PINN prediction method for mechanical response of bulk superconductor under electromagnetic field distribution and electromagnetic load

Abstract

The invention discloses a PINN prediction method for electromagnetic field distribution and mechanical response under an electromagnetic load of a superconducting block. The method comprises the following steps of: 1, establishing a PINN model for predicting the electromagnetic field of the bulk superconductor under the excitation of an external magnetic field by taking a coordinate point and a time point as input and magnetic field intensity and electric field intensity as output; 2, training the PINN model for predicting the electromagnetic field of the superconducting block under the excitation of the external magnetic field, achieving the distribution prediction of the electromagnetic field, and obtaining the prediction of the electromagnetic force; 3, establishing a PINN model for predicting the mechanical response of the bulk superconductor under the action of the electromagnetic force, wherein the coordinate point is used as input, and the displacement field is used as output; and step 4, training the PINN model for the mechanical response prediction of the bulk superconductor under the action of the electromagnetic force, and finally realizing the electromagnetic field distribution and mechanical response prediction of the bulk superconductor. The method can predict the electromagnetic field distribution of the superconducting block and the mechanical response under the electromagnetic force, and does not need grid division and a large amount of experimental data.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-field coupling analysis of superconducting materials, and in particular to a PINN prediction method for electromagnetic field distribution and mechanical response of a superconducting bulk material under electromagnetic load. Background Art

[0002] Superconducting materials are widely used in high-tech fields such as energy, medical treatment and transportation, and have attracted much attention due to their unique electromagnetic properties such as zero resistance. 7-δ (REBCO) high-temperature superconducting materials are considered to be the preferred materials for various superconducting device applications in the future due to their high critical current density. In order to effectively guide the design and operation of superconducting devices with REBCO as the core, it is very important to carry out accurate and efficient prediction of the electromagnetic field distribution of REBCO superconductors and the mechanical response under electromagnetic loads.

[0003] In recent years, data-driven artificial intelligence technology has provided new opportunities for solving the problem of accurately predicting the magnetic field and mechanical response of superconductors. Based on finite element simulation data or experimental data, researchers have developed proxy models of machine learning methods such as multi-layer artificial neural networks to predict the central magnetic field of superconducting coil magnets and the stress in torsional superconducting tapes. However, data-driven technology also has significant limitations in the application of superconducting problems. Due to the difficulty in obtaining experimental data on superconducting materials, model training usually faces the problem of data scarcity; in addition, the "black box" nature of the data-driven proxy model makes it difficult to predict physical field variables outside the data set, which greatly limits its ability to predict the electromagnetic and mechanical behavior of superconductors.

[0004] The Physics-Informed Neural Network (PINN) successfully solves complex physical fields by embedding physical laws directly into the loss function. With its significant advantages of low data dependence, no need for meshing and inverse problem solving, PINN provides a new solution for solving the electromagnetic field distribution and mechanical response of superconductors. In summary, how to develop an efficient and accurate multi-field coupling analysis method for superconducting materials based on advanced physical information neural network technology to solve the problem of insufficient physical field prediction accuracy caused by the black box nature of traditional data-driven methods has become one of the key challenges of current research. Summary of the invention

[0005] In order to overcome the defects of the above prior art, the purpose of the present invention is to provide a PINN prediction method for the electromagnetic field distribution of superconducting bulk materials and the mechanical response under electromagnetic load, which predicts the electromagnetic field distribution of superconducting bulk materials and the mechanical response under electromagnetic force by considering the PINN calculation framework of Maxwell's equations, the minimum potential energy principle and the linear elastic mechanics equilibrium equation. The prediction method has the characteristics of not requiring grid division and not requiring a large amount of experimental data.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is:

[0007] The PINN prediction method of the electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic loads comprises the following steps;

[0008] Step 1: Based on the physical information neural network model of superconducting electric and magnetic fields, a PINN model is established to predict the electromagnetic field of superconducting bulk materials under external magnetic field excitation, with coordinate points and time points as inputs and magnetic field intensity and electric field intensity as outputs;

[0009] Step 2: training the PINN model for predicting the electromagnetic field of the superconducting bulk material under the external magnetic field excitation to achieve the prediction of the electromagnetic field distribution, and then obtain the prediction of the electromagnetic force;

[0010] Step 3: Based on the physical information neural network model and the electromagnetic force distribution obtained in step 2, a PINN model is established to predict the mechanical response of superconducting bulk materials under the electromagnetic force, which takes coordinate points as input and displacement field as output;

[0011] Step 4: training the PINN model for predicting the mechanical response of the superconducting bulk material under the electromagnetic force, and finally realizing the electromagnetic field distribution and mechanical response prediction of the superconducting bulk material.

