A deep learning method combining physical information neural network and finite element

Abstract

The present invention relates to the technical field of the intersection of computational mechanics and deep learning, and specifically to a deep learning method that combines a physical information neural network with a finite element method; specifically, partial differential equations are embedded into a neural network, comprising the following steps: S1, meshing the solution domain, and calculating the stiffness matrix and the node load array; S2, collecting observation data and boundary conditions; S3, using the energy represented by the stiffness matrix and the fitting error of the observation data as a loss function to train the neural network; S4, executing an optimization algorithm to optimize the neural network; S5, and repeating the training and optimization steps until convergence; the deep finite element method proposed in the present invention combines a physical information neural network with a finite element method, and exhibits significant advantages and beneficial effects in terms of the fusion of physical knowledge and observation data, modeling process and loss function design, model training stability and simplicity, computational efficiency and accuracy, pre- and post-processing technology and transfer learning capability, and applicability.

Description

Technical Field

[0001] The present invention relates to the technical field of the intersection of computational mechanics and deep learning, and specifically to a deep learning method that combines physical information neural networks with finite elements. Background Art

[0002] In science and engineering, partial differential equations (PDEs) are important tools for describing physical phenomena such as elasticity, fluid mechanics, and heat conduction. However, solving PDEs under complex structures and boundary conditions often faces the problems of large computational costs and difficulty in solving. Traditional numerical methods, such as the finite element method (FEM), although able to effectively solve PDEs, usually require complex meshing and pre- and post-processing, and are computationally expensive when dealing with high-dimensional and nonlinear problems.

[0003] In recent years, Physics-Informed Neural Networks (PINN), as an emerging deep learning technology, have achieved the fusion of data and physical knowledge by encoding the physical laws of PDEs into neural networks, providing new ideas for solving PDEs. PINN utilizes the nonlinear expression ability of neural networks to approximate complex solution spaces and considers both physical laws and observational data during training, thereby obtaining solutions that are both consistent with physical laws and close to actual observations. However, PINN often faces problems such as poor convergence and sensitivity to hyperparameters during training. In particular, its computational efficiency and accuracy are limited when dealing with PDEs with large degrees of freedom or complex geometries.

[0004] Therefore, how to combine the advantages of traditional numerical methods and deep learning to develop an efficient and stable PDEs solution method has become a hot topic of current research; the finite element method is widely used in engineering practice due to its powerful pre- and post-processing technology and adaptability to complex geometric shapes, while the physical information neural network has attracted attention due to its flexibility and ability to learn physical laws; combining the two is expected to form a new PDEs solution method that inherits the numerical stability and mesh convergence of the finite element method and has the nonlinear expression ability and data fusion ability of the neural network. Summary of the Invention

[0005] In response to the problems pointed out in the above background technology, the present invention proposes a deep learning method that combines physical information neural network with finite element method, called Deep Finite Element Method (DeepFEM); this method integrates the physical laws of PDEs with observation data, utilizes the nonlinear expression ability of neural network and the numerical stability of finite element method, and aims to efficiently and accurately solve the physical field distribution and response in complex structural problems, providing new ideas and methods for solving PDEs.

[0006] The technical solution adopted by the present invention to solve the technical problem is: a deep learning method combining physical information neural network and finite element, embedding partial differential equations into the neural network, including the following steps:

[0007] S1. Divide the solution domain into grids and calculate the stiffness matrix and node load array;

[0008] S2, collect observation data and boundary conditions;

[0009] S3, using the energy represented by the stiffness matrix and the fitting error of the observed data as the loss function to train the neural network;

[0010] S4, executing the optimization algorithm to optimize the neural network;

[0011] S5, and repeat the training and optimization steps until convergence.

[0012] The neural network is a multi-layer perceptron, comprising an input layer, a hidden layer and an output layer, wherein the hidden layer applies a nonlinear activation function.

[0013] The general form of the partial differential equation can be expressed as follows:

[0014]

[0015] Where L represents the differential operator, u(x) represents the vector field function defined on the domain Ω, and R n represents n-dimensional Euclidean space, q(x) represents an excitation term; in addition, B is the operator corresponding to the boundary condition, Γ is a part of the boundary, indicating the area of ​​action of B, represents the boundary of the domain Ω;

[0016] Assuming that the order of operator L is 2m, the boundary conditions of order 0 to m-1 are called forced boundary conditions, and the boundary conditions of order m to 2m-1 are called natural boundary conditions. The natural variation method requires that the trial function u satisfy the forced boundary conditions in advance; the boundary conditions are decomposed into the following two types:

[0017]

[0018] where Γ u is the mandatory boundary, Γ σ is a natural boundary, and B u is a mandatory boundary condition operator with an order of 0 to m-1. σ It is a natural boundary condition operator with an order of m to 2m-1.

