A penalty-free calculation method for motor interface problems based on physical information neural network

Abstract

The invention discloses a penalty-free calculation method for motor interface problems based on a physical information neural network. The calculation method comprises the following steps: S1, establishing a computational physical model of a motor; S2, preprocessing a partial differential equation by dimensionless transformation; S3, constructing an energy functional by a variational principle; S4, using vertex coordinate information as an intermediate representation of a geometric object to achieve gridless sampling; S5, constructing a neural network; S6, enhancing interface recognition; S7, ensuring that boundary conditions are naturally satisfied; S8, training a neural network, selecting Adam as an optimizer, and selecting Tanh as a linear layer activation function; the invention uses a neural network to parameterize partial differential equation solutions, and solves interface problems by finding the minimum value of a corresponding optimization problem. Meanwhile, a grid sampler only needs to act on a domain and a boundary, and does not need to perform special processing on the interface within the domain, thereby effectively reducing the amount of calculation and facilitating use.

Description

Technical Field

[0001] The invention belongs to the technical field of motor electromagnetic field calculation, and in particular is a penalty-free calculation method for motor interface problems based on physical information neural network. Background Art

[0002] Numerical calculation of electromagnetic fields plays an important role in the fields of electricity, electronics, and communications. However, traditional numerical calculation methods take a long time to solve and are difficult to meet practical needs. Some studies have tried to use data-driven models to calculate electromagnetic fields, such as the Kriging method and the U-net convolutional neural network method. Such methods often require a large amount of high-precision data to ensure the accuracy of the model, and due to the lack of physical constraints, they may produce results that do not conform to physical laws when dealing with physical problems.

[0003] Physics-Informed Neural Networks (PINNs) can accurately calculate physical field solutions without the need for labeled data, while providing parameterized solutions and supporting transfer learning, especially showing potential in the fast calculation of electromagnetic fields. Although PINNs have been initially applied in electromagnetic field analysis, they still have some significant limitations when dealing with objects such as motors that are composed of multiple materials and have complex interfaces and boundary conditions, which affect the accuracy of their analysis. First, since neural networks mainly perform smooth approximations, they perform poorly when dealing with non-smooth or discontinuous characteristics at the interface, especially under conditions of high contrast coefficients or weak discontinuities. Second, due to the spectral bias of neural networks, it is difficult for the network to capture high-frequency changes at the interface. In addition, the adjustment process of hyperparameters is complicated, especially the weight coefficients of the penalty terms of interface conditions and boundary conditions are difficult to properly balance, which increases the difficulty of optimization. In actual engineering applications, these problems lead to technical defects such as high computational complexity and low solution efficiency.

[0004] Therefore, the applicant proposed a penalty-free calculation method for motor interface problems based on physical information neural network. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides a penalty-free calculation method for motor interface problems based on physical information neural network. The method uses neural network to parameterize the partial differential equation solution and solves the interface problem by finding the minimum value of the corresponding optimization problem. At the same time, the grid sampler only needs to act on the domain and boundary, and no special processing is required for the interface within the domain, which effectively reduces the amount of calculation and facilitates use.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A penalty-free calculation method for motor interface problems based on a physical information neural network, the calculation method comprising the following steps:

[0008] S1. Establish a computational physics model of a motor;

[0009] S2, preprocessing partial differential equations using dimensionless transformation;

[0010] S3. Construct energy functional by variational principle;

[0011] S4, using vertex coordinate information as an intermediate representation of geometric objects to achieve gridless sampling;

[0012] S5. Construct a neural network;

[0013] S6, Enhance interface recognition;

[0014] S7. Ensure that boundary conditions are naturally satisfied;

[0015] S8. Train the neural network, select Adam as the optimizer, and select Tanh as the linear layer activation function.

[0016] As a further solution of the present invention: the specific steps of step S1 are as follows:

[0017] The structure of the physical model includes the stator, air gap, rotor, inter-pole air area and excitation winding area. The no-load magnetic field is only established by the excitation current and is not affected by the armature reaction. The model is analyzed as a two-dimensional field. The magnetic permeability of the ferromagnetic medium is constant in the calculation.