[0012] The step 1 is specifically as follows:

[0013] Build a physical information neural network Net of superconducting magnetic field EM , to predict the electromagnetic field distribution HE p

[0014] HE p =Net EM (X;θ EM ) (1)

[0015] Among them, Net EM represents a fully connected neural network, the coordinate point and time point of the superconducting block are X, HE p =(H p ,E p ) is the predicted value of magnetic field intensity H and electric field intensity E, θ EM Net EMThe trainable parameters of , here refers to the weight w and bias b of the fully connected neural network;

[0016] Using the physical information neural network Net EM The output of the network is combined with the automatic differentiation of the network to calculate the relevant physical constraints satisfied by the physical information network according to the electromagnetic Maxwell equations of the embedded superconducting model;

[0017] The constraints include three constraints, which act together on the training process of the electromagnetic field neural network. Constraint 1 (Maxwell's equations) ensures that the model satisfies the basic equations of the electromagnetic field in terms of physical laws. Constraint 2 (boundary conditions) ensures that the predicted magnetic field meets the requirements of the external magnetic field at the boundary. Constraint 3 (initial conditions) ensures that the predicted magnetic field is consistent with the external magnetic field at the initial moment. Through the combined effect of these three constraints, accurate prediction of the electromagnetic field distribution of superconducting bulk materials can be achieved.

[0018] The constraints are specifically:

[0019] Constraint 1: Maxwell equations for embedded superconducting model

[0020]

[0021] ▽ × H = J (3)

[0022] B=μ0H (4)

[0023] E=ρ HTS J (5)

[0024] Where E is the electric field intensity vector, J is the current density vector, H is the magnetic field intensity vector, B is the magnetic induction intensity vector, μ0 is the vacuum magnetic permeability, ρ HTS is the resistivity of the superconducting bulk material. Substituting equation (4) into equation (2), and equation (5) into equation (3), we obtain the equation containing the predicted magnetic field intensity H p and the electric field strength E p The equation

[0025]

[0026] f2=E p -ρ HTS ▽×H p =0 (7)

[0027] Among them, the resistivity ρ of the superconducting bulk material is HTS Obtained from the superconducting EJ power law model

[0028]

[0029] Among them, E cis the critical electric field strength, n is the power law index, J p is the predicted value of current density, J c To consider the critical current density of magnetic field dependence, the superconducting Kim model gives

[0030]

[0031] Among them, J c0 is the critical current density under zero field, and B0 is the fitting parameter.

[0032] Constraint 2: The predicted magnetic field of the superconducting bulk material satisfies the boundary conditions

[0033]

[0034] in, is the boundary of the superconducting bulk, H ext (t) is the external magnetic field that varies with time.

[0035] Constraint 3: The predicted magnetic field of the superconducting bulk material satisfies the initial conditions

[0036] H p (X) = H ext (0), X∈Ω (11)

[0037] Among them, H p (X,0) is the predicted value of the initial magnetic field of the superconducting bulk material, H ext (0) is the value of the external magnetic field at time t=0.

[0038] The physical information neural network Net EM It embeds the constraints of physical equations such as Maxwell's equations into the loss function and trains by reducing the total loss function Loss, so that the trained model satisfies the basic equations of superconducting electromagnetic field; for predicting the total loss function Loss of the PINN framework of superconducting bulk electromagnetic field EM , defined as the mean square error Loss of the Maxwell equations with the embedded superconducting model MAXWELL ; Mean square error Loss of magnetic field boundary conditions BC and the mean square error Loss of the initial conditions of the magnetic field IC

[0039] Loss EM =λ MAXWELL Loss MAXWELL +λ BC Loss BC +λ IC Loss IC (12)

[0040] Among them, λ MAXWELL ,λBC and λ IC are the weight values ​​corresponding to each loss function term, and the loss function terms of Maxwell equations, magnetic field boundary and initial conditions are

[0041]

[0042] in, is the internal sample point, is the boundary sample point, is the initial sample point; N int , N bc and N ic is the number of corresponding sampling points.

[0043] The step 2 is specifically as follows:

[0044] When the superconducting magnetic field physics information neural network Net in step 1 EM Loss function EM When the training times of the two networks reach the maximum at the same time, the network training is stopped; the final output of the neural network is the predicted value of the electromagnetic field distribution of the superconducting block under the excitation of the external magnetic field. Based on the predicted values ​​of the magnetic field intensity and the electric field intensity, the predicted value of the current density can be obtained, and finally the prediction of the electromagnetic force distribution can be realized.

