An electromagnetic field intelligent calculation method based on deep learning

Abstract

The application discloses an electromagnetic field intelligent calculation method based on deep learning, specifically, a space-time input vector is input into an MFF-PINN neural network, the MFF-PINN neural network comprises a plurality of parallel subnetworks and a linear superposition module, the first subnetwork comprises a scale transformation module, a Fourier feature transformation module and an MLP processing module, the space-time input vector is subjected to scale transformation first, then the vector subjected to scale transformation is subjected to Fourier feature transformation, the Fourier feature transformation module performs Fourier transformation on the vector subjected to scale transformation based on an effective frequency matrix; all the subnetworks share the Fourier feature transformation, the output obtained through the Fourier feature transformation is input into the MLP processing module in the first subnetwork; and the outputs of the subnetworks are subjected to linear superposition. The application significantly enhances the expression capability of the network on high-frequency components and multi-scale features of electromagnetic fields.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic field calculation, and in particular relates to an intelligent electromagnetic field calculation method based on deep learning. Background Technology

[0002] The exact solution to Maxwell's equations plays a fundamental and crucial role in modern science and engineering fields such as antenna design, electromagnetic compatibility analysis, and electromagnetic stealth. Currently, electromagnetic field simulation and prediction techniques for these equations mainly include traditional numerical calculation methods and emerging methods based on deep learning that have developed rapidly in recent years.

[0003] Among traditional numerical methods, the Finite Difference Time Domain (FDTD), Finite Element Method (FEM), and Method of Moments (MoM) are the most widely used. Their core idea is to discretize the continuous physical space into meshes or patch elements, thereby approximating the Maxwell equations in an algebraic form. Therefore, the reliability and efficiency of the computational results are highly dependent on the quality of the mesh or patch generation. Taking the FDTD algorithm based on the differential form of Maxwell's equations as an example, in multi-scale electromagnetic problems, to reduce the impact of inherent numerical dispersion errors on solution accuracy, Yee mesh elements much smaller than the electrical size are typically required. This meticulous meshing often results in hundreds of millions or even more mesh elements, posing a severe challenge to memory consumption and computational speed. Furthermore, under the constraint of the Courant–Friedrichs–Lewy stability condition, excessively dense spatial discretization inevitably leads to hundreds of millions of time iteration steps, further amplifying the pressure on simulation time. Furthermore, for target objects with complex curved surface geometry, traditional FDTD is unable to avoid stepped approximation errors. Even with conformal mesh technology, it is difficult to completely eliminate this effect, which makes the accuracy and reliability of simulation results still face significant challenges.

[0004] To thoroughly address the impact of discrete element generation on the numerical solution of Maxwell's equations, the rapidly developing Physics-Informed Neural Network (PINN) based on deep learning has gradually emerged as a novel research paradigm to replace traditional numerical algorithms and has attracted widespread attention. The core idea of ​​PINN is to explicitly embed the governing equations, boundary conditions, and initial conditions into the loss function of the neural network, driving the network to learn solutions to partial differential equations (PDEs) by minimizing the physical residuals. The Multilayer Perceptron (MLP) is considered a universal function approximator within this framework, enabling the solution of various mathematical physics equations in a unified and flexible manner. Due to its meshless nature, PINN has been widely applied in various fields such as fluid dynamics, acoustics, electrostatics, low-frequency electromagnetic problems, and single-frequency electromagnetic analysis, demonstrating significant development potential. However, when the classic PINN architecture is used to solve high-frequency transient electromagnetic problems, its inherent spectral bias often leads to decreased solution efficiency, convergence difficulties, and even an inability to accurately capture high-frequency details.

[0005] Therefore, there is an urgent need in this field for a novel technical solution that can fundamentally overcome the spectral bias problem of PINN, enabling it to efficiently characterize rapidly changing transient responses and wide-spectrum components in time-domain electromagnetic fields. This invention addresses these technical challenges by constructing an innovative PINN architecture. This architecture, through the introduction of multi-scale Fourier eigenmaps, significantly enhances the neural network's ability to learn and represent high-frequency transient field components, thereby achieving high-precision, meshless, and complex electromagnetic process simulation performance when solving Maxwell's equations in the time domain. Summary of the Invention

[0006] Purpose of the invention: In order to solve the problems existing in the prior art, the present invention provides an intelligent electromagnetic field calculation method based on deep learning.

