Static electromagnetic calculation method of physical information neural network based on Fourier mapping

Abstract

The invention discloses a static electromagnetic calculation method of a physical information neural network based on Fourier mapping, and the method comprises the steps: constructing a full-connection network structure fusing a Fourier mapping mechanism, and designing a parallel modeling architecture of a plurality of three-output sub-networks to adapt to multi-region medium division, network output predicts vector magnetic potential, normal magnetic field intensity and tangential magnetic field intensity at the same time, the problem of network training instability caused by medium mutation is relieved, and meanwhile the network training process has higher physical interpretation performance; vector magnetic potential and magnetic field component continuity are introduced as interface coupling physical constraints, and global physical consistency of solutions is ensured. A Fourier feature mapping mechanism is introduced into a physical information neural network structure, so that the expression capability of a neural network on high-frequency spatial features is enhanced, and efficient modeling of complex boundaries, medium distribution and multi-source coupling problems in a two-dimensional static electromagnetic field is realized; explicit annotation data is not needed, and the method is suitable for modeling and prediction of various static electromagnetic problems.

Description

Technical Field

[0001] The present invention relates to the cross-application of electromagnetic field numerical calculation and artificial intelligence technology, in particular to a physical information neural network solution method based on the Fourier mapping mechanism for modeling static two-dimensional electromagnetic problems, which is suitable for unsupervised modeling and accurate prediction of electromagnetic fields in multi-region, multi-medium, and multi-source field problems. Background Art

[0002] In the design and analysis of electromagnetic devices, the accuracy of the distribution of electrostatic and magnetic fields plays a decisive role in performance evaluation. While the traditional finite element method (FEM) can solve electromagnetic partial differential equations relatively accurately, its computational efficiency and accuracy are affected by the refinement of the discrete mesh when dealing with complex boundary and multi-medium coupled field problems. In recent years, physical information neural network methods have garnered widespread attention in the field of intelligent neural network solutions for partial differential equations due to their meshless, physical constraint-driven solution advantages.

[0003] However, traditional PINNs methods are prone to training difficulties when dealing with high-gradient regions, such as sudden changes in the dielectric or the introduction of current sources. This can lead to slow convergence or local inaccuracies. This is particularly evident in the discontinuity of the physical field solutions for multi-region coupled electromagnetic problems. Therefore, improving the network's adaptability to physical boundary characteristics and its training stability is an urgent issue that needs to be addressed. Summary of the Invention

[0004] Technical problem: The purpose of the present invention is to provide a static electromagnetic calculation method based on a physical information neural network of Fourier mapping, aiming to solve the following technical problems: 1. Insufficient ability to express jump characteristics in electromagnetic field problems; 2. Discontinuity and physical inconsistency of network prediction values in multi-region medium environments; 3. Gradient instability caused by medium differences during network optimization; 4. Insufficient solution accuracy in key areas.

[0005] Technical solution: The present invention is a static electromagnetic calculation method based on the physical information neural network of Fourier mapping, which introduces the Fourier mapping mechanism between the input layer and the hidden layer to map the input spatial features to the high-frequency Fourier feature space; construct multiple sub-region neural networks, each of which inputs spatial feature information and outputs the vector magnetic potential A in the region. Z , normal magnetic field strength H n , tangential magnetic field strength H t ;

[0006] Construct vector magnetic potential A at the interface of the region Z and the tangential magnetic field strength H t The loss of physical continuity can be achieved by coupling physical information across networks;

[0007] A combined unsupervised objective function of electromagnetic partial differential equations (PDEs), residual loss, continuity loss, and boundary condition loss is constructed to train all sub-networks to obtain global electromagnetic field prediction results that meet physical constraints.

[0008] The Fourier mapping mechanism includes a combination of sine and cosine functions, and the frequency parameters can be customized to adjust and scale, and the frequency parameters do not participate in network updates.

[0009] The Fourier mapping is as follows: a Fourier mapping module is introduced between the input layer and the first hidden layer, the input coordinates are expanded into high-frequency spatial feature vectors through the sine and cosine functions of the frequency parameters, and the network is converted from the spatial domain to the frequency domain. It is specifically expressed by the following formula:

[0010]

[0011] Z (1) =φ(Φ(X)W (1) +b (1) )

[0012] Z (2) =φ(Z (1) W (2) +b (2) )

[0013] Z (L) =φ(Z (L-1) W (L) +b (L) )

[0014]

[0015] Among them, Φ(X) is the feature result after Fourier mapping, H is the frequency matrix, d is the first dimension of the frequency matrix, and the number of frequency extensions can be adjusted according to actual needs; L is the number of hidden layers in the network; W (i) 、b (i) are the weight and bias of the fully connected neural network in layer i; φ is the nonlinear activation function; Z (i) is the output result of the i-th layer network; A is the output layer result, the shape of A is n×k, n is the number of input features, and k is the target output dimension.

[0016] The construction of multiple sub-region neural networks is as follows: a three-output neural network is deployed in each sub-region, and the output physical quantities are vector magnetic potential A z , normal magnetic field strength H n , tangential magnetic field strength H t , the regions are divided and the network models are deployed independently according to the distribution of incentives.

