of the radar system. Target and environmental parameters are continuously and randomly sampled within the training set range, generating 50 valid validation samples. These samples are used for model hyperparameter tuning and fitting accuracy verification under similarly distributed scenarios, avoiding model overfitting. The parameter value ranges of the radar data before validation set standardization are shown in Table 2.

[0065] Table 2. Parameter table of radar data before validation set standardization

[0066]

[0067] All parameters in the test set exceed the parameters of the training set, representing a completely non-uniformly distributed extrapolation scenario. A total of 50 valid test samples were generated for pure extrapolation performance testing, verifying the model's generalization ability under extreme conditions not covered by training. The parameter value ranges of the radar data before standardization are shown in Table 3.

[0068] Table 3. Parameters of radar data before test set standardization.

[0069]

[0070] Step S2: Construct a PC-BPNN network model, which includes an input layer, hidden layers, and an output layer. Input features enter the network from the input layer, undergo linear transformation and nonlinear activation function mapping in the hidden layer, and finally obtain the predicted radar range from the output layer.

[0071] like Figure 2 As shown, the PC-BPNN network model is a radar detection performance inference model that integrates physical information and data-driven approaches. Its goal is to add radar detection performance indicators to the input features of the BPNN, specifically calculating the standardized radar range as a physical prior feature and fusing it with the standardized radar features to obtain an 11-dimensional fused feature. This fused feature is then input into the PC-BPNN network model, allowing the model to be guided by physical trend information during data fitting. At the loss function optimization level, a gradient monotonicity physical regularization constraint is added to penalize gradients in the model output that violate the radar range variation law, thus enhancing the physical consistency of the extrapolated predictions while maintaining data fitting ability. Its performance on non-independent and identically distributed test sets outperforms analytical inference methods and conventional data-driven inference methods.

[0072] Step S3: Construct the loss function of PC-BPNN, and add gradient monotonicity physical regularization constraints to the loss function.

[0073] Specifically, the loss function of PC-BPNN is expressed as follows:

[0074]

[0075] in, Represents the loss function of PC-BPNN; This represents the loss in the basic mean squared error fitting. Represents the physical monotonicity regularity term; These represent the physical constraint weighting coefficients, used to balance fitting accuracy and physical constraint strength. .

[0076] The basic mean squared error fitting loss uses the mean squared error loss to ensure the model's fitting accuracy to the training data, as shown below:

[0077]

[0078] in, Indicates the number of samples; This represents the true value, referring to the calculated standardized radar range. This represents the predicted value, which is... Indicates the first The predicted range of radar for each sample.

[0079] The input features exhibit a monotonic relationship with the radar's effective range, serving as the core theoretical basis for physical constraints. Specific constraint rules are shown in Table 4.

[0080] Table 4 Monotonicity Physical Constraint Rules

[0081]

[0082] Based on TensorFlow's automatic differentiation mechanism, this invention calculates the gradient of the input features with respect to the output radar range in real time during training, and penalizes gradients that violate monotonicity constraints. The physical monotonicity regularization term is expressed as follows:

[0083]

[0084] in, Indicates the dimension of the input features; Indicates the number of samples; Indicates the first Monotonicity constraint symbol for dimensional features; Indicates the first The first sample 3D input features; Indicates the first Predicted radar range for each sample; Indicates model output pairs The gradient.

[0085] By incorporating gradient monotonicity regularization for each input feature and output into the loss function, penalties are imposed on gradients that violate physical laws. This ensures that the model maintains a mapping relationship between input and output that conforms to the physical principles of radar throughout the entire feature space, further enhancing the model's ability to learn core physical laws.

[0086] Step S4: Based on the loss function of PC-BPNN, train the PC-BPNN network model using the training set to obtain the trained PC-BPNN network model.

[0087] Step S41: Initialize the PC-BPNN network structure (number of layers, number of neurons), learning rate, number of iterations, regularization parameters, and other hyperparameters.