[0007] Technical solution: This invention provides a deep learning-based intelligent electromagnetic field calculation method, specifically as follows:

[0008] Spatiotemporal input vector The input is fed into the MFF-PINN neural network, which includes... A parallel subnetwork and a linear superposition module, wherein , Indicates time The coordinates below;

[0009] No. Each sub-network includes a scaling module, a Fourier feature transform module, and an MLP processing module. First, a scaling transformation is performed on the spatiotemporal input vector. Then, a Fourier feature transform is performed on the scaled vector. The Fourier feature transform module uses the associative law of matrix multiplication and scalar multiplication to define an effective frequency matrix, and performs a Fourier transform on the scaled vector based on this effective frequency matrix. All sub-networks share this Fourier feature transform, and the output obtained from the Fourier feature transform is... Enter to the number MLP processing module in each sub-network;

[0010] Will The outputs of each subnetwork are linearly superimposed to obtain the time step. coordinates below The electromagnetic field vector at that location, .

[0011] Furthermore, the first The scaling module of each subnetwork uses a scalar scaling factor. Multiply the input vector element by element to obtain the scaled vector; A scalar scaling factor sequence consisting of scalar scaling factors For geometric series, scalar scaling factor From the fundamental amplitude factor and the Individual scale base factors Jointly determined:

[0012] .

[0013] Furthermore, the Fourier feature transformation module processes the scaled vector as follows:

[0014] ;

[0015] in, This is the vector after scaling. It is the Fourier feature transform module. The output of the Fourier feature transform module In order to be with the first The Fourier feature transformation module performs a Fourier transform on the scaled vectors output by the corresponding sub-networks based on the effective frequency matrix of each sub-network. ,in It is a Gaussian frequency matrix; The elements follow a Gaussian distribution. , The elements will follow a Gaussian distribution. , It is a hyperparameter with a fixed base scale.

[0016] Furthermore, the loss function for training the MFF-PINN neural network. The expression is:

[0017] ;

[0018] in, Represents the physical information loss function. Represents the initial condition loss function. Represents the boundary condition loss function; It is a weighted hyperparameter that balances the loss of physical information. These are the weight hyperparameters that balance the initial condition loss. It is the weight hyperparameter for balancing the boundary condition loss.

[0019] Furthermore, the physical information loss function is constructed using the following method:

[0020] Construct Maxwell's equations to describe two-dimensional TE waves:

[0021] ;

[0022] in, , , These represent the electromagnetic field components along the x-axis, y-axis, and z-axis, respectively. Represents the dielectric constant. Let represent the permeability. To measure how well the neural network output fits the equations used to describe electromagnetic laws, the following residual function is constructed for the Maxwell equations described above:

[0023] ;

[0024] in, , , These are the first, second, and third residual functions, respectively. , , These represent the electromagnetic field components predicted by the MFF-PINN neural network on the x-axis, y-axis, and z-axis, respectively.

[0025] To make the mean square values ​​of the three residual functions approach zero, in the spatiotemporal solution domain... Internal random sampling Configuration points Therefore, the following physical information loss function is constructed:

[0026] ;

[0027] in, , , They represent the first The first, second, and third residual functions corresponding to each configuration point.

[0028] Furthermore, the specific method for constructing the initial condition loss function is as follows: sampling within the region where the initial conditions apply. Data points Based on these data points, construct the initial conditional loss function:

[0029] ;

[0030] in, The first prediction of the MFF-PINN neural network The physical quantity at each data point For the first The actual physical quantity corresponding to each data point.