[0018] The Poisson equation is established in the field area as follows:

[0019] (1)

[0020] is the current density function perpendicular to the field plane. In the field winding area, it corresponds to the no-load excitation current density of the motor field winding, and is zero in other areas. is the magnetic permeability of the material in the region, is the magnetic vector potential, is the Hamiltonian operator;

[0021] Both the rotor and the stator are made of ferromagnetic materials. The magnetic field cannot pass through the inner boundary of the rotor and the outer boundary of the stator yoke. Under two-dimensional conditions, the boundary conditions can be written as:

[0022] (2).

[0023] As a further solution of the present invention: the specific steps of step S2 are as follows:

[0024] For a known static magnetic field PDE current density , the dimension can be expressed as:

[0025] (3)

[0026] in, , As the length scale, select the region in Direction and Maximum distance in direction , is the magnetic permeability scale, is the current density scale, the vector magnetic potential The scale for , are the coordinates and magnetic permeability corresponding to different regions respectively; is the corresponding physical quantity after dimensionless conversion,

[0027] The resulting dimensionless equation is:

[0028] (4)

[0029] (5)

[0030] In the formula, is the dimensionless Hamiltonian operator.

[0031] As a further solution of the present invention: the specific steps of step S3 are as follows:

[0032] Considering the variational principle, given the magnetostatic equation (4), the corresponding variational problem can be derived as:

[0033] (6)

[0034] In the formula, For Sobolev space, by solving the energy functional of Poisson's equation, the interface conditions can be automatically satisfied. is the dimensionless region. The solution of equation (6) can be used to find the minimized energy functional through equation (7) to find the magnetic vector potential solution that needs to be solved;

[0035] (7).

[0036] As a further solution of the present invention: the specific steps of step S4 are as follows:

[0037] First, extract vertex information from the stereolithography STL, organize the point set of all vertices and the face set of triangle patches in the STL model, extract the boundary elements of all triangles from the face set, and eliminate duplicate edges. Then, distinguish the inner and outer boundary edges, and sort the edges in clockwise and counterclockwise directions respectively to correctly define the direction of the mesh boundary. Finally, store the vertices in the order of counterclockwise outer boundary and clockwise inner boundary. Then use the vertex information to calculate the signed distance field SDF of different geometric entities. The SDF of the line segment can be obtained by the following equation:

[0038] (8)

[0039] (9)

[0040] (10)

[0041] in, and are the two endpoints of the line segment, are the parameters of the parametric equation, is any point on the line. is a point in two-dimensional space, used to calculate its distance from the polygon or the SDF between line segments, Equation (9) gives the point and The distance between To find the second norm, the SDF can be calculated using equation (10), which yields the following result:

[0042] (11)

[0043] (12)

[0044] (13)

[0045] The magnitude of the SDF of a convex polygon can be obtained by minimizing the SDFs of all its straight line segments. Then the sign of the SDF is determined by the wind number algorithm. The wind number of point Q relative to the polygon is calculated. If the wind number is non-zero, then point Q is inside the polygon. The wind number is defined as the point Q emitting a ray in any direction. ,for In the positive direction of the axis, if the ray intersects the edge on the polygon and blows from the wind downward to upward, add 1, otherwise subtract 1. The geometric SDF function is obtained through the above steps. Next, the regular area is sampled iteratively until the required number of samples is obtained.

[0046] As a further solution of the present invention: the specific steps of step S5 are as follows:

[0047] The input of the neural network is the coordinate value of the sampling point, and the coordinate after scaling is obtained by dimensionless transformation according to step 2. and The output of the neural network is the magnetic vector position of each sampling point. , in order to make the neural network learn the high-frequency area faster, the coordinate values ​​are firstly transformed by Fourier transform, as shown in formula (14):

[0048] (14)

[0049] in, is the input of the network, B satisfies the two-dimensional normal distribution with mean 0, covariance matrix (15), and correlation coefficient 0.

[0050] (15)

[0051] Among them, X, Y are variables, that is, the sampling points , coordinate, for and The variance of .