[0045] J p =▽×H p (14)

[0046] f p =J p ×μ0H p (15)

[0047] Among them, f p Predicted values ​​of the electromagnetic force.

[0048] The step three is specifically as follows:

[0049] Building a physical information neural network Net for superconducting solid mechanics SM , to predict the displacement field distribution U p

[0050] U p =Net SM (X;θ SM ) (16)

[0051] Among them, Net SM represents a fully connected neural network, the coordinate point and time point of the superconducting block are X = (X, Y, t), U p is the predicted value of the displacement field, θ SM Net SMThe trainable parameters of , here refers to the weight w and bias b of the fully connected neural network;

[0052] Using the physical information neural network Net SM The output of the network is combined with the automatic differentiation technology of the network, and according to the principle of minimum potential energy and the equilibrium equation of linear elasticity, the relevant physical constraints satisfied by the physical information network are calculated.

[0053] The constraints are specifically:

[0054] Constraint 1: Principle of Minimum Potential Energy

[0055] Π=E in -E ext (17)

[0056] Where Π is the total potential energy, E in and E ext The internal potential energy and the external potential energy are

[0057]

[0058] Among them, U p For neural network Net SM Predicted displacement field, f p is the predicted value of the electromagnetic force obtained in step 2, σ p and ε p are the predicted values ​​of the stress and strain components, respectively, given by the predicted value of the displacement field

[0059]

[0060] σ=λθI+2Gε (21)

[0061] Where λ and G are the Lame constants, and θ is the volume strain.

[0062] Constraint 2: Equilibrium equations for linear elastic materials

[0063] f3=▽·σ p +f p =0 (22)

[0064] Since the loss function based on minimum potential energy does not involve the equilibrium equation containing stress differential terms, inaccurate stress-strain prediction values ​​may occur under complex electromagnetic forces. Therefore, the present invention also uses the equilibrium equation as a physical equation constraint condition.

[0065] Constraint 3: The predicted displacement satisfies the boundary conditions

[0066]

[0067] in, is the displacement constraint boundary, U p (X) is the given displacement constraint.

[0068] Physical Information Neural Network Net SM It embeds physical equation constraints such as the minimum potential energy principle and the linear elastic mechanics equilibrium equation into the loss function, and is trained by reducing the total loss function Loss, so that the trained model satisfies the basic equations of two-dimensional solid mechanics. The total loss function Loss of the PINN framework is used to predict the mechanical response of superconducting bulk materials under electromagnetic force. SM , defined as the total potential energy mean square error Loss based on the minimum potential energy principle ENERGY ; Mean square error Loss that satisfies the linear elastic equilibrium equation PDE and the mean square error Loss of the displacement boundary conditions B

[0069] Loss SM =λ ENERGY Loss ENERGY +λ PDE Loss PDE +λ B Loss B (twenty four)

[0070] Among them, λ ENERGY ,λ PDE and λ B are the weight values ​​corresponding to each loss function term, and the total potential energy, linear elastic equilibrium equation and displacement boundary condition loss function terms based on the minimum potential energy principle are

[0071]

[0072] in, is the internal sample point, is the boundary sample point; N int and N bc is the number of corresponding sampling points, is the boundary sample point The actual displacement value on .

[0073] These three constraints act together in the training process of the solid mechanics neural network. Constraint 1 (the principle of minimum potential energy) ensures that the mechanical response of the superconducting bulk material will reach a minimum total potential energy state under the interaction of internal and external forces. Constraint 2 (linear elastic equilibrium equation) ensures that the relationship between stress and displacement is correctly expressed under the action of complex electromagnetic loads, thereby avoiding inaccurate stress prediction due to the lack of equilibrium equation constraints. Constraint 3 (displacement boundary condition) ensures that the predicted displacement field meets the actual constraint conditions at the boundary. Through the combined effect of these three constraints, the PINN model can more accurately predict the mechanical response of superconducting bulk materials under the action of electromagnetic body forces and satisfy the mutual coupling relationship between mechanics and electromagnetism.

[0074] The step 4 is specifically as follows:

[0075] When the physical information neural network Net SM Loss function SM When the training times of the two networks reach the maximum at the same time, the network training is stopped; the final output of the neural network is the predicted value of the mechanical response of the superconducting bulk material under the action of the electromagnetic force, and finally the force-electromagnetic multi-field coupling analysis of the superconducting bulk material is realized.