[0088] In this embodiment, there are 3 hidden layers, with 48 neurons in each layer. The training parameters of the PC-BPNN network model are shown in Table 5.

[0089] Table 5 Training parameters for the PC-BPNN network model

[0090]

[0091] Step S42: Randomly initialize the weight matrix and bias vector of each layer of the network, and set the convergence threshold and the maximum number of iterations.

[0092] Step S43: Forward Propagation: Input the input features of the training set into the network, calculate the output of each neuron in the network layer according to the calculation formula, and finally obtain the model prediction value. .

[0093] Specifically, no. The formula for calculating the output of neurons in a layer is:

[0094]

[0095] in, Indicates the first The neuron output of the layer, For the first The weight matrix of the layer, For bias vectors, The hidden layer uses the ReLU activation function, and the output layer uses the linear activation function to adapt to the regression task.

[0096] Step S44: Loss Calculation: Calculate the basic mean squared error fitting loss and the physical monotonicity regularization term to obtain the model training loss;

[0097] Step S45: Backpropagation: Based on the chain rule, calculate the gradient of the loss function with respect to the weights and biases of each layer from the output layer to the input layer.

[0098] Step S46: Parameter Update: Use the gradient descent optimizer to update the weights and biases of each layer along the negative gradient direction of the loss function;

[0099] Step S47: Determine if the stopping condition is met: the loss value is lower than the convergence threshold, or the maximum number of iterations has been reached; if not, return to step 3 to continue iterating.

[0100] Step S48: The iteration ends, save the trained network weights and standardized parameters, and output the trained PC-BPNN network model.

[0101] Step S5: Based on the trained PC-BPNN network model, use a test set that is not independent and identically distributed with the training samples to perform an equivalent calculation of the radar performance.

[0102] To comprehensively and objectively quantify the performance of different inference models, the coefficient of determination (R-squared, R0) was selected from three dimensions: fitting accuracy, explanatory power, and error level. 2 The four core evaluation indicators—Explained Variance (EV), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE)—are analyzed, and the results are shown in Table 6.

[0103] Table 6. Prediction Performance of the PC-BPNN Model (Test Set)

[0104]

[0105] Comparison of actual and predicted values Figure 3 As shown, the pattern between predicted values ​​(orange) and actual values ​​(blue) conforms to real physical laws, indicating that the PC-BPNN model has good fitting ability. The scatter plots of the actual and predicted values ​​are shown below. Figure 4 As shown, most of the blue sample points are closely distributed around the red ideal fitting line. As the true value increases, the predicted value increases synchronously, and the predicted value is significantly positively correlated with the true value. The overall prediction trend of the model is correct.

[0106] Table 7 shows a comparison of the test set with analytical inference methods and conventional data-driven methods:

[0107] Table 7 Comparison of Model Extrapolation Generalization Performance

[0108]

[0109] Experimental results show that the method of this invention has higher extrapolation accuracy than analytical extrapolation methods and conventional data-driven methods, combining the advantages of both, and significantly improving all evaluation indicators. The physical prior feature fusion scheme introduces the radar range calculated by the radar equation as a prior feature, providing the model with physical supervision information gain. Based on the physical prior, monotonic regularization penalizes gradients that violate physical laws, ensuring that the model maintains a mapping relationship between input and output that conforms to radar physical principles throughout the entire feature space. This further enhances the model's ability to learn core physical laws, maintaining stable extrapolation accuracy and physical consistency even in extrapolation scenarios.