[0017] The vector magnetic potential A Z , normal magnetic field strength H n , tangential magnetic field strength H t It is used to directly construct the distribution of vector magnetic potential and magnetic field components for the derivation of magnetic induction intensity.

[0018] The electromagnetic field PDEs residual loss: in Maxwell's equations, the curl of the magnetic induction intensity H and the vector magnetic potential A z The curl of is used as the electromagnetic control equation, and the residual after automatic differentiation is calculated;

[0019]

[0020] Where J is the current density, B is the magnetic field strength, in two-dimensional problems A only has a component in the z direction, H is the magnetic field strength, is the curl operator.

[0021] The boundary condition loss includes Dirichlet boundary condition (first type boundary condition) or Neumann boundary condition (second type boundary condition) type boundary condition loss, directly utilizing the forward output structure of the three-output model, reducing the differential link of the single-output model in boundary condition processing, and reducing the computational cost.

[0022] The continuity loss is the cross-network continuity loss. For adjacent regions, the vector magnetic potential A is constrained on the interface. z , tangential component H t The continuity of the whole physical field solution is ensured.

[0023] The training is as follows: using an optimizer to minimize the weighted sum of the above losses as the optimization goal, and jointly optimizing all sub-networks.

[0024] The global electromagnetic field prediction results are ultimately output as the vector magnetic potential and magnetic field intensity distribution within the entire computational domain, and further calculations of engineering physical quantities such as magnetic induction intensity B, magnetic field line distribution, and electromagnetic force density are performed to enable electromagnetic performance evaluation and optimized design in the preliminary design phase of electromagnetic equipment such as motors, transformers, and relays.

[0025] Beneficial effects: Enhance the model's high-frequency information expression capability through Fourier mapping, improve accuracy and convergence speed; z 、H n 、H tThree physical quantities are explicitly modeled to address the impact of sudden changes in the medium on the network optimization process. An interface continuity strategy ensures consistency in physical field solutions across multiple networks, adapting to multi-medium coupled structures. An unsupervised approach reduces reliance on high-precision simulation data, improving the method's applicability and scalability. This invention provides a novel deep learning solution strategy for modeling static electromagnetic problems. It possesses both theoretical value and engineering applicability, and can be widely applied to the electromagnetic performance prediction and optimization design of devices such as motors, transformers, and relays. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 This is the overall architecture diagram of the present invention.

[0027] Figure 2 This is a diagram of the network forward process with the Fourier mapping mechanism introduced.

[0028] Figure 3 Schematic diagram of the PDEs loss construction for a three-output network. DETAILED DESCRIPTION

[0029] The static electromagnetic calculation method of the present invention based on the physical information neural network of Fourier mapping:

[0030] A Fourier mapping mechanism is introduced between the input layer and the hidden layer to map the input spatial features to the high-frequency Fourier feature space; multiple sub-region neural networks are constructed, each network input is spatial feature information, and the output is the vector magnetic potential A in the region. Z , normal magnetic field strength H n , tangential magnetic field strength H t ;

[0031] Construct vector magnetic potential A at the interface of the region Z and the tangential magnetic field strength H t The loss of physical continuity can be achieved by coupling physical information across networks;

[0032] Construct an unsupervised objective function of electromagnetic partial differential, residual loss, continuity loss, and boundary condition loss, train all sub-networks, and obtain global electromagnetic field prediction results that meet physical constraints.

[0033] Taking a two-dimensional static problem as an example, in the Cartesian coordinate system, the input feature dimension is 2, that is, The output dimension is three-dimensional, that is The frequency matrix dimension is d=1, the 3-layer hidden layer network has L=3, and the number of frequency extensions is 64, i.e. d=64.

[0034] Step 1:

[0035] Construct Fourier feature map: Introduce the Fourier map module between the input layer and the first hidden layer, expand the input coordinates into high-frequency spatial feature vectors through the sine and cosine functions of the frequency parameters, and transform the network from the spatial domain to the frequency domain, thereby improving the network's high-frequency expression ability and detail extraction. It can be expressed by the following formula:

[0036]

[0037] Z (1) =φ(Φ(X)W (1) +b (1) )

[0038] Z (2) =φ(Z (1) W (2) +b (2) )

[0039] Z (3) =φ(Z (2) W (3) +b (3) )

[0040]

[0041] Among them, X is n two-dimensional input features, the frequency matrix H is 2×64, and after Fourier mapping transformation After 3 layers of linear transformation (·)W (i) +b (i) Together with the nonlinear transformation φ(·), the final target output A contains three target physical quantities.

[0042] Step 2:

[0043] Deployment of multi-output network construction and solution model: A three-output neural network is deployed separately in each sub-area, and the output physical quantities are vector magnetic potential A z , normal magnetic field strength H n , tangential magnetic field strength H t , the regions are divided and the network models are deployed independently according to the distribution of incentives.

