A three-dimensional electromagnetic target modeling method and device based on a graph neural network

Abstract

The application discloses a three-dimensional electromagnetic target modeling method and device based on a graph neural network, and the method comprises the following steps: discretizing a three-dimensional electromagnetic target by using a non-uniform grid, discretely representing unknown physical quantities to be solved of the three-dimensional electromagnetic target based on the non-uniform grid, and converting physical equations satisfied by the three-dimensional electromagnetic target into a linear equation set; on the basis, the three-dimensional electromagnetic target discretized by the non-uniform grid is described by using a graph data structure, a graph neural network is trained to learn an iterative solving process of the linear equation set, and in each iteration step, an iterative update amount of a candidate solution is predicted according to a residual error of the linear equation set, so that fast and accurate three-dimensional electromagnetic target modeling is realized.

Description

Technical Field

[0001] This invention relates to the field of wireless communication technology, and in particular to a method and apparatus for modeling three-dimensional electromagnetic targets based on graph neural networks. Background Technology

[0002] Electromagnetic modeling is a crucial tool for electromagnetic field analysis, enabling efficient and accurate solution of Maxwell's equations. It has been widely applied in electromagnetic engineering and other scientific and technological fields. In computational electromagnetics, a variety of numerical methods have been proposed and studied, forming the core of electromagnetic modeling. These methods include the finite difference method, the finite element method, and the method of moments. These methods discretize Maxwell's equations with certain boundary conditions and transform them into a set of linear equations for solution. However, in practical applications, the transformed linear equations often contain a large number of unknowns, especially in three-dimensional electromagnetic modeling. This results in very high computational costs and memory consumption for solving the equations. Therefore, real-time and accurate electromagnetic modeling remains a long-standing challenge.

[0003] To reduce the computational complexity of electromagnetic modeling, fast algorithms have seen rapid development, such as adaptive integration, conjugate gradient fast Fourier transform, and multilevel fast multipole algorithms. The development of fast algorithms effectively reduces computation time and memory consumption, further expanding the capability for complex, large-scale electromagnetic modeling. In addition, two-step methods are also commonly used acceleration techniques, which accelerate electromagnetic modeling through offline and online computation. Two-step methods pre-compute and offline store computationally intensive parts, while sacrificing computational memory to speed up online computation; examples include basis subtraction methods.

[0004] Artificial intelligence (AI) technology is also being used to accelerate electromagnetic modeling, and current applications can be divided into two categories: fully data-driven and physics-inspired approaches. The fully data-driven approach involves building and training deep neural network models to abstract and learn physical laws from massive amounts of physical data, achieving accurate simulation of physical phenomena. The physics-inspired approach leverages the mathematical similarity between deep learning technology and traditional algorithms to construct highly interpretable deep neural network models for rapid electromagnetic calculations. Although AI technology has achieved some success in electromagnetic modeling, current applications are mainly concentrated on two-dimensional electromagnetic problems, with a smaller number of applications targeting three-dimensional electromagnetic problems based on uniform meshes. This is primarily because current AI technologies are designed for structured data and typically require fixed input and output dimensions. While three-dimensional modeling based on non-uniform meshes can more accurately represent objects of arbitrary shapes, the resulting data is unstructured, and the number of unknowns varies depending on the target, making most existing AI technologies unusable. Summary of the Invention

[0005] The present invention aims to at least partially solve one of the technical problems in the related art.

[0006] To address this, this invention proposes a three-dimensional electromagnetic target modeling method based on graph neural networks. The three-dimensional electromagnetic target is discretized using a non-uniform grid, and the physical equations satisfied by the three-dimensional electromagnetic target are transformed into a system of linear equations. A graph data structure is used to describe the non-uniformly grid-discrete three-dimensional electromagnetic target, and the graph neural network is trained to learn the iterative solution process of the linear equations, achieving fast and accurate three-dimensional electromagnetic target modeling.

[0007] Another objective of this invention is to propose a three-dimensional electromagnetic target modeling device based on graph neural networks.

[0008] To achieve the above objectives, this invention proposes a three-dimensional electromagnetic target modeling method based on graph neural networks, comprising:

[0009] Three-dimensional electromagnetic targets are discretized using non-uniform meshes;

[0010] The physical equations satisfied by the discretized three-dimensional electromagnetic target are transformed into a system of linear equations.