[0031] Furthermore, the specific method for constructing the boundary condition loss function is as follows: sampling is performed on the region where the boundary conditions apply. sampling points By calculating the boundary operators of the MFF-PINN neural network at these sampling points The mean square error between the result after the action and zero is used to obtain the boundary condition loss function:

[0032] ;

[0033] in, The first prediction of the MFF-PINN neural network The physical quantity at each sampling point.

[0034] Furthermore, a gradient-based optimization algorithm is used to train the model end-to-end. After the backpropagation step begins, the total loss is calculated. Relative to the entire parameter set The gradient of each trainable parameter is calculated, and then the Adam optimizer performs parameter updates.

[0035] A computer device includes a memory, a processor, and a computer program stored in the memory and capable of running on the processor, wherein the processor executes the computer program to implement the steps of the deep learning-based electromagnetic field intelligent computing method.

[0036] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the deep learning-based intelligent electromagnetic field computing method.

[0037] Beneficial Effects: This invention effectively overcomes the inherent spectral bias problem of the PINN model by applying a geometrically distributed multi-scale scaling to the input coordinates, significantly enhancing the network's ability to express high-frequency components and multi-scale features of the electromagnetic field, thereby improving solution accuracy and accelerating the training convergence process. Furthermore, all sub-networks in this invention's architecture share the same Fourier feature mapping module. Compared to schemes that configure Fourier modules separately for different scales, this significantly reduces the number of model parameters and memory overhead, making the overall structure more lightweight and efficient. This invention improves the traditional feature encoding structure. Compared to traditional mapping methods that only include trigonometric function transformations, this scheme adopts a full-scale feature integration mechanism. By using the original input coordinates... By concatenating linear components with high-frequency components obtained through Fourier transform, the neural network can simultaneously decouple the global evolution trend and local subtle fluctuations of the electromagnetic field. This invention scales the shared Fourier frequency matrix using a multi-scale scaling factor to form an effective frequency matrix. This design allows each sub-network to automatically cover multiple frequency scales from low to high frequencies, achieving decoupled learning and collaborative fusion of multi-scale features. Furthermore, the parallel multi-sub-network design of this invention assigns physical components of different scales to corresponding sub-networks for separate learning, resulting in a clear structure that facilitates model interpretability analysis. It also possesses good scalability, allowing the number of sub-networks to be flexibly adjusted according to the scale complexity of the problem, thus significantly improving the overall modeling flexibility. This invention provides an efficient and accurate new paradigm for deep learning in solving partial differential equations with complex multi-scale characteristics, and has broad application prospects in scientific and engineering computing fields such as fluid mechanics, electromagnetics, and acoustics. Attached Figure Description

[0038] Figure 1 This is a flowchart illustrating the overall process framework of the present invention.

[0039] Figure 2 This is a diagram of the MFF-PINN neural network structure of the present invention;

[0040] Figure 3 A comparison of the field thermodynamics of different models at the same time;

[0041] Figure 4 A graph showing the error comparison of the exact solutions between different models;

[0042] Figure 5 The graphs show the waveforms of different models at specific observation points, where (a) represents the electric field components. The waveform comparison diagram shows that (b) represents the electric field components. The waveform comparison diagram shows that (c) represents the magnetic field component. Waveform comparison chart;

[0043] Figure 6 The training error convergence curves for different models;

[0044] Figure 7 The graph shows the error evolution curves of different models in long-term prediction, where (a) represents the electric field components. The L2 relative error evolution curve is shown in (b), where the electric field components are shown. The L2 relative error evolution curve, (c) is the magnetic field component. The L2 relative error evolution curve. Detailed Implementation

[0045] This invention proposes a physical information neural network architecture for solving Maxwell's equations. The overall process framework of the invention is as follows: Figure 1 As shown. This framework employs the MFF-PINN neural network. , Represents a mapping from 3D to 3D, from spacetime to field quantities, in the input layer. After neural network calculation, the output is the electromagnetic field component corresponding to that spatiotemporal point. Parameterized displacement field, here Representative network parameter set:

[0046] (1);