[0052] As a further solution of the present invention: the specific steps of step S6 are as follows:

[0053] By introducing a level set function to accurately classify spatial coordinates into subdomains, the ability of neural networks to effectively approximate functions is improved, and a continuous level set function is constructed through a differentiable distance field. , where the zero level set defines the location of the interface between different media, within the geometric region except for vertices and boundaries, The calculation method of the differentiable distance field of the vertex polygon is as follows:

[0054] in (16)

[0055] (17)

[0056] (18)

[0057] (19)

[0058] In the formula, for To the top The Euclidean distance of and are the dot product and cross product of vectors, respectively, and polygons have vertices, The first Vertices, is a coordinate point in the region, Indicates the definition symbol;

[0059] By using the level set function Added to the input of the neural network, and the energy functional in equation (6) is modified to form the following PDE:

[0060] (20)

[0061] In the formula, To solve for the square of the two norm.

[0062] As a further solution of the present invention: the specific steps of step S7 are as follows:

[0063] The boundary conditions are hard constrained by differentiable distance fields to ensure that the boundary conditions are naturally satisfied, thereby solving the gradient pathological problem that occurs when PINNs optimize boundary conditions. For non-uniform or mixed boundary conditions, the region The upper boundary , defining the upper shard boundary in The corresponding boundary conditions are , by generalized interpolation we can get exist The interpolation value is:

[0064] (twenty one)

[0065] In the formula, The calculation method is as follows:

[0066] (twenty two)

[0067] In the formula, For the The boundaries of the slices, and Used as index in operations, For the shard boundary The associated differentiable distance field, is the number of shard boundaries, Used to construct mixed boundary conditions when When, function exist Interpolation, when When, function and exist Interpolation on, is a coordinate point in the region;

[0068] By imposing hard constraints on boundary conditions, the structure of the neural network solution can be expressed as:

[0069] (twenty three)

[0070] In the formula, For the geometric area The differentiable distance field of is the solution of the modified neural network, is the neural network solution before transformation, is the boundary condition;

[0071] Select Function , so that it satisfies the Dirichlet boundary conditions of the equation. Due to the zero boundary conditions as shown in equation (2), the constructor , so that it is zero at the boundary and positive inside, and the neural network result is corrected by formula (23) to naturally meet the boundary conditions and avoid the optimization problem caused by considering the boundary condition loss penalty term.

[0072] As a further solution of the present invention: the specific steps of step S8 are as follows:

[0073] Use the traditional finite element method to solve the model and obtain the finite element solution , the accuracy of the trained model is evaluated by the L2 relative error of the test set, as shown in formula (24):

[0074] (twenty four)

[0075] In the formula, is the neural network output, is the finite element solution, is the L2 relative error, To find the second norm.

[0076] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention introduces the variational principle to reformulate the control equation into a variational form, so that the interface conditions are naturally satisfied and the penalty term of the interface loss is eliminated; a differentiable distance field is added to the input feature to overcome the smoothness of the neural network, and the interface information is used as a priori of the neural network to enhance the recognition of the interface; the differentiable distance field in complex geometry is calculated, and the neural network is corrected using the differentiable distance field, which can naturally satisfy the boundary conditions and eliminate the penalty term of the boundary loss; the solution of the partial differential equation is parameterized by a neural network, and the interface problem is solved by finding the minimum value of the corresponding optimization problem. At the same time, the grid sampler only needs to act on the domain and the boundary, and no special processing is required for the interface within the domain, which effectively reduces the amount of calculation and is convenient for use. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the prior art descriptions. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the structures shown in these drawings without creative work. Among them:

[0078] Figure 1 A cross-sectional view of a computational physical model of the motor of the present invention;

[0079] Figure 2 Constructing a continuous level set function for the differentiable distance field of the present invention Schematic diagram of

[0080] Figure 3 The microscopic distance field of the present invention Schematic diagram of

[0081] Figure 4 A schematic diagram of the magnetic vector potential obtained by the finite element solution of the present invention;

[0082] Figure 5 A schematic diagram of the magnetic vector potential solved by the neural network of the present invention;

[0083] Figure 6 Schematic diagram of the magnetic vector potential solved for the existing point-by-point error solution;

[0084] Figure 7 is a schematic diagram of the wind number algorithm of the present invention;