[0076] Beneficial effects of the present invention:

[0077] The method proposed in the present invention is based on the Maxwell equations with an embedded superconducting model, combined with the training of the superconducting electric magnetic field physics information neural network, to achieve accurate prediction of the electromagnetic field distribution of superconducting bulk materials under external magnetic field excitation. Further based on the minimum potential energy principle and the linear elastic mechanics equilibrium equation, combined with the electromagnetic force value predicted by the superconducting electric magnetic field physics information neural network, combined with the training of the superconducting solid mechanics physics information neural network, the mechanical response of superconducting bulk materials under the action of electromagnetic force is accurately predicted, and finally the force-electromagnetic multi-field coupling analysis of superconducting bulk materials is realized.

[0078] The present invention does not rely on experimental data, but is driven by pure physical information to ensure that the output of the network conforms to the real physical laws. Since it does not rely on experimental data of specific materials, the model has stronger applicability and can be widely used in different types of superconducting materials and their performance analysis in various electromagnetic environments.

[0079] The method of the present invention can accurately analyze the complex response of superconducting materials under force-electromagnetic coupling by learning the laws of physics, and further promote the design and optimization of superconducting materials. Through this deep learning method based on physical information, the performance of materials under different conditions can be predicted and analyzed, providing a scientific basis for future engineering applications. In addition, compared with traditional complex finite element numerical models, this method does not rely on meshing and can be solved in a continuous space. This feature is particularly suitable for complex geometric problems.

[0080] In summary, the PINN-based superconducting bulk material mechanical-electromagnetic multi-field coupling analysis method provides a new way to solve the superconducting multi-field coupling problem, and is expected to be applied to more complex superconducting problems in the future. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 Schematic diagram of the process flow of the PINN-based superconducting bulk material mechanical-electromagnetic multi-field coupling analysis method.

[0082] Figure 2 Superconducting bulk material test case based on this method, schematic diagram of electromagnetic field distribution prediction and mechanical response prediction. DETAILED DESCRIPTION

[0083] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0084] The PINN prediction method of the electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic loads of the present invention realizes multi-field coupling analysis of superconducting bulk materials based on deep learning by combining physical equations.

[0085] The following is a detailed description of the present invention in steps. Figure 1 As shown:

[0086] Step 1: Physical information neural network of superconducting electric field Net EM Build. By building a neural network Net EM , performs the mapping of space-time coordinate input to electromagnetic field output, and can automatically learn and predict the distribution of complex electromagnetic fields without relying on grid division. The execution of step one provides an accurate electromagnetic field prediction tool for the entire technical solution.

[0087] Establish a fully connected neural network NN(w,b) to predict the electromagnetic field distribution of superconducting bulk materials under external magnetic field excitation HE p , where the neural network input is the two-dimensional coordinate point and time point X = (X, Y, t) of the superconducting block, and the output is the magnetic field intensity and electric field intensity HE p =(H p ,E x p ,E y p ), as attached Figure 1 shown.

[0088] HE p =Net EM (X,Y,t;θ EM ) (1)

[0089] Among them, Net EM represents a fully connected neural network, θ EMNet EM The trainable parameters here refer to the weights w and bias b of the fully connected neural network.

[0090] Using the physical information neural network Net EM The output of the network is combined with the automatic differentiation technology of the network to calculate the relevant physical constraints satisfied by the physical information network according to the electromagnetic Maxwell equations of the embedded superconducting model. The physical constraints ensure that the electromagnetic field distribution predicted by the neural network conforms to the actual physical behavior of the superconducting material, and guide the network to learn the correct physical laws.

[0091] The constraints are specifically:

[0092] Constraint 1: Maxwell equations for embedded superconducting model

[0093]

[0094] ▽ × H = J (3)

[0095] B=μ0H (4)

[0096] E=ρ HTS J (5)

[0097] Where E is the electric field intensity vector, J is the current density vector, H is the magnetic field intensity vector, B is the magnetic induction intensity vector, μ0 is the vacuum magnetic permeability, ρ HTS Substituting equation (29) into equation (27), equation (30) into equation (28), and simplifying the problem for two-dimensional superconducting bulk materials, we obtain the predicted magnetic field intensity H and electric field intensity E x and E y The equation

[0098]

[0099] Among them, the resistivity ρ of the superconducting bulk material is HTS Obtained from the superconducting EJ power law model

[0100]

[0101] Among them, E c is the critical electric field strength, n is the power law index, J p is the predicted value of current density. J c To consider the critical current density of magnetic field dependence, the superconducting Kim model gives

[0102]

[0103] Among them, J c0 is the critical current density under zero field, and B0 is the fitting parameter.