[0044] Step 3:

[0045] Loss function design: For each sub-network, the following three types of loss functions are constructed:

[0046] 1. Electromagnetic field partial differential residual loss: Based on Maxwell's equations, the curl of the magnetic induction intensity H and the vector magnetic potential A z The curl of is used as the electromagnetic control equation, and the residual after automatic differentiation is calculated;

[0047]

[0048] Where J is the current density, B is the magnetic field intensity, and in two-dimensional problems, A only has a component in the z direction.

[0049] 2. Boundary condition loss: This includes Dirichlet or type boundary condition loss, which can directly use the forward output structure of the three-output model, reducing the differentiation step of the single-output model when processing boundary conditions, and thus reducing the computational cost.

[0050] 3. Cross-network continuity loss: For adjacent regions, constrain the vector magnetic potential A on the interface z , tangential component H t The continuity of the whole physical field solution is ensured.

[0051] Step 4:

[0052] Training and optimization process: Use the optimizer to minimize the weighted sum of the above losses as the optimization goal, and jointly optimize all sub-networks.

[0053] Step 5:

[0054] Model output and application: The final output is the vector magnetic potential and magnetic field intensity distribution within the entire computational domain. This can be used to further calculate engineering physical quantities such as magnetic induction intensity B, magnetic field line distribution, and electromagnetic force density, enabling electromagnetic performance evaluation and optimization design in the preliminary design phase of electromagnetic equipment such as motors, transformers, and relays.

Claims

1. A static electromagnetic calculation method based on a physical information neural network of Fourier mapping, characterized in that: A Fourier mapping mechanism is introduced between the input layer and the hidden layer to map the input spatial features to the high-frequency Fourier feature space; multiple sub-region neural networks are constructed, each network input is spatial feature information, and the output is the vector magnetic potential A in the region. Z , normal magnetic field strength H n , tangential magnetic field strength H t ; Construct vector magnetic potential A at the interface of the region Z and the tangential magnetic field strength H t The loss of physical continuity can be achieved by coupling physical information across networks; Construct an unsupervised objective function of electromagnetic partial differential, residual loss, continuity loss, and boundary condition loss, train all sub-networks, and obtain global electromagnetic field prediction results that meet physical constraints.

2. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The Fourier mapping mechanism includes a combination of sine and cosine functions, and the frequency parameters can be customized to adjust and scale, and the frequency parameters do not participate in network updates.

3. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 2, characterized in that: The Fourier mapping is as follows: a Fourier mapping module is introduced between the input layer and the first hidden layer, the input coordinates are expanded into high-frequency spatial feature vectors through the sine and cosine functions of the frequency parameters, and the network is converted from the spatial domain to the frequency domain. It is specifically expressed by the following formula: Z (1) =φ(Φ(X)W (1) +b (1) ) WITH (2) =φ(Z (1) IN (2) +b (2) ) WITH (L) =φ(Z (L-1) IN (L) +b (L) ) Among them, Φ(X) is the feature result after Fourier mapping, H is the frequency matrix, d is the first dimension of the frequency matrix, and the number of frequency extensions can be adjusted according to actual needs; L is the number of hidden layers in the network; W (i) 、b (i) are the weight and bias of the fully connected neural network in layer i; φ is the nonlinear activation function; Z (i) is the output result of the i-th layer network; A is the output layer result, the shape of A is n×k, n is the number of input features, and k is the target output dimension.

4. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The construction of multiple sub-region neural networks is as follows: a three-output neural network is deployed in each sub-region, and the output physical quantities are vector magnetic potential A z , normal magnetic field strength H n , tangential magnetic field strength H t , the regions are divided and the network models are deployed independently according to the distribution of incentives.

5. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The vector magnetic potential A Z , normal magnetic field strength H n , tangential magnetic field strength H t It is used to directly construct the distribution of vector magnetic potential and magnetic field components for the derivation of magnetic induction intensity.

6. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The electromagnetic field PDEs residual loss: in Maxwell's equations, the curl of the magnetic induction intensity H and the vector magnetic potential A z The curl of is used as the electromagnetic control equation, and the residual after automatic differentiation is calculated; Where J is the current density, B is the magnetic field strength, in two-dimensional problems A only has a component in the z direction, H is the magnetic field strength, is the curl operator.

7. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The boundary condition loss includes Dirichlet boundary condition (first type boundary condition) or Neumann boundary condition (second type boundary condition) type boundary condition loss, directly utilizing the forward output structure of the three-output model, reducing the differential link of the single-output model in boundary condition processing, and reducing the computational cost.

8. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The continuity loss is the cross-network continuity loss. For adjacent regions, the vector magnetic potential A is constrained on the interface. z , tangential component H t The continuity of the whole physical field solution is ensured.

9. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The training is as follows: using an optimizer to minimize the weighted sum of the above losses as the optimization goal, and jointly optimizing all sub-networks.

10. The static electromagnetic calculation method based on the physical information neural network of Fourier mapping according to claim 1, characterized in that: The global electromagnetic field prediction results are ultimately output as the vector magnetic potential and magnetic field intensity distribution within the entire computational domain, and further calculations of engineering physical quantities such as magnetic induction intensity B, magnetic field line distribution, and electromagnetic force density are performed to enable electromagnetic performance evaluation and optimized design in the preliminary design phase of electromagnetic equipment such as motors, transformers, and relays.

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