[0085] Figure 8 Schematic diagram of the differentiable distance field of the present invention. DETAILED DESCRIPTION

[0086] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0087] Example

[0088] See also Figure 1-Figure 8 As shown, the present invention provides a penalty-free calculation method for motor interface problems based on physical information neural network, and the calculation method comprises the following steps:

[0089] 1. Establish a computational physics model of a motor:

[0090] The structure of the physical model includes the stator, air gap, rotor, inter-pole air area and excitation winding area. The no-load magnetic field is only established by the excitation current and is not affected by the armature reaction. The calculation model of the motor is based on a 150 kW, 4-pole salient synchronous motor, such as Figure 1 As shown, it is a salient pole synchronous motor, which is composed of ferromagnetic material (steel), conductor (copper) and surrounding air. The current is 270A, the outer diameter of the stator core is 49cm, the inner diameter of the stator core is 33.6cm, the pole pitch is 26.39cm, the stator core length is 26cm, the air gap spacing is 0.15cm, and the magnetic permeability of the ferromagnetic material is higher than that of the air. 1000 times higher, assuming that the conductor material and air can be regarded as the same material based on their linear magnetic properties, i.e. , the model is analyzed as a two-dimensional field, and the magnetic permeability of the ferromagnetic medium is constant in the calculation:

[0091] The Poisson equation is established in the field area as follows:

[0092] (1)

[0093] is the current density function perpendicular to the field plane. In the field winding area, it corresponds to the no-load excitation current density of the motor field winding, and is zero in other areas. is the magnetic permeability of the material in the region, is the magnetic vector potential, is the Hamiltonian operator;

[0094] Both the rotor and the stator are made of ferromagnetic materials. The magnetic field cannot pass through the inner boundary of the rotor and the outer boundary of the stator yoke. Under two-dimensional conditions, the boundary conditions can be written as:

[0095] (2).

[0096] 2. Perform dimensionless processing on the motor parameters to obtain the dimensionless equation that needs to be solved:

[0097] For a known static magnetic field PDE current density , the dimension can be expressed as:

[0098] (3)

[0099] in, , As the length scale, select the region in Direction and Maximum distance in direction , is the magnetic permeability scale, is the current density scale, the vector magnetic potential The scale for , are the coordinates and magnetic permeability corresponding to different regions respectively; is the corresponding physical quantity after dimensionless conversion,

[0100] The resulting dimensionless equation is:

[0101] (4)

[0102] (5)

[0103] In the formula, is the dimensionless Hamiltonian operator.

[0104] 3. Construct energy functional through variational principle:

[0105] Considering the variational principle, given the magnetostatic equation (4), the corresponding variational problem can be derived as:

[0106] (6)

[0107] In the formula, For Sobolev space, by solving the energy functional of Poisson's equation, the interface conditions can be automatically satisfied. is the dimensionless region. The solution of equation (6) can be used to find the minimized energy functional through equation (7) to find the magnetic vector potential solution that needs to be solved;

[0108] (7).

[0109] 4. Use vertex coordinate information as the intermediate representation of geometric objects to implement gridless sampling to iteratively sample regular areas. Use the Holden sequence quasi-random method for sampling. 5300 points are sampled for the motor air gap, 600 points for the winding, 3000 points for the rotor core, and 3000 points for the stator core:

[0110] First, extract vertex information from the stereolithography STL, organize the point set of all vertices and the face set of triangle patches in the STL model, extract the boundary elements of all triangles from the face set, and eliminate duplicate edges. Then, distinguish the inner and outer boundary edges, and sort the edges in clockwise and counterclockwise directions respectively to correctly define the direction of the mesh boundary. Finally, store the vertices in the order of counterclockwise outer boundary and clockwise inner boundary. Then use the vertex information to calculate the signed distance field SDF of different geometric entities. The SDF of the line segment can be obtained by the following equation:

[0111] (8)

[0112] (9)

[0113] (10)

[0114] in, and are the two endpoints of the line segment, are the parameters of the parametric equation, is any point on the line. is a point in two-dimensional space, used to calculate its distance from the polygon or the SDF between line segments, Equation (9) gives the point and The distance between To find the second norm, the SDF can be calculated using equation (10), which yields the following result:

[0115] (11)

[0116] (12)

[0117] (13)

[0118] The magnitude of the SDF of a convex polygon can be obtained by minimizing the SDFs of all its straight line segments. Then the sign of the SDF is determined by the wind number algorithm. The wind number of point Q relative to the polygon is calculated. If the wind number is non-zero, then point Q is inside the polygon. The wind number is defined as the point Q emitting a ray in any direction. ,for The positive direction of the axis. If the ray intersects the edge of the polygon and blows from the bottom to the top, add 1, otherwise subtract 1. The definition of wind direction is as follows Figure 7 As shown, the geometric SDF function is obtained through the above steps, and then the regular area is sampled iteratively until the required number of samples is obtained.

[0119] 5. Build a fully connected neural network, select Tanh as the activation function, the grid depth is 3, and each layer has 512 neurons:

[0120] The input of the neural network is the coordinate value of the sampling point, and the coordinate after scaling is obtained by dimensionless transformation according to step 2. and The output of the neural network is the magnetic vector position of each sampling point. , in order to make the neural network learn the high-frequency area faster, the coordinate values ​​are firstly transformed by Fourier transform, as shown in formula (14):

[0121] (14)

[0122] in, is the input of the network, B satisfies the two-dimensional normal distribution with mean 0, covariance matrix (15), and correlation coefficient 0.

[0123] (15)

[0124] Among them, X, Y are variables, that is, the sampling points , coordinate, for and The variance of .

[0125] 6. In order to enhance interface recognition, a continuous level set function is constructed through a differentiable distance field ,like Figure 2 As shown, the level set Add to the input of the neural network:

[0126] By introducing a level set function to accurately classify spatial coordinates into subdomains, the ability of neural networks to effectively approximate functions is improved, and a continuous level set function is constructed through a differentiable distance field. , where the zero level set defines the location of the interface between different media, such as Figure 8 As shown, in the geometric area excluding vertices and boundaries, The calculation method of the differentiable distance field of the vertex polygon is as follows:

[0127] in (16)

[0128] (17)

[0129] (18)

[0130] (19)

[0131] In the formula, for To the top The Euclidean distance of and are the dot product and cross product of vectors, respectively, and polygons have vertices, The first Vertices, is a coordinate point in the region, Indicates the definition symbol;

[0132] By using the level set function Added to the input of the neural network, and the energy functional in equation (6) is modified to form the following PDE:

[0133] (20)

[0134] In the formula, To solve for the square of the two norm.

[0135] 7. To ensure that the boundary conditions are naturally satisfied, construct a differentiable distance field ,like Figure 3 As shown, hard constraints are imposed on the boundary conditions and the solution is reformulated:

[0136] The boundary conditions are hard constrained by differentiable distance fields to ensure that the boundary conditions are naturally satisfied, thereby solving the gradient pathological problem that occurs when PINNs optimize boundary conditions. For non-uniform or mixed boundary conditions, the region The upper boundary , defining the upper shard boundary in The corresponding boundary conditions are , by generalized interpolation we can get exist The interpolation value is:

[0137] (twenty one)

[0138] In the formula, The calculation method is as follows:

[0139] (twenty two)

[0140] In the formula, For the The boundaries of the slices, and Used as index in operations, For the shard boundary The associated differentiable distance field, is the number of shard boundaries, Used to construct mixed boundary conditions when When, function exist Interpolation, when When, function and exist Interpolation on, is a coordinate point in the region;

[0141] By imposing hard constraints on boundary conditions, the structure of the neural network solution can be expressed as:

[0142] (twenty three)

[0143] In the formula, For the geometric area The differentiable distance field of is the solution of the modified neural network, is the neural network solution before transformation, is the boundary condition;

[0144] Select Function , so that it satisfies the Dirichlet boundary conditions of the equation. Due to the zero boundary conditions as shown in equation (2), the constructor , so that it is zero at the boundary and positive inside, and the neural network result is corrected by formula (23), which naturally satisfies the boundary conditions and avoids the optimization problem caused by considering the boundary condition loss penalty term.