[0104] Constraint 2: The predicted magnetic field of the superconducting bulk material satisfies the boundary conditions

[0105]

[0106] in, is the boundary of the superconducting bulk, H ext (t) is the external magnetic field that varies with time.

[0107] Constraint 3: The predicted magnetic field of the superconducting bulk material satisfies the initial conditions

[0108] H p (X,Y,0)=H ext (0),(X,Y)∈Ω

[0109] Among them, H p (X, Y, 0) is the predicted value of the initial magnetic field of the superconducting bulk material, H ext (0) is the value of the external magnetic field at time t=0.

[0110] Physical Information Neural Network Net EM The training is done by embedding the constraints of physical equations such as Maxwell's equations into the loss function and reducing the total loss function Loss, so that the trained model satisfies the basic equations of the superconducting electromagnetic field. EM , defined as the mean square error Loss of the Maxwell equations with the embedded superconducting model MAXWELL ; Mean square error Loss of magnetic field boundary conditions BC and the mean square error Loss of the initial conditions of the magnetic field IC

[0111] Loss EM =λ MAXWELL Loss MAXWELL +λ BC Loss BC +λ IC Loss IC (12)

[0112] Among them, λ MAXWELL ,λ BC and λ IC are the weight values ​​corresponding to each loss function term, and the loss function terms of Maxwell equations, magnetic field boundary and initial conditions are

[0113]

[0114] in, is the internal sample point, is the boundary sample point, is the initial sample point; N int , N bc and N ic is the number of corresponding sampling points. It is worth noting that in order to reduce the complexity of training, we deal with the way the initial conditions are applied. The output layer of the neural network for predicting the magnetic field strength is defined as

[0115] [H']=N(X,Y,t),H p =H'·t+H ext (0)

[0116] Among them, it can be found that when t = 0, the predicted value of the magnetic field strength H p The condition of being equal to the external magnetic field value at the initial moment can be automatically satisfied.

[0117] Step 2: When the physical information neural network Net in step 1 EM Loss function EM When the training times of the two training methods reach the maximum at the same time, the network training is stopped. The final output of the neural network is the predicted value of the electromagnetic field distribution of the superconducting block under the excitation of the external magnetic field. Based on the predicted values ​​of the magnetic field intensity and the electric field intensity, the predicted value of the current density can be obtained, and finally the prediction of the electromagnetic force distribution can be realized.

[0118]

[0119] in, and is the predicted value of current density, and is the predicted value of the electromagnetic force. Through the training of the superconducting magnetic field physics information neural network, the electromagnetic field distribution prediction is finally realized, and then the electromagnetic force distribution at a certain moment is obtained, as shown in the attached figure. Figure 1 shown.

[0120] Step 3: Physical Information Neural Network Net of Superconducting Solid Mechanics SM Based on the electromagnetic force distribution obtained in step 2, a fully connected neural network NN(w,b) is established to predict the displacement field U of the superconducting block under the action of the electromagnetic force. p , where the neural network input is the two-dimensional coordinate point X = (X, Y) of the superconducting bulk material, and the output is the displacement field U p =(u p ,v p ), as attached Figure 1 shown.

[0121] U p =Net SM (X,Y;θ SM )

[0122] Among them, Net SM represents a fully connected neural network, θ SM Net SM The trainable parameters here refer to the weights w and bias b of the fully connected neural network.

[0123] Using the physical information neural network Net SM The output of the network is combined with the automatic differentiation technology of the network, and the relevant physical constraints satisfied by the physical information network are calculated according to the principle of minimum potential energy and the equilibrium equation of linear elasticity. The physical constraints ensure that the mechanical response (such as displacement, stress and strain) predicted by the neural network conforms to the actual physical behavior of superconducting materials under electromagnetic loads, and guide the network to learn the correct physical laws.

[0124] The constraints are specifically:

[0125] Constraint 1: Principle of Minimum Potential Energy

[0126] Π=E in -E ext (17)

[0127] Where Π is the total potential energy, E in and E ext The internal potential energy and the external potential energy are

[0128]

[0129] Among them, u and v are neural network Net SM Predicted displacement field, f x and f y is the predicted value of the electromagnetic force obtained in step 2, σ x , σ y , σ xy and ε x , ε y , ε xy are the predicted values ​​of the stress and strain components, respectively, given by the predicted value of the displacement field

[0130]

[0131] Wherein, E and ν are the elastic modulus and Poisson's ratio of the superconducting bulk material. Equation (48) is a physical equation for a plane stress problem, while the superconducting bulk material problem targeted by the present invention is a plane strain problem, so E is replaced by ν is replaced by

[0132] Constraint 2: Equilibrium equations for linear elastic materials

[0133]

[0134] Since the loss function based on minimum potential energy does not involve the equilibrium equation containing stress differential terms, inaccurate stress-strain prediction values ​​may occur under complex electromagnetic forces. Therefore, the present invention also uses the equilibrium equation as a physical equation constraint condition.