[0047] in, It is an electromagnetic field vector predicted by a neural network, containing along... electric field components of the axis ,along electric field components of the axis and along Magnetic field components of the axis The basic module used in the architecture of this invention is a classic fully connected network, consisting of an input layer, multiple hidden layers, and an output layer. Its forward propagation process can be defined by the following recursive formula:

[0048] (2);

[0049] in, Represents the input coordinate vector Its dimensions are , Representing the Feature mapping of layers, It is the first The weight matrix of the layer, It is the first The layer's bias vector, It is the first The nonlinear Sine activation function of the layer, It is the set of all trainable parameters in the network. Represents the real number field. Indicates the neural network's first... Number of neurons in a layer, bias vector Due to the There are layers There are 10 neurons, each requiring a bias value and a weight matrix. Its function is to transform the previous layer 3D feature mapping to the current In the 3D feature space. This represents the total number of hidden layers.

[0050] The neural network structure designed in this invention is as follows: Figure 2 As shown, its final output for Parallel subnetworks Linear superposition of outputs:

[0051] (3);

[0052] Each subnetwork Specifically designed for fitting function components within a specific frequency scale range, it consists of a cascaded combination of the following three operations, the mathematical expression of which is:

[0053] (4);

[0054] in, It is a scale-specific input coordinate transformation. It is the basic Fourier feature module shared by all sub-networks, and It is the first The subnetworks correspond to independent MLPs. This linear superposition method ensures that information from different scales can be effectively and decoupledly combined to form a complete solution to the objective function. "" indicates function composition.

[0055] Input layer

[0056] For a given input spatiotemporal vector This vector is not directly fed into subsequent operations. The first parallel subnetwork. For the second... Each sub-network first processes the input vector Apply a dedicated scaling transformation The transformation is performed through a scalar scaling factor. Perform element-wise multiplication on the input vector:

[0057] (5);

[0058] The scale factor sequence Designed as a geometric series, consisting of a fundamental amplitude factor and a scale basis factor Together, their general term formula is determined to be:

[0059] (6);

[0060] By employing the aforementioned geometric series transformation, each sub-network receives a coordinate system scaled at a specific scale. This makes the... Subnetworks can focus on learning and scaling factors. The relevant feature components at specific frequencies or spatial scales. This task decomposition mechanism greatly enhances the network's overall ability to capture and reconstruct multi-scale details, laying a solid foundation for subsequent Fourier feature encoding and high-precision field prediction.

[0061] Feature mapping layer

[0062] Shared feature mapping module

[0063] Coordinate vector after scaling It is fed into a single, fundamental Fourier feature map module shared by all subnetworks. This module maps scaled coordinates to a high-dimensional feature space, which contains a fixed, non-learned random Gaussian frequency matrix. ,in It is half the dimension of the Fourier feature. Matrix elements Generated by sampling from a Gaussian distribution in one go. This represents the dimension index of the vector after scaling. The component index representing the random frequency, i.e. ,in It is a hyperparameter with a fixed base scale. Will ( The mapping is transformed into an aggregated feature vector containing the original input and Fourier features. :

[0064] (7);

[0065] Equivalent matrix

[0066] No. The Fourier features in a subnetwork are Using the associative laws of matrix multiplication and scalar multiplication Then an effective frequency matrix can be defined. Therefore, for the first The features generated by each subnetwork are mathematically equivalent to applying this effective matrix to the original coordinates. Given the fundamental matrix The elements follow a Gaussian distribution. Based on the scaling properties of the Gaussian distribution, the effective matrix... The elements will follow a new Gaussian distribution. This means that the first part of the present invention Each subnetwork is functionally equivalent to an independent PINN branch, which has its own frequency sampling scale. Automatically enlarged to Fourier feature module:

[0067] (8)

[0068] Through sharing The module significantly reduces the number of model parameters and memory usage because it eliminates the need to store a separate, large frequency matrix for each scale.

[0069] In this embodiment, shared resources are utilized. Module for the first The Fourier feature transform is performed on each subnetwork, and its mapping function is defined as follows:

[0070] (9);

[0071] This mapping is achieved by inputting coordinates. The transformed high-frequency features are then concatenated to achieve the fusion of multi-scale features.