[0110] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for equivalent estimation of radar performance under non-uniformly distributed test conditions based on physical constraints, characterized in that, The method includes the following steps: Step S1: Acquire radar data and calculate the radar range based on the radar data; standardize the radar data and radar range to obtain standardized radar features and standardized radar range; use the standardized radar range as a physical prior feature and together with the standardized radar features to construct the training set, validation set, and test set; Step S2: Construct a PC-BPNN network model, which includes an input layer, a hidden layer, and an output layer; add radar detection performance index features to the input features, that is, calculate the standardized radar range as a physical prior feature, fuse it with the standardized radar features to obtain an 11-dimensional fusion feature, and input the fusion feature as the input feature into the PC-BPNN network model so that the model can obtain physical trend information guidance during the data fitting process; Step S3: Construct the loss function of PC-BPNN, and add gradient monotonicity physical regularization constraint to the loss function to penalize gradients in the model output that violate the radar range variation law; Step S4: Based on the loss function of PC-BPNN, train the PC-BPNN network model using the training set to obtain the trained PC-BPNN network model. Step S5: Based on the trained PC-BPNN network model, use a test set that is not independent and identically distributed with the training samples to perform an equivalent calculation of the radar performance.

2. The radar performance equivalent estimation method based on physical constraints under non-uniformly distributed test conditions according to claim 1, characterized in that, The formula for calculating radar range is as follows: in, Indicates the radar's effective range; This represents the peak power of the radar transmitter. Antenna gain; For radar operating wavelength, The speed of light in a vacuum. The operating frequency of the radar; For target RCS; J / K is the Boltzmann constant; Standard room temperature is typically 290K. This is the receiver's equivalent noise bandwidth; The receiver noise figure; This represents the total loss of the radar system. The minimum output signal-to-noise ratio required to satisfy a given detection probability and false alarm probability.

3. The radar performance equivalent estimation method based on physically constrained non-uniformly distributed test conditions according to claim 1, characterized in that, Standardization is achieved using the following formula: in, Indicates the first Radar characteristics after standardization of radar data; Indicates the first Radar data; Indicates the first The mean of the radar dataset; Indicates the first The variance of the radar data set.

4. The radar performance equivalent estimation method based on physical constraints under non-uniformly distributed test conditions according to claim 1, characterized in that, The loss function of PC-BPNN is expressed as follows: in, Represents the loss function of PC-BPNN; This represents the fitting loss based on the mean squared error. Represents the physical monotonicity regularity term; This represents the physical constraint weight coefficient.

5. The radar performance equivalent estimation method based on physically constrained non-uniformly distributed test conditions according to claim 4, characterized in that, The basic mean squared error fitting loss uses the mean squared error loss to ensure the model's fitting accuracy to the training data, as shown below: in, Indicates the number of samples; This represents the true value, referring to the calculated standardized radar range. This represents the predicted value, which is... Indicates the first The predicted range of radar for each sample.

6. The radar performance equivalent estimation method based on physical constraints under non-uniformly distributed test conditions according to claim 4, characterized in that, The physical monotonicity regularization term is expressed as follows: in, Indicates the dimension of the input features; Indicates the number of samples; Indicates the first Monotonicity constraint symbol for dimensional features; Indicates the first The first sample 3D input features; Indicates the first Predicted radar range for each sample; Indicates model output pairs The gradient.

7. The radar performance equivalent estimation method based on physically constrained non-uniformly distributed test conditions according to claim 1, characterized in that, Step S4 includes the following steps: Step S41: Initialize the PC-BPNN network structure, learning rate, number of iterations, and regularization parameters; Step S42: Randomly initialize the weight matrix and bias vector of each layer of the network, and set the convergence threshold and the maximum number of iterations; Step S43: Forward Propagation: Input the input features of the training set into the network, calculate the output of each neuron in the network layer according to the calculation formula, and finally obtain the model prediction value. ; Step S44: Loss Calculation: Calculate the basic mean squared error fitting loss and the physical monotonicity regularization term to obtain the model training loss; Step S45: Backpropagation: Based on the chain rule, calculate the gradient of the loss function with respect to the weights and biases of each layer from the output layer to the input layer. Step S46: Parameter Update: Use the gradient descent optimizer to update the weights and biases of each layer along the negative gradient direction of the loss function; Step S47: Determine if the stopping condition is met: The stopping condition is that the loss value is lower than the convergence threshold, or the maximum number of iterations is reached; if not met, return to step 3 to continue iterating; Step S48: The iteration ends, save the trained network weights and standardized parameters, and output the trained PC-BPNN network model.

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