[0145] 8. Use the traditional finite element method to solve the model and obtain the finite element solution As the test set, the neural network is trained, the optimizer is Adam, and the L2 relative error is used to evaluate the accuracy. The average value of the test set is 0.34%, and the accuracy meets the requirements. The magnetic vector potentials obtained by the finite element solution, neural network solution and point-by-point error solution are as follows: Figure 4 , Figure 5 and Figure 6 As shown:

[0146] Use the traditional finite element method to solve the model and obtain the finite element solution , the accuracy of the trained model is evaluated by the L2 relative error of the test set, as shown in formula (24):

[0147] (twenty four)

[0148] In the formula, is the neural network output, is the finite element solution, is the L2 relative error, To find the second norm.

[0149] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A penalty-free calculation method for motor interface problems based on physical information neural network, characterized by: The calculation method includes the following steps: S1. Establish a computational physics model of a motor; S2, preprocessing partial differential equations using dimensionless transformation; S3. Construct energy functional by variational principle; S4, using vertex coordinate information as an intermediate representation of geometric objects to achieve gridless sampling; S5. Construct a neural network; S6, Enhance interface recognition; S7. Ensure that boundary conditions are naturally satisfied; S8, train the neural network, select Adam as the optimizer, and select Tanh as the linear layer activation function; Step S1 comprises: The Poisson equation is established in the field area as follows: （1） is the current density function perpendicular to the field plane. In the field winding area, it corresponds to the no-load excitation current density of the motor field winding, and is zero in other areas. is the magnetic permeability of the material in the region, is the magnetic vector potential, is the Hamiltonian operator; Both the rotor and the stator are made of ferromagnetic materials. The magnetic field cannot pass through the inner boundary of the rotor and the outer boundary of the stator yoke. Under two-dimensional conditions, the boundary conditions are written as: （2）； The specific steps of step S3 are as follows: Considering the variational principle, given the equation obtained in step S2, the corresponding variational problem can be derived as: （6） In the formula, For Sobolev space, by solving the energy functional of Poisson's equation, the interface conditions can be automatically satisfied. is the dimensionless region. The solution of equation (6) can be used to find the minimized energy functional through equation (7) to find the magnetic vector potential solution that needs to be solved; （7）； By introducing a level set function to accurately classify spatial coordinates into subdomains, the ability of neural networks to effectively approximate functions is improved. It is added to the input of the neural network and the energy functional in equation (6) is modified to form a PDE.

2. The non-penalty calculation method for motor interface problems based on physical information neural network according to claim 1 is characterized by: The structure of the physical model in step S1 includes a stator, an air gap, a rotor, an inter-pole air region and an excitation winding region. The no-load magnetic field is established only by the excitation current and is not affected by the armature reaction. The model is analyzed as a two-dimensional field. In the calculation, the magnetic permeability of the ferromagnetic medium is constant.

3. The non-penalty calculation method for motor interface problems based on physical information neural network according to claim 2 is characterized by: The specific steps of step S2 are as follows: For a known static magnetic field PDE current density , the dimension can be expressed as: （3） in, , As the length scale, select the region in Direction and Maximum distance in direction , is the magnetic permeability scale, is the current density scale, the vector magnetic potential The scale for , are the coordinates and magnetic permeability corresponding to different regions respectively; is the corresponding physical quantity after dimensionless conversion, The resulting dimensionless equation is: （4） （5） In the formula, is the dimensionless Hamiltonian operator.