[0072] Sub-MLP processing

[0073] From shared modules The output feature vector that carries specific scale information He was then sent to a special room belonging to the first Subnetworks of independent branches Nonlinear processing is performed to ultimately produce the output component at this scale. The activation function for each subnet is Sine, to enhance its ability to fit complex functions.

[0074] loss function

[0075] To make the output of the MFF-PINN neural network To obtain an effective solution to the target partial differential equation, the training process of this invention minimizes the composite loss function. To achieve this, namely:

[0076] (10);

[0077] in Represents the physical information loss function. Represents the initial condition loss function. This represents the boundary condition loss function. It is a weighted hyperparameter that balances the loss of physical information. These are the weight hyperparameters that balance the initial condition loss. It is the weight hyperparameter for balancing the boundary condition loss.

[0078] Physical information loss function

[0079] The Maxwell equations used to describe two-dimensional TE waves are as follows:

[0080] (11);

[0081] in, , , These represent the electromagnetic field components along the x-axis, y-axis, and z-axis, respectively. Represents the dielectric constant. It represents the magnetic permeability.

[0082] To measure the degree of fit of the neural network output to the description of electromagnetic laws, the present invention defines the following residual functions for equation (11):

[0083] (12);

[0084] in, , , These are the first, second, and third residual functions, respectively. , , This represents the electromagnetic field components predicted by the MFF-PINN neural network in the x, y, and z axes.

[0085] The optimization objective of the physical information neural network is to drive the mean square value of these three residuals to approach zero. Therefore, in the spatiotemporal solution domain... Internal random sampling Configuration points Construct the physical information loss function:

[0086] (13);

[0087] in, , , They represent the first The first, second, and third residual functions corresponding to each configuration point.

[0088] Initial condition loss function

[0089] This loss term is used to constrain the network's behavior at the initial moment of the physical process, sampling within the region where the initial conditions apply. Data points ,in These are the coordinates at the initial moment. These are the known physical quantities at these points. The initial conditional loss is constructed by calculating the mean squared error between the network's predictions and the actual values ​​at these points:

[0090] (14);

[0091] in, The first prediction of the MFF-PINN neural network The physical quantity at each data point.

[0092] Boundary condition loss function

[0093] This loss term is responsible for ensuring that the network's output remains within the spatial boundaries of the physical solution domain. The given boundary conditions are satisfied. The general form can be represented by a boundary operator. To express this, where, in order to implement the constraint, sampling is performed on the region where the boundary condition acts. Points By calculating the boundary operators of the network at these points. The mean square error between the result after the action and zero is used to obtain the boundary condition loss.

[0094] (15);

[0095] in, The first prediction of the MFF-PINN neural network The physical quantity at each sampling point.

[0096] Training and optimization process

[0097] A total loss function was constructed that includes physical information loss, initial condition loss, and boundary condition loss. Subsequently, this invention employs a gradient-based optimization algorithm to train the model end-to-end, with the Adam optimizer as the core driver. The core of the training lies in a continuously repeating iterative loop. At the beginning of each iteration, the system first samples points within the corresponding solution domain, initial domain, and boundary domain, and calculates the neural network's predicted values ​​and corresponding total loss at these sampled points through a single forward propagation. .

[0098] Backpropagation

[0099] Building upon this, the crucial backpropagation step begins. Leveraging the core technology of automatic differentiation, the system can efficiently and accurately calculate the total loss in scalar form. Relative to the entire parameter set The gradient of each trainable parameter, i.e. This gradient vector indicates the direction in which the loss function grows the fastest, while its negative direction is the target direction for optimization.

[0100] Parameter update

[0101] After taking the gradient, the Adam optimizer immediately performs parameter updates based on the calculated gradient. By combining its internal momentum and learning rate information, an effective update is calculated and applied to all parameters, thereby adjusting the parameters. Towards total loss Move to a lower point.