[0011] The graph data structure describes the discretized three-dimensional electromagnetic target to obtain the graph data description result, and the graph neural network is trained based on the graph data description result to learn the iterative solution process of the linear equation system, so as to realize the modeling of the three-dimensional electromagnetic target.

[0012] In addition, the three-dimensional electromagnetic target modeling method based on graph neural networks according to the above embodiments of the present invention may also have the following additional technical features:

[0013] Furthermore, in one embodiment of the present invention, the physical equations satisfied by the three-dimensional electromagnetic target are transformed into a system of linear equations:

[0014]

[0015] in, is the coefficient matrix of the linear equation system, b is the right-hand side term of the linear equation system, u is the unknown quantity to be solved, and each element in u corresponds to the physical quantity on each grid cell.

[0016] Furthermore, in one embodiment of the present invention, the structural information of each grid cell of the non-uniform mesh includes vertices, edges, and corresponding physical quantities. A graph data structure is used to represent the cell structural information of the non-uniform mesh, consisting of three sets (V, E, F), where V = {v1, ..., v...}. N} is a set containing N nodes. It is a set of edges formed by nodes. The information contained in each node is represented by its feature vector. The set of node feature vectors is represented as F = {f(v1), ..., f(v2)}. N )}.

[0017] Furthermore, in one embodiment of the present invention, the solution is obtained using a three-dimensional electromagnetic modeling method based on the graph neural network, and the iterative formula for the k-th step is:

[0018]

[0019]

[0020]

[0021]

[0022] Among them, u k R represents a candidate solution to the system of linear equations at the k-th iteration. k It is the residual of the linear equation system at the k-th iteration, where ⊕ represents R. k and u k The connection, u k and R k It is based on graph data structure representation, Ψ r and Ψ i These are two independent graph neural networks, with corresponding network parameters Θ and Θ, respectively. r and Θ i .

[0023] Furthermore, in one embodiment of the present invention, the training of the graph neural network to learn the iterative solution process of the linear equation system includes supervised training and self-supervised training, wherein,

[0024] The objective function of the supervised training method measures the root mean square error between the results of the graph neural network and the true results:

[0025]

[0026] Among them, u m and u g These represent the actual results and the results obtained from the graph neural network, respectively. [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements;

[0027] The objective function of the self-supervised training method uses the physical equations satisfied by the three-dimensional electromagnetic target to measure the solution results of the graph neural network:

[0028]

[0029] Where ug This represents the solution obtained from the graph neural network, [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements.

[0030] To achieve the above objectives, another aspect of the present invention proposes a three-dimensional electromagnetic target modeling device based on graph neural networks, comprising:

[0031] Discretization module, used to discretize three-dimensional electromagnetic targets using non-uniform grids;

[0032] The equation transformation module is used to transform the physical equations satisfied by the discretized three-dimensional electromagnetic target into a linear system of equations.

[0033] The electromagnetic target modeling module is used to describe the discretized three-dimensional electromagnetic target based on the graph data structure to obtain the graph data description result, and to train a graph neural network based on the graph data description result to learn the iterative solution process of the linear equation system, so as to realize the three-dimensional electromagnetic target modeling.

[0034] The three-dimensional electromagnetic target modeling method and apparatus based on graph neural networks of the present invention describe three-dimensional electromagnetic targets with non-uniform grids based on graph data structures, train graph neural networks to learn the iterative solution process of three-dimensional electromagnetic targets, and realize fast and accurate three-dimensional electromagnetic target modeling.

[0035] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0036] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0037] Figure 1 This is a flowchart of a three-dimensional electromagnetic target modeling method based on graph neural networks according to an embodiment of the present invention;

[0038] Figure 2 This is a schematic diagram of the non-uniform grid discretization process of a three-dimensional electromagnetic target according to an embodiment of the present invention;

[0039] Figure 3 This is a schematic diagram of RWG basis functions according to an embodiment of the present invention;

[0040] Figure 4 This is a schematic diagram of a three-dimensional electromagnetic target based on a graph data structure representation of a non-uniform grid, according to an embodiment of the present invention.