[0019] The optimization algorithm includes SGD, Adam or L-BFGS.

[0020] The loss function consists of physical term loss and data term loss. The data term loss is used to introduce observation data or boundary conditions. The general form of the loss function is shown in the following formula:

[0021] L=L phys. (θ)+L dat. (θ) (12)

[0022] The data item loss is as follows:

[0023]

[0024] The derivative of the physical loss term with respect to the neural network parameters is as follows:

[0025]

[0026] When the loss function converges, the solution obtained is the correct solution to the partial differential equation, and the second-order derivative of the loss function with respect to the parameter is greater than 0, which has local convexity.

[0027] The stiffness matrix and the node load array provide a simple form for calculating energy, which is expressed as a quadratic function of the node displacement.

[0028] The method also includes decomposing the boundary conditions into forced boundary conditions and natural boundary conditions, and satisfying the forced boundary conditions; or introducing boundary conditions by modifying the neural network structure or modifying the stiffness matrix and the node load vector.

[0029] The neural network expresses the test function on the unit into an interpolation form through shape functions to handle complex structural problems.

[0030] When processing partial differential equations, the method performs conversion through the Galerkin form and obtains a global stiffness matrix and a global node response vector.

[0031] The technical effect achieved by the present invention is as follows: the Deep Finite Element Method (DFEM) proposed in the present invention combines the Physics-Informed Neural Networks (PINN) with the Finite Element Method (FEM), and shows significant advantages and beneficial effects in the fusion of physical knowledge and observation data, modeling process and loss function design, model training stability and simplicity, computational efficiency and accuracy, pre- and post-processing technology and transfer learning ability, and applicability, which are specifically reflected in the following aspects:

[0032] (1) The present invention successfully integrates efficient physical knowledge with observational data: The present invention inherits the advantages of PINN and constructs a flexible learning framework that can effectively integrate physical laws and observational data into the neural network; this integration not only improves the prediction accuracy of the model, but also gives the model stronger generalization ability, enabling it to quickly predict the response performance of any location.

[0033] (2) The present invention innovatively establishes a unified modeling process and a concise loss function: By drawing on the idea of ​​the finite element method, the present invention proposes a unified modeling process that is applicable to large-scale, complex geometric elasticity problems; at the same time, the design of the loss function is concise and clear, and has a unified form for simple and complex problems, which greatly simplifies the complexity of model construction and optimization.

[0034] (3) The present invention has excellent model training stability and simplicity: The DFEM proposed in the present invention shows extremely high stability and simplicity in model training; the loss function has extreme values ​​near the true solution, making it difficult for the converged model to jump out of the optimal solution, thereby ensuring the stability and efficiency of training; in addition, the same hyperparameters and training strategies are applicable to problems of different complexities, eliminating the need for tedious parameter adjustment for different problems.

[0035] (4) The present invention has higher computational efficiency and accuracy: Compared with the traditional PINN, DFEM has higher computational efficiency and accuracy; it inherits the mesh convergence of the finite element method, and as the mesh is refined, the model accuracy can gradually approach the exact solution; while the general PINN does not have this property, its model accuracy cannot be increased indefinitely with the increase in the number of configuration points.

[0036] (5) The present invention has powerful pre- and post-processing technology and transfer learning capabilities: The present invention makes full use of the powerful pre- and post-processing technology of the finite element method, making the solution of complex structural problems universal and standardized; at the same time, the trained model can be used as a pre-trained model for transfer learning. When the load changes, it can be quickly adjusted and optimized based on the original model, greatly improving the calculation efficiency; DFEM is not only suitable for simple two-dimensional problems, but can also effectively handle complex three-dimensional large degree of freedom problems; through example verification, the present invention has demonstrated excellent performance and accuracy in complex problems such as flat plate stretching with holes, asymmetric stretching, and crane booms. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a schematic flow chart of the stiffness matrix modification method in the present invention;

[0038] Figure 2 (a) Problem description and (b) Schematic diagram of mesh division in Example 1 of the present invention;

[0039] Figure 3 This is the diagram of (a) problem description, (b) grid division, and (c) configuration points and their weights in Example 2 of the present invention;

[0040] Figure 4 (a) Problem description and (b) Schematic diagram of mesh division in Example 3 of the present invention;

[0041] Figure 5 This is a training curve diagram of Example 1 in the present invention;

[0042] Figure 6 This is the training curve diagram of Example 2 in the present invention;

[0043] Figure 7 This is the training curve diagram of Example 3 in the present invention;

[0044] Figure 8 This is a diagram showing the predicted results of the displacement field model in Example 1 of the present invention;

[0045] Figure 9 This is the prediction result diagram of the x-direction displacement field model in Example 2 of the present invention;

[0046] Figure 10 This is the prediction result diagram of the y-direction displacement field model in Example 2 of the present invention;