[0102] Repeated iterations

[0103] The complete process from sampling, forward propagation, loss calculation, backpropagation to parameter update constitutes an independent training iteration. The essence of the method in this invention lies in repeating this iterative loop tens of thousands of times. This repeated optimization process continues until the total loss is reached. The training converges to a sufficiently small tolerance range, or reaches the preset maximum number of training iterations. When training terminates, the final parameter set obtained is... A continuous and differentiable neural network function is defined that can accurately approximate the solution of a partial differential equation.

[0104] Model training evaluation

[0105] This numerical experiment uses a transient electromagnetic pulse radiation example in a two-dimensional space. Due to the wide spectrum characteristics of the pulse source, its field solution simultaneously contains low-frequency slowly varying components and high-frequency transient components, exhibiting significant multi-scale characteristics, which poses a severe challenge to the fitting ability of neural networks.

[0106] To comprehensively evaluate the performance of the MFF-PINN model proposed in this invention, this numerical experiment compares and analyzes it with the following three existing benchmark models:

[0107] a. Classical-PINN: It adopts a standard fully connected feedforward neural network structure, with the input being the original spatial coordinates, and does not introduce Fourier features or multi-scale mechanisms.

[0108] b. Fourier Feature PINN (FF-PINN): A single-scale Fourier feature mapping module is added to the input of the classic PINN, which can be regarded as a special case of the number of sub-networks N = 1 in the architecture of this invention.

[0109] c. Multi-scale PINN (MS-PINN): Similar to the present invention, N parallel sub-networks are superimposed in parallel. Each sub-network receives the original coordinate input and performs scale transformation of the input coordinates. It does not include a shared Fourier feature mapping module.

[0110] To ensure fairness in the comparison, all models maintained a similar total number of network parameters and used the same network depth, width, activation function, optimizer, learning rate, and training period. The number and distribution of region sampling points and boundary / initial condition sampling points that satisfy the physical information loss condition were also completely consistent.

[0111] Accuracy and time comparison analysis

[0112] To verify the superiority of the novel architecture based on physical information neural networks proposed in this invention, Table 1 compares the L2 relative error and inference time between different models used to compute Maxwell's equations and FDTD, serving as evaluation indicators for prediction accuracy and computational efficiency.

[0113] Table 1

[0114]

[0115] The experimental data in Table 1 show that the MFF-PINN model proposed in this invention achieves the lowest L2 relative error across all measured physical field components. This value is significantly lower than all other comparative models, demonstrating the accuracy advantage of this invention in solving such multi-scale, wide-spectrum wave problems. The classic PINN model, due to its inherent spectral bias, has an L2 relative error around 1, indicating its complete inability to capture the complex high-frequency waves and multi-scale electromagnetic field structures generated by differential Gaussian pulse excitation. While the FF-PINN model incorporating Fourier features significantly reduces the error, its error is still approximately 137.5% higher than that of the MFF-PINN model in this invention. This indicates that while single-scale Fourier features can improve high-frequency fitting ability, they cannot simultaneously and efficiently account for both the global low-frequency structure and local high-frequency details of the solution. The MS-PINN model, employing multi-scale connections alone, also achieves significant progress, even outperforming FF-PINN, demonstrating the effectiveness of the multi-scale decomposition approach. However, its error is still about 35% higher than that of the MFF-PINN of the present invention, which reveals that in ordinary multi-scale structures, each sub-network is still limited by standard coordinate input and has insufficient ability to capture high-frequency components within its respective scale range.

[0116] As shown in Table 1, although the prediction time for electromagnetic fields varies among PINN models with different structures, their prediction times are all within 1 second. The prediction time of this invention is nearly 70 times faster than that of FDTD, demonstrating the efficiency advantage of this method in the prediction stage. Furthermore, traditional numerical methods belong to the numerical solver paradigm, requiring a complete iterative calculation for each query. However, once trained, the neural network itself becomes a functional model of the physical field, eliminating the need for repeated iterative calculations. Moreover, any modification to the simulation settings means that FDTD requires a complete and time-consuming recalculation, while the model trained by this invention can make predictions instantly within its learned parameter space.