[0041] Figure 5 This is a schematic diagram of a three-dimensional electromagnetic target modeling method based on graph neural networks according to an embodiment of the present invention;

[0042] Figure 6 This is a schematic diagram comparing the surface current of a conducting sphere solved by the method of moments according to an embodiment of the present invention and the three-dimensional electromagnetic target modeling method based on graph neural networks.

[0043] Figure 7 This is a schematic diagram comparing the RWG basis function coefficients of a conducting sphere solved by the method of moments and the three-dimensional electromagnetic target modeling method based on graph neural networks according to an embodiment of the present invention.

[0044] Figure 8 This is a schematic diagram of the structure of a three-dimensional electromagnetic target modeling device based on a graph neural network according to an embodiment of the present invention. Detailed Implementation

[0045] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0046] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments. 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 should fall within the scope of protection of the present invention.

[0047] The following description, with reference to the accompanying drawings, describes a method and apparatus for modeling three-dimensional electromagnetic targets based on graph neural networks, according to embodiments of the present invention.

[0048] Figure 1 This is a flowchart of a three-dimensional electromagnetic target modeling method based on graph neural networks according to an embodiment of the present invention.

[0049] like Figure 1 As shown, the method includes, but is not limited to, the following steps:

[0050] S1, using non-uniform grid discretization to process three-dimensional electromagnetic targets;

[0051] S2 transforms the physical equations satisfied by the discretized three-dimensional electromagnetic target into a system of linear equations.

[0052] S3 describes the discretized 3D electromagnetic target based on the graph data structure to obtain the graph data description result, and trains a graph neural network based on the graph data description result to learn the iterative solution process of the linear equation system, so as to realize the 3D electromagnetic target modeling.

[0053] Specifically, for a given three-dimensional electromagnetic target, a suitable non-uniform mesh and corresponding basis functions are selected for discretization, and the physical equations it satisfies are transformed into linear equations:

[0054]

[0055] in, Let be the coefficient matrix of the linear equation system, b be the right-hand side term of the system, and u be the unknown quantity to be solved. Here, we can assume that the unknown quantity u belongs to the imaginary field, and can express it as u = u r +ju i , where u r and u i Let represent the real and imaginary parts of u, and j be the imaginary unit. For u within the real number field, the imaginary part can be ignored during calculation.

[0056] It is understandable that the grid cells of a non-uniform grid have different sizes and can accurately describe the shape of a three-dimensional electromagnetic target under certain numerical errors.

[0057] It is understandable that the calculation of the residual R of a system of linear equations is based on a graph data structure.

[0058] Understandably, this refers to a class of artificial neural networks used to process data that can be represented as graph data structures.

[0059] Furthermore, a graph data structure is used to represent the three-dimensional electromagnetic target described by a non-uniform mesh. The structural information of each mesh cell in the non-uniform mesh includes vertices, edges, and corresponding physical quantities. The graph data structure is used to represent the structural information of the non-uniform mesh, consisting of three sets (V, E, F), where V = {v1, ..., v}. N} is a set containing N nodes. It is a set of edges composed of nodes. The information contained in each node can be represented by the feature vector of that node. The set of node feature vectors can be represented as F = {f(v1), ..., f(v2)}. N )}.

[0060] As an example, this invention considers the electromagnetic modeling problem of a three-dimensional metallic conductor. For a three-dimensional conductor, its surface current satisfies the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE):

[0061]

[0062]

[0063] Among them, S o It is a three-dimensional conductor surface, E inc and H inc These are the incident electric field and the incident magnetic field, J s It is the surface current of the conductor. Z0 is the equivalent surface current, and Z0 is the wave impedance. It is the normal vector of the conductor's surface. The expression is:

[0064]

[0065] The expression is:

[0066]

[0067] To overcome the internal resonance problem in three-dimensional conductors, the combined-field integral equation (CFIE) is generally used. The CFIE is a weighted sum of the EFIE and CFIE, and can be expressed as:

[0068]

[0069] Where α is the weighting coefficient, which is 0.5 in this embodiment. The three-dimensional conductor target is discretized using the Rao–Wilton–Glisson (RWG) basis function, as follows: Figure 2 As shown, Figure 2 The paper presents a conducting sphere with a radius of 1m, which is discretized using RWG basis functions to obtain a discrete representation of the conducting sphere. The method of moments (MoM) is then applied to transform the CFIE into a matrix equation.