[0135] Constraint 3: The predicted displacement satisfies the boundary conditions

[0136] U p (X,Y)=U(X,Y),

[0137] in, is the displacement constraint boundary, and U(X,Y) is the given displacement constraint.

[0138] Physical Information Neural Network Net SM The training is carried out by embedding physical equation constraints such as the minimum potential energy principle and the linear elastic mechanics equilibrium equation into the loss function, and then reducing the total loss function Loss to make the trained model satisfy the basic equations of two-dimensional solid mechanics. SM , defined as the total potential energy mean square error Loss based on the minimum potential energy principle ENERGY ; Mean square error Loss that satisfies the linear elastic equilibrium equation PDE and the mean square error Loss of the displacement boundary conditions B

[0139] Loss SM =λ ENERGY Loss ENERGY +λ PDE Loss PDE +λ B Loss B (twenty four)

[0140] Among them, λ ENERGY ,λ PDE and λ B are the weight values ​​corresponding to each loss function term, and the total potential energy, linear elastic equilibrium equation and displacement boundary condition loss function terms based on the minimum potential energy principle are

[0141]

[0142] in, is the internal sample point, is the boundary sample point; N int and N bc is the number of corresponding sampling points, is the boundary sample point The actual displacement value on the displacement boundary condition is also processed similarly, and the output layer of the neural network for predicting displacement is defined as

[0143] [u',v']=N(X,Y),u p =u'·X,v p =v'·Y

[0144] Among them, it can be found that when it is located at the displacement constraint boundary, the displacement field prediction value u p and v p The given displacement boundary conditions can be automatically satisfied.

[0145] Step 4: When the physical information neural network Net in step 3 SM Loss function SM When the number of trainings reaches the maximum at the same time, the network training is stopped. The final output of the neural network is the displacement prediction value of the superconducting bulk material under the action of the electromagnetic force. By automatically differentiating the displacement prediction value and combining it with the constitutive equation of mechanics, the stress distribution result of the superconducting bulk material is obtained. By training the neural network of superconducting solid mechanics physical information, the accurate prediction of the mechanical response of the superconducting bulk material is finally achieved, as shown in the attached figure. Figure 1 shown.

[0146] Example:

[0147] The invention discloses an application of the PINN prediction superconducting bulk multi-field coupling analysis scheme to rectangular superconducting bulk materials with an external falling field.

[0148] As attached Figure 2 As shown in (a), an external decreasing magnetic field H is applied to a rectangular superconducting block with a size of 30 mm × 30 mm. ext (t), the external magnetic field gradually decreases from 10 T to 5 T within 0.05 s.

[0149] The electromagnetic field distribution of the superconducting bulk material in this case is predicted by the superconducting electric field physics information neural network of the present invention, wherein the electromagnetic field parameter of the superconducting model is E c =1×10 -6 Vm -1 , n=8,B0=5T,J c0 =1.462×10 7 Am -2 .

[0150] The hidden layer of the feedforward neural network is iteratively optimized to finally obtain the predicted values ​​of the electromagnetic field and electromagnetic force, such as Figure 2 As shown in (b), it can be seen that the network prediction results are basically consistent with the finite element numerical simulation results.

[0151] The electromagnetic force prediction results of the superconducting magnetic field physics information neural network at the end time are further input into the superconducting solid mechanics physics information neural network to predict the mechanical response of the superconducting bulk material under the electromagnetic force. Constraints are imposed on the left and lower boundaries of the superconducting bulk material, and the mechanical parameters are E = 103GPa, ν = 0.3.

[0152] The hidden layer of the feedforward neural network is iteratively optimized to finally obtain the predicted value of the mechanical response, such as Figure 2 As shown in (c), it can be seen that the network prediction results are basically consistent with the finite element numerical simulation results.

[0153] The solution proposed in the present invention is based on the Maxwell equations with an embedded superconducting model, and through physical information neural network training, the accurate prediction of the electromagnetic field distribution of the superconducting bulk structure under the action of an external magnetic field is achieved. Further based on the minimum potential energy principle and the linear elastic mechanics equilibrium equation, combined with the electromagnetic force prediction results, the physical information neural network training is used to achieve the accurate prediction of the mechanical response of the superconducting bulk structure under the electromagnetic force. The method provided by the present invention is the first to apply PINN to the simulation of superconducting electric magnetic fields and the prediction of mechanical responses under electromagnetic forces.