[0117] To more intuitively demonstrate the advantages of this precision, Figure 3 The electromagnetic field thermograms of different models at the same time are shown. It can be seen intuitively that the classic PINN model cannot fit the correct field diagram, while FF-PINN, MS-PINN and the MFF-PINN of this invention can all effectively fit the field diagram. Figure 4 The error heatmaps of different models relative to the FDTD reference solution are further presented. The error heatmaps show that the classic PINN model exhibits a large area of ​​warm color, failing to predict the correct results, indicating a significant prediction bias. While FF-PINN and MS-PINN can fit the field morphology, they have large errors in the details of the electromagnetic field. The field heatmap predicted by the method of this invention closely matches the exact solution in terms of propagation details, and its corresponding error heatmap is the darkest, indicating that the prediction error is at an extremely low level throughout the simulation region.

[0118] Dynamic feature capture capability analysis

[0119] To test the model's ability to capture the dynamic evolution of the field over time, such as Figure 5 As shown, the figure compares the electromagnetic field waveforms at observation points (0.5m, 1.5m) in space. Due to its inherent spectral bias, the classical PINN model failed to converge to the correct waveform solution in all test cases, demonstrating its fundamental flaw in solving such high-frequency electrodynamic problems. FF-PINN and MS-PINN exhibit significant performance uncertainties and dependence on specific problems. and In terms of predictions, MS-PINN may be slightly better than FF-PINN, while in another physical quantity... In terms of predictions, the situation may be the opposite. In stark contrast to the models mentioned above, the MFF-PINN model proposed in this invention achieves prediction accuracy closest to the FDTD benchmark solution. This strongly demonstrates that by synergistically fusing Fourier features and multi-scale networks, the proposed solution overcomes the limitations of a single strategy and provides a high-precision, highly stable solution.

[0120] Convergence speed analysis

[0121] To examine the convergence speed of different models during training, Figure 6 The graphs show the L2 relative error of different models as a function of iteration number. The classic PINN model has a constant error of 1, indicating that it cannot converge at all in solving high-frequency transient electromagnetic problems. Compared with the single improvement strategies FF-PINN and MS-PINN, the MFF-PINN model of this invention not only exhibits the fastest convergence speed, significantly shortening the training cycle and saving computational costs, but more importantly, it ultimately achieves the lowest convergence error, obtaining the most accurate physics prediction model. Figure 6 It has been effectively demonstrated that fusing Fourier features with a multi-scale network architecture enables the model to surpass any single technical solution in both training efficiency and final performance, achieving a comprehensive improvement in both accuracy and efficiency.

[0122] Generalization ability and stability analysis

[0123] To further verify the long-term stability of the present invention in solving dynamic problems Figure 7 This figure shows a comparison of the evolution of prediction errors of each model over physical time after training. The figure is presented from three key physical components. , , The above consistently demonstrates that at all time points, the classical PINN model has a constant error of 1, rendering it completely ineffective. FF-PINN exhibits the highest error among the effective models, while MS-PINN falls in the middle. The MFF-PINN model of this invention consistently demonstrates significantly lower L2 relative error than all comparative models, showcasing its consistent superiority. The figure shows that the errors of all models increase over time, revealing the technical challenge of error accumulation in long-term predictions for neural networks. However, the error curve of the MFF-PINN model consistently remains at the bottom, proving that it most effectively suppresses this error accumulation and exhibits the strongest long-term prediction stability. Through the synergistic effect of Fourier features and multi-scale networks, not only is the instantaneous accuracy of the model improved, but more importantly, its generalization ability and stability over the time dimension are enhanced.