[0070]

[0071] in, Let be the coefficient matrix of the linear equation system, b be the right-hand side term of the system, and u be the unknown quantity to be solved. Here, the unknown quantity u belongs to the imaginary field and can be expressed as u = u r+ju i , where u r and u i Let be the real and imaginary parts of u, and j be the imaginary unit.

[0072] like Figure 3 As shown, the RWG basis functions are defined as follows:

[0073]

[0074] Where o is the origin of the coordinate system, l m It is the length of the m-th edge. The triangles represent the two sides of the m-th edge. Let r be the area of ​​the triangle. i and r j It is the position vector of the vertex opposite to the m-th edge. It is a triangle The position vector of a certain point,

[0075] Based on the discretization of the three-dimensional conductor target, a graph data structure is used to represent the three-dimensional electromagnetic target described by the non-uniform grid. This target consists of three sets (V, E, F), and the process is as follows: Figure 4 As shown. First, a set of RWG basis functions is generated, such as... Figure 4 As shown in (a) in the figure; the RWG basis functions correspond to Figure 4 The sides of the triangle in (a) can be considered as independent nodes in the graph structure, forming a node set V = {v1, ..., v...}. N} is a set of N nodes, where each node represents a physical quantity that forms the node feature vector set F = {f(v1), ..., f(v2)}. N )},like Figure 4 As shown in (b) in the figure; connecting two adjacent nodes forms an edge set in the graph structure. like Figure 4 As shown in (c), the principle for determining whether two nodes are connected is whether the RWG basis functions represented by the nodes have a common endpoint; Figure 4 (d) in the figure shows the graph structure constructed from a three-dimensional electromagnetic target described by a non-uniform grid.

[0076] Furthermore, after representing 3D electromagnetic targets with graph data structures based on non-uniform mesh descriptions, a 3D electromagnetic target modeling method based on graph neural networks is constructed, such as... Figure 5As shown, this method assumes K iterations. GraphConv-[a,b,mean] represents the graph convolutional layer, with input channels, output channels, and neighbor node aggregation functions a, b, and the mean function, respectively. Linear-[c,d] represents the linear layer with input and output channels c and d, respectively. Dropout-[0.5] represents the dropout rate of the linear layer, which is 0.5. The formula for the k-th iteration is:

[0077]

[0078]

[0079]

[0080]

[0081] Among them, u k R represents a candidate solution to the system of linear equations at the k-th iteration. k It is the residual of the linear equation system at the k-th iteration. Represents R k and u k The connection should be noted that u k and R k It is based on graph data structure representation, Ψ r and Ψ i These are two independent graph neural networks, with corresponding network parameters Θ and Θ, respectively. r and Θ i Ψ r and Ψ i These are two independent graph convolutional neural networks with the same structure and independent parameter sets. For the i-th node of the graph structure, the iterative update formula for the graph convolution from the l-th layer to the (l+1)-th layer is:

[0082]

[0083] in and These are the feature vectors of the i-th node in the l-th and (l+1)-th layers. It is a node The index of the neighbor node, It is a node The number of neighboring nodes, e j, It is a node and For the edge of vertex W s and W a It is a node The trainable parameters of the feature vectors of neighboring nodes. The above equation shows that the neighbor aggregation function used in graph convolution is an averaging function.

[0084] Furthermore, a method for training a 3D electromagnetic modeling system based on a graph neural network is constructed, employing two training methods: supervised training and self-supervised training. The objective function in the supervised training method measures the root mean square error between the graph neural network's result and the actual result.

[0085]

[0086] Among them, u m and u g These represent the actual results and the results obtained from the graph neural network, respectively. [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements. The objective function of the self-supervised training method uses the physical equations satisfied by the three-dimensional electromagnetic target to measure the results of the graph neural network:

[0087]

[0088] Where u g This represents the solution obtained from the graph neural network, [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements.