4. The method for calculating the motor interface problem without penalty based on physical information neural network according to claim 3 is characterized by: The specific steps of step S4 are as follows: First, extract vertex information from the stereolithography STL, organize the point set of all vertices and the face set of triangle patches in the STL model, extract the boundary elements of all triangles from the face set, and eliminate duplicate edges. Then, distinguish the inner and outer boundary edges, and sort the edges in clockwise and counterclockwise directions respectively to correctly define the direction of the mesh boundary. Finally, store the vertices in the order of counterclockwise outer boundary and clockwise inner boundary. Then use the vertex information to calculate the signed distance field SDF of different geometric entities. The SDF of the line segment can be obtained by the following equation: （8） （9） （10） in, and are the two endpoints of the line segment, are the parameters of the parametric equation, is any point on the line. is a point in two-dimensional space, used to calculate its distance from the polygon or the SDF between line segments, Equation (9) gives the point and The distance between To find the second norm, the SDF can be calculated using equation (10), which yields the following result: （11） （12） （13） The magnitude of the SDF of a convex polygon can be obtained by minimizing the SDFs of all its straight line segments. Then the sign of the SDF is determined by the wind number algorithm. The wind number of point Q relative to the polygon is calculated. If the wind number is non-zero, then point Q is inside the polygon. The wind number is defined as the point Q emitting a ray in any direction. ,for In the positive direction of the axis, if the ray intersects the edge on the polygon and blows from the wind downward to upward, add 1, otherwise subtract 1. The geometric SDF function is obtained through the above steps. Next, the regular area is sampled iteratively until the required number of samples is obtained.

5. The method for calculating the motor interface problem without penalty based on physical information neural network according to claim 4 is characterized by: The specific steps of step S5 are as follows: The input of the neural network is the coordinate value of the sampling point, and the coordinate after scaling is obtained by dimensionless transformation according to step 2. and The output of the neural network is the magnetic vector position of each sampling point. In order to make the neural network learn the high-frequency area faster, the coordinate values ​​are firstly transformed by Fourier transform, as shown in formula (14): （14） in, is the input of the network, B satisfies the two-dimensional normal distribution with mean 0, covariance matrix (15), and correlation coefficient 0. （15） Among them, X, Y are variables, that is, the sampling points , coordinate, for and The variance of .

6. The non-penalty calculation method for motor interface problems based on physical information neural network according to claim 1 is characterized by: The specific steps of step S6 are as follows: Constructing continuous level set functions via differentiable distance fields , where the zero level set defines the location of the interface between different media, within the geometric region except for vertices and boundaries, The calculation method of the differentiable distance field of the vertex polygon is as follows: in (16) （17） （18） （19） In the formula, for To the top The Euclidean distance of and are the dot product and cross product of vectors, respectively, and polygons have vertices, The first Vertices, is a coordinate point in the region, Indicates the definition symbol; By using the level set function Added to the input of the neural network, and the energy functional in equation (6) is modified to form the following PDE: （20） In the formula, To solve for the square of the two norm.

7. The method for calculating the motor interface problem without penalty based on physical information neural network according to claim 6 is characterized by: The specific steps of step S7 are as follows: The boundary conditions are hard constrained by differentiable distance fields to ensure that the boundary conditions are naturally satisfied, thereby solving the gradient pathological problem that occurs when PINNs optimize boundary conditions. For non-uniform or mixed boundary conditions, the region The upper boundary , defining the upper shard boundary in The corresponding boundary conditions are , by generalized interpolation we can get exist The interpolation value is: （21） In the formula, The calculation method is as follows: （22） In the formula, For the The boundaries of the slices, and Used as index in operations, For the shard boundary The associated differentiable distance field, is the number of shard boundaries, Used to construct mixed boundary conditions when When, function exist Interpolation, when When, function and exist Interpolation on, is a coordinate point in the region; By imposing hard constraints on boundary conditions, the structure of the neural network solution can be expressed as: （23） In the formula, For the geometric area The differentiable distance field of is the solution of the modified neural network, is the neural network solution before transformation, is the boundary condition; Select Function , so that it satisfies the Dirichlet boundary conditions of the equation. Due to the zero boundary conditions as shown in equation (2), the constructor , so that it is zero at the boundary and positive inside, and the neural network result is corrected by formula (23) to naturally meet the boundary conditions and avoid the optimization problem caused by considering the boundary condition loss penalty term.

8. The non-penalty calculation method for motor interface problems based on physical information neural network according to claim 7 is characterized by: The specific steps of step S8 are as follows: Use the traditional finite element method to solve the model and obtain the finite element solution , the accuracy of the trained model is evaluated by the L2 relative error of the test set, as shown in formula (24): （24） In the formula, is the neural network output, is the finite element solution, is the L2 relative error, To find the second norm.

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