[0047] Figure 11 This is a diagram showing the predicted results of the displacement field model in Example 3 of the present invention;

[0048] Figure 12 This is a training curve diagram of model transfer learning in Example 1 of the present invention;

[0049] Figure 13This is the training curve of model transfer learning in Example 3 of the present invention;

[0050] Figure 14 It is a schematic diagram of the model training process in the present invention. DETAILED DESCRIPTION

[0051] The following describes in detail the specific implementation of the present invention in conjunction with specific embodiments and accompanying drawings:

[0052] like Figure 1-14 As shown, the present invention combines the advantages of physical induced neural networks and finite element methods to develop a deep finite element method that can stably embed partial differential equations into neural networks; this method draws on the idea of ​​finite elements, simplifies the loss function through the grid, and expresses the energy as a quadratic function of the node displacement, avoiding the derivative representation in the loss function, making the model optimization more stable.

[0053] Specifically, a deep learning method that combines physical information neural networks with finite element methods, embeds partial differential equations into the neural network, and includes the following steps:

[0054] S1, mesh the solution domain and calculate the stiffness matrix and node load array;

[0055] S2, collects data consisting of observations and boundary conditions;

[0056] S3, using the energy represented by the stiffness matrix and the fitting error of the observation data as the loss function to train the neural network, so that the neural network satisfies both the physical laws and the observation data, forming a physical information fusion model;

[0057] S4, executes the optimization algorithm. Common optimization algorithms include SGD, Adam, and L-BFGS;

[0058] S5, repeat the training and optimization steps until convergence.

[0059] The neural network is generally a multi-layer perceptron, whose basic components are: input layer, hidden layer, and output layer; the input layer is the layer that receives input data, and each input feature corresponds to an input neuron; the hidden layer is one or more layers of neurons between the input layer and the output layer, and each hidden layer is composed of multiple neurons. Each neuron receives the output of the previous layer, and performs a nonlinear transformation after weighted summation through an activation function; the output layer generates the output of the model, which usually corresponds to the prediction result of the task; the neurons in the hidden layer usually apply nonlinear activation functions, such as ReLU (Rectified Linear Unit), Sigmoid, Tanh, etc., to introduce nonlinear characteristics.

[0060] The stiffness matrix and the nodal load matrix provide a simple form for calculating energy, which is expressed as a quadratic function of the nodal displacement.

[0061] The general form of the partial differential equation can be expressed as follows:

[0062]

[0063] Where L represents the differential operator, u(x) represents the vector field function defined on the domain Ω, and R n represents n-dimensional Euclidean space, q(x) represents an excitation term; in addition, B is the operator corresponding to the boundary condition, Γ is a part of the boundary, indicating the area of ​​action of B, represents the boundary of the domain Ω.

[0064] Assuming that the order of operator L is 2m, the boundary conditions of order 0 to m-1 are called forced boundary conditions, and the boundary conditions of order m to 2m-1 are called natural boundary conditions. The natural variation method requires that the trial function u satisfy the forced boundary conditions in advance; the boundary conditions are decomposed into the following two types:

[0065]

[0066] where Γ u is the mandatory boundary, Γ σ is a natural boundary, and B u is a mandatory boundary condition operator with an order of 0 to m-1. σ It is a natural boundary condition operator with an order of m to 2m-1.

[0067] When dealing with complex structural problems, the existing powerful pre- and post-processing technology of finite element method makes the solution of complex structural problems universal and standardized; by using piecewise continuous function as the test function and decomposing the domain Ω into several units, each unit consists of several node variables; different from the existing PINN (physics-informed neural network) method, the displacement of the node variables is approximated by artificial neural network:

[0068] u(x i )=N(x i ;θ), i=1,2,...,n (3)

[0069] Where n represents the number of nodes, x i is the coordinate of node i, and the shape function is used to express the test function on element e into an interpolation form:

[0070]

[0071] in is the interpolation shape function of the jth node in element e, ej represents the global number of the jth node in unit e, and k is the total number of nodes in unit e.

[0072] Assuming that the mandatory boundary conditions are satisfied, the Galerkin form of the partial differential equation can be written as follows:

[0073]

[0074] By integrating by parts, it can be converted into the following form:

[0075]

[0076] After finishing, we can get:

[0077] δJ e =0 (7)

[0078] in:

[0079]

[0080] If Denoted as N e (θ), it is not difficult to find that the above formula is N e The quadratic function of (θ) can be rearranged into the following form:

[0081]

[0082] The above formula can be organized into the following form:

[0083]

[0084] Where K is the global stiffness matrix, which is a symmetric semi-positive definite matrix. It is generally not full-rank, but can be converted into a full-rank positive definite matrix by introducing boundary conditions; N(θ) = [N(x1;