[0089] Figure 6 The method of moments (MoM) and a graph neural network-based three-dimensional electromagnetic target modeling method are used to compare the solution of the surface current of a conducting sphere. Figure 6 (a) in the middle, Figure 6 (b) in the middle Figure 6 In the figure, (c) represents the surface currents solved by the method of moments (MoM) and the 3D electromagnetic target modeling method based on graph neural networks, respectively, along with the absolute error distributions between the two methods. In the legend, AE, MoM, and S-PhiGRL represent the absolute error, the MoM method, and the 3D electromagnetic target modeling method based on graph neural networks, respectively. It can be seen that the surface currents solved by the graph neural network agree very well with those solved by the MoM method, with a very low error level. Figure 7 The method of moments (MoM) and a graph neural network-based three-dimensional electromagnetic target modeling method are used to compare the RWG basis function coefficients of a conducting sphere. Figure 7 The RWG basis function coefficients in the middle correspond to respectively Figure 6 The surface current is solved using the method of moments and the three-dimensional electromagnetic target modeling method based on graph neural networks. Figure 7 (a) and Figure 7In the figure, (b) represents the real and imaginary parts of the RWG basis function coefficients, respectively. In the legend, MoM and S-PhiGRL represent the method of moments and the three-dimensional electromagnetic target modeling method based on graph neural networks, respectively. It can be seen that the RWG basis function coefficients solved by the graph neural network agree very well with the results solved by the method of moments, and the calculation accuracy is very high.

[0090] According to the embodiment of the present invention, the three-dimensional electromagnetic target modeling method based on graph neural network uses non-uniform grid to discretize the three-dimensional electromagnetic target, transforms the physical equations satisfied by the three-dimensional electromagnetic target into a system of linear equations, describes the non-uniform grid-discrete three-dimensional electromagnetic target based on graph data structure, trains the graph neural network to learn the iterative solution process of the system of linear equations, and realizes fast and accurate three-dimensional electromagnetic target modeling.

[0091] To achieve the above embodiments, such as Figure 8 As shown, this embodiment also provides a weakly supervised learning device 10 based on duplex similarity measurement. The device 10 includes a discretization processing module 100, an equation transformation module 200, and an electromagnetic target modeling module 300.

[0092] Discretization module 100 is used to discretize three-dimensional electromagnetic targets using a non-uniform grid;

[0093] The equation transformation module 200 is used to transform the physical equations satisfied by the discretized three-dimensional electromagnetic target into a linear equation system.

[0094] The electromagnetic target modeling module 300 is used to describe the discretized three-dimensional electromagnetic target based on the graph data structure to obtain the graph data description result, and to train a graph neural network based on the graph data description result to learn the iterative solution process of the linear equation system in order to realize the three-dimensional electromagnetic target modeling.

[0095] Furthermore, the above-mentioned equation transformation module 200 is also used for:

[0096] The physical equations satisfied by the discretized three-dimensional electromagnetic target are transformed into a system of linear equations:

[0097]

[0098] in, is the coefficient matrix of the linear equation system, b is the right-hand side term of the linear equation system, u is the unknown quantity to be solved, and each element in u corresponds to the physical quantity on each grid cell.

[0099] Furthermore, the structural information of each grid cell in the non-uniform mesh includes vertices, edges, and corresponding physical quantities. A graph data structure is used to represent the cell structure information of the non-uniform mesh, consisting of three sets (V, E, F), where V = {v1, ..., v...}. N} is a set containing N nodes. It is a set of edges formed by nodes. The information contained in each node is represented by its feature vector. The set of node feature vectors is represented as F = {f(v1), ..., f(v2)}. N )}.

[0100] Furthermore, a 3D electromagnetic modeling method based on graph neural networks is used to solve the problem. The iterative formula for the k-th step is:

[0101]

[0102]

[0103]

[0104]

[0105] Among them, u k R represents a candidate solution to the system of linear equations at the k-th iteration. k It is the residual of the linear equation system at the k-th iteration, where ⊕ represents R. k and u k The connection, u k and R k It is based on graph data structure representation, Ψ r and Ψ i These are two independent graph neural networks, with corresponding network parameters Θ and Θ, respectively. r and Θ i .

[0106] Furthermore, the electromagnetic target modeling module includes supervised training and self-supervised training methods, wherein...

[0107] The objective function of the supervised training method measures the root mean square error between the results of the graph neural network and the true results:

[0108]

[0109] Among them, u m and u g These represent the actual results and the results obtained from the graph neural network, respectively. [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements;

[0110] The objective function of the self-supervised training method uses the physical equations satisfied by the three-dimensional electromagnetic target to measure the solution results of the graph neural network:

[0111]

[0112] Where u g This represents the solution obtained from the graph neural network, [*] r and[*] i These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements.