Claims

1. A PINN prediction method for the electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic loads, characterized in that: The steps include: Step 1: Based on the physical information neural network model of superconducting electric and magnetic fields, a PINN model is established to predict the electromagnetic field of superconducting bulk materials under external magnetic field excitation, with coordinate points and time points as inputs and magnetic field intensity and electric field intensity as outputs; Step 2: training the PINN model for predicting the electromagnetic field of the superconducting bulk material under the external magnetic field excitation to achieve the prediction of the electromagnetic field distribution, and then obtain the prediction of the electromagnetic force; Step 3: Based on the physical information neural network model and the electromagnetic force distribution obtained in step 2, a PINN model is established to predict the mechanical response of superconducting bulk materials under the electromagnetic force, which takes coordinate points as input and displacement field as output; Step 4: training the PINN model for predicting the mechanical response of the superconducting bulk material under the electromagnetic force, and finally realizing the electromagnetic field distribution and mechanical response prediction of the superconducting bulk material.

2. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 1, characterized in that: The step 1 is specifically as follows: Build a physical information neural network Net of superconducting magnetic field EM , to predict the electromagnetic field distribution HE p HE p =No EM (X;θ EM ) (1) Among them, Net EM represents a fully connected neural network, the coordinate point and time point of the superconducting block are X, HE p =(H p ,E p ) is the predicted value of magnetic field intensity H and electric field intensity E, θ EM Net EM The trainable parameters of , here refers to the weight w and bias b of the fully connected neural network; Using the physical information neural network Net EM The output of the network is combined with the automatic differentiation of the network to calculate the relevant physical constraints satisfied by the physical information network based on the electromagnetic Maxwell equations of the embedded superconducting model.

3. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 2, characterized in that: The constraints include three constraints, which act together on the training process of the electromagnetic field neural network. Constraint 1 ensures that the model satisfies the basic equations of the electromagnetic field in terms of physical laws, constraint 2 ensures that the predicted magnetic field meets the requirements of the external magnetic field at the boundary, and constraint 3 ensures that the predicted magnetic field is consistent with the external magnetic field at the initial moment. Through the combined effect of these three constraints, accurate prediction of the electromagnetic field distribution of superconducting bulk materials can be achieved.

4. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 3, characterized in that: The constraints are specifically: Constraint 1: Maxwell equations for embedded superconducting model B=μ0H (4) E=ρ HTS J (5) Where E is the electric field intensity vector, J is the current density vector, H is the magnetic field intensity vector, B is the magnetic induction intensity vector, μ0 is the vacuum magnetic permeability, ρ HTS is the resistivity of the superconducting bulk material. Substituting equation (4) into equation (2), and equation (5) into equation (3), we obtain the equation containing the predicted magnetic field intensity H p and the electric field strength E p The equation Among them, the resistivity ρ of the superconducting bulk material is HTS Obtained from the superconducting EJ power law model Among them, E c is the critical electric field strength, n is the power law index, J p is the predicted value of current density, J c To consider the critical current density of magnetic field dependence, the superconducting Kim model gives Among them, J c0 is the critical current density under zero field, B0 is the fitting parameter; Constraint 2: The predicted magnetic field of the superconducting bulk material satisfies the boundary conditions in, is the boundary of the superconducting bulk, H ext (t) is the external magnetic field that varies with time; Constraint 3: The predicted magnetic field of the superconducting bulk material satisfies the initial conditions Among them, H p (X,0) is the predicted value of the initial magnetic field of the superconducting bulk material, H ext (0) is the value of the external magnetic field at time t=0.

5. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 2, characterized in that: The physical information neural network Net EM The physical equation constraints of Maxwell's equations are embedded into the loss function, and the training is performed by reducing the total loss function Loss, so that the trained model satisfies the basic equations of the superconducting electromagnetic field; the total loss function Loss of the PINN framework for predicting the electromagnetic field of superconducting bulk materials is EM , defined as the mean square error Loss of the Maxwell equations with the embedded superconducting model MAXWELL ; Mean square error Loss of magnetic field boundary conditions BC and the mean square error Loss of the initial conditions of the magnetic field IC Loss EM =λ MAXWELL Loss MAXWELL +λ BC Loss BC +λ IC Loss IC (12) Among them, λ MAXWELL ,λ BC and λ IC are the weight values ​​corresponding to each loss function term, and the loss function terms of Maxwell equations, magnetic field boundary and initial conditions are in, is the internal sample point, is the boundary sample point, is the initial sample point; N int , N bc and N ic is the number of corresponding sampling points.

6. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 5, characterized in that: The step 2 is specifically as follows: When the superconducting magnetic field physics information neural network Net in step 1 EM Loss function EM When the training times of the two networks reach the maximum at the same time, the network training is stopped; the final output of the neural network is the predicted value of the electromagnetic field distribution of the superconducting block under the excitation of the external magnetic field. Based on the predicted values ​​of the magnetic field intensity and the electric field intensity, the predicted value of the current density can be obtained, and finally the prediction of the electromagnetic force distribution can be realized. f p =J p ×μ0H p (15) Among them, f p Predicted values ​​of the electromagnetic force.

7. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 1, characterized in that: The step three is specifically as follows: Building a physical information neural network Net for superconducting solid mechanics SM , to predict the displacement field distribution U p U p =No SM (X;θ SM ) (16) Among them, Net SM represents a fully connected neural network, the coordinate point and time point of the superconducting block are X = (X, Y, t), U p is the predicted value of the displacement field, θ SM Net SM The trainable parameters of , here refers to the weight w and bias b of the fully connected neural network; Using the physical information neural network Net SM The output of the network is combined with the automatic differentiation of the network, and according to the principle of minimum potential energy and the equilibrium equation of linear elasticity, the relevant physical constraints satisfied by the physical information network are calculated.

8. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 7, characterized in that: The constraints are specifically: Constraint 1: Principle of Minimum Potential Energy Π=E in -E ext (17) Where Π is the total potential energy, E in and E ext The internal potential energy and the external potential energy are Among them, U p For neural network Net SM Predicted displacement field, f p is the predicted value of the electromagnetic force obtained in step 2, σ p and ε p are the predicted values ​​of the stress and strain components, respectively, given by the predicted value of the displacement field σ=λθI+2Gε (21) Where λ and G are Lame constants, and θ is the volume strain; Constraint 2: Equilibrium equations for linear elastic materials The equilibrium equations are also used as constraints of the physical equations; Constraint 3: The predicted displacement satisfies the boundary conditions in, is the displacement constraint boundary, U p (X) is the given displacement constraint.

9. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 7, characterized in that: The physical information neural network Net SM The physical equation constraints of the minimum potential energy principle and the linear elastic equilibrium equation are embedded into the loss function, and the training is performed by reducing the total loss function Loss, so that the trained model satisfies the basic equations of two-dimensional solid mechanics. The total loss function Loss of the PINN framework is used to predict the mechanical response of superconducting bulk materials under electromagnetic force. SM , defined as the total potential energy mean square error Loss based on the minimum potential energy principle ENERGY ; Mean square error Loss that satisfies the linear elastic equilibrium equation PDE and the mean square error Loss of the displacement boundary conditions B Loss SM =λ ENERGY Loss ENERGY +λ PDE Loss PDE +λ B Loss B (24) Among them, λ ENERGY ,λ PDE and λ B are the weight values ​​corresponding to each loss function term, and the total potential energy, linear elastic equilibrium equation and displacement boundary condition loss function terms based on the minimum potential energy principle are in, is the internal sample point, is the boundary sample point; N int and N bc is the number of corresponding sampling points, is the boundary sample point The true displacement value on ; These three constraints work together in the training process of the solid mechanics neural network. Constraint 1 ensures that the mechanical response of the superconducting bulk material will reach a minimum total potential energy state under the interaction of internal and external forces. Constraint 2 ensures that the relationship between stress and displacement is correctly expressed under the action of complex electromagnetic loads. Constraint 3 ensures that the predicted displacement field meets the actual constraint conditions at the boundary. Through the combined effect of these three constraints, the PINN model can more accurately predict the mechanical response of superconducting bulk materials under the action of electromagnetic body forces and satisfy the mutual coupling relationship between mechanics and electromagnetism.

10. The PINN prediction method for electromagnetic field distribution and mechanical response of superconducting bulk materials under electromagnetic load according to claim 1, characterized in that: The step 4 is specifically as follows: When the physical information neural network Net SM Loss function SM When the number of trainings reaches the maximum at the same time, stop network training; The final output of the neural network is the predicted value of the mechanical response of the superconducting bulk material under the action of electromagnetic force, which ultimately realizes the force-electromagnetic multi-field coupling analysis of the superconducting bulk material.

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