[0124] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

Claims

1. A deep learning-based intelligent electromagnetic field calculation method, characterized in that, Specifically: Spatiotemporal input vector The input is fed into the MFF-PINN neural network, which includes... A parallel subnetwork and a linear superposition module, wherein , Indicates time The coordinates below; No. Each sub-network includes a scaling module, a Fourier feature transform module, and an MLP processing module. First, a scaling transformation is performed on the spatiotemporal input vector. Then, a Fourier feature transform is performed on the scaled vector. The Fourier feature transform module uses the associative law of matrix multiplication and scalar multiplication to define an effective frequency matrix, and performs a Fourier transform on the scaled vector based on this effective frequency matrix. All sub-networks share this Fourier feature transform, and the output obtained from the Fourier feature transform is... Enter to the number MLP processing module in each sub-network; Will The outputs of each subnetwork are linearly superimposed to obtain the time step. coordinates below The electromagnetic field vector at that location, ; Loss function for training MFF-PINN neural network The expression is: ； in, Represents the physical information loss function. Represents the initial condition loss function. Represents the boundary condition loss function; It is a weighted hyperparameter that balances the loss of physical information. These are the weight hyperparameters that balance the initial condition loss. It is the weight hyperparameter that balances the boundary condition loss; The physical information loss function is constructed using the following method: Construct Maxwell's equations to describe two-dimensional TE waves: ； in, , , These represent the electromagnetic field components along the x-axis, y-axis, and z-axis, respectively. Represents the dielectric constant. Let represent the permeability. To measure how well the neural network output fits the equations used to describe electromagnetic laws, the following residual function is constructed for the Maxwell equations described above: ； in, , , These are the first, second, and third residual functions, respectively. , , These represent the electromagnetic field components predicted by the MFF-PINN neural network on the x-axis, y-axis, and z-axis, respectively. To make the mean square values ​​of the three residual functions approach zero, in the spatiotemporal solution domain... Internal random sampling Configuration points Therefore, the following physical information loss function is constructed: ； in, , , They represent the first The first, second, and third residual functions corresponding to each configuration point.

2. The electromagnetic field intelligent calculation method based on deep learning according to claim 1, characterized in that, The first The scaling module of each subnetwork uses a scalar scaling factor. Element-wise multiplication of the spatiotemporal input vector yields the scaled vector. A scalar scaling factor sequence consisting of scalar scaling factors For geometric series, scalar scaling factor From the fundamental amplitude factor and the Individual scale base factors Jointly determined: 。 3. The electromagnetic field intelligent calculation method based on deep learning according to claim 2, characterized in that, The Fourier feature transformation module processes the scaled vector as follows: ； in, This is the vector after scaling. It is the Fourier feature transform module. This is the output of the Fourier feature transform module. In order to be with the first The Fourier feature transformation module performs Fourier transform on the corresponding scaled vectors based on the effective frequency matrix of each sub-network. ,in It is a Gaussian frequency matrix; The elements follow a Gaussian distribution. , The elements follow a Gaussian distribution. , It is a hyperparameter with a fixed base scale.

4. The electromagnetic field intelligent calculation method based on deep learning according to claim 1, characterized in that, The specific method for constructing the initial condition loss function is as follows: sampling within the region where the initial conditions apply. Data points ; Construct an initial conditional loss function based on these data points: ； in, The first prediction of the MFF-PINN neural network The physical quantity at each data point For the first The actual physical quantity corresponding to each data point.

5. The electromagnetic field intelligent calculation method based on deep learning according to claim 1, characterized in that, The specific method for constructing the boundary condition loss function is as follows: sampling is performed on the region where the boundary conditions apply. sampling points ; By calculating the boundary operators of the MFF-PINN neural network at these sampling points The mean square error between the result after the action and zero is used to obtain the boundary condition loss function: ； in, The first prediction of the MFF-PINN neural network The physical quantity at each sampling point.

6. The electromagnetic field intelligent calculation method based on deep learning according to claim 1, characterized in that, Gradient-based optimization algorithms train the model end-to-end. After the backpropagation step begins, the total loss is calculated. Relative to the entire parameter set The gradient of each trainable parameter is calculated, and then the Adam optimizer performs parameter updates.

7. A computer device comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the deep learning-based intelligent electromagnetic field computing method as described in any one of claims 1 to 6.

8. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the deep learning-based intelligent electromagnetic field computing method as described in any one of claims 1 to 6.

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