[0113] The three-dimensional electromagnetic target modeling device based on graph neural network according to an embodiment of the present invention uses non-uniform grid to discretize the three-dimensional electromagnetic target, transforms the physical equations satisfied by the three-dimensional electromagnetic target into a system of linear equations, describes the non-uniform grid-discrete three-dimensional electromagnetic target based on graph data structure, trains the graph neural network to learn the iterative solution process of the system of linear equations, and realizes fast and accurate three-dimensional electromagnetic target modeling.

[0114] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0115] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

Claims

1. A method for modeling three-dimensional electromagnetic targets based on graph neural networks, characterized in that, Includes the following steps: Three-dimensional electromagnetic targets are discretized using a non-uniform mesh; each mesh cell of the non-uniform mesh contains vertex, edge, and corresponding physical quantity information. The physical equations satisfied by the discretized three-dimensional electromagnetic target are transformed into a system of linear equations: in, It is the coefficient matrix of a system of linear equations. It is the right-hand term of the system of linear equations. It is the unknown quantity to be solved. Each element in the table corresponds to a physical quantity on each grid cell. The discretized three-dimensional electromagnetic target is described using a graph data structure, which consists of three sets. Composition, in which It includes A set of nodes It is a set of edges formed by nodes. The information contained in each node is represented by its feature vector. The set of node feature vectors is represented as follows: ; Based on the graph data, the linear equation system is iteratively solved using a graph neural network. The step-by-step iterative formula is: in, Representing the Candidate solutions to the linear equation system during step iteration. It is the first The residuals of the linear equation system during step iteration, represent and The connection, and It is based on graph data structure representation. and These are two independent graph neural networks, with corresponding network parameters as follows: and ; The iterative solution process of linear equations is learned by training a graph neural network to achieve three-dimensional electromagnetic target modeling.

2. The method according to claim 1, characterized in that, The trained graphical neural network learns the iterative solution process of the linear equation system, including supervised training and self-supervised training. The objective function of the supervised training method measures the root mean square error between the results of the graph neural network and the true results: in, and These represent the actual results and the results obtained from the graph neural network, respectively. and These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements; The objective function of the self-supervised training method uses the physical equations satisfied by the three-dimensional electromagnetic target to measure the solution results of the graph neural network: in This represents the solution obtained by the graph neural network. and These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements.

3. A three-dimensional electromagnetic target modeling device based on graph neural networks, characterized in that, include: The discretization module is used to discretize three-dimensional electromagnetic targets using a non-uniform mesh; each mesh cell of the non-uniform mesh contains vertices, edges, and corresponding physical quantity information. The equation transformation module is used to transform the physical equations satisfied by the discretized three-dimensional electromagnetic target into a linear system of equations: in, It is the coefficient matrix of a system of linear equations. It is the right-hand term of the system of linear equations. It is the unknown quantity to be solved. Each element in the table corresponds to a physical quantity on each grid cell. The electromagnetic target modeling module is used to describe the discretized three-dimensional electromagnetic target based on a graph data structure, which consists of three sets. Composition, in which It includes A set of nodes It is a set of edges formed by nodes. The information contained in each node is represented by its feature vector. The set of node feature vectors is represented as follows: ; The electromagnetic target modeling module is also used to iteratively solve the linear equation system based on the graph data structure using a graph neural network. The step-by-step iterative formula is: in, Representing the Candidate solutions to the linear equation system during step iteration. It is the first The residuals of the linear equation system during step iteration, represent and The connection, and It is based on graph data structure representation. and These are two independent graph neural networks, with corresponding network parameters as follows: and ; Furthermore, the iterative solution process of linear equations is learned through training a graph neural network to achieve three-dimensional electromagnetic target modeling.

4. The apparatus according to claim 3, characterized in that, The electromagnetic target modeling module includes supervised training and self-supervised training methods, wherein... The objective function of the supervised training method measures the root mean square error between the results of the graph neural network and the true results: in, and These represent the actual results and the results obtained from the graph neural network, respectively. and These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements; The objective function of the self-supervised training method uses the physical equations satisfied by the three-dimensional electromagnetic target to measure the solution results of the graph neural network: in This represents the solution obtained by the graph neural network. and These represent taking the real and imaginary parts of the imaginary number, respectively. and represent and The number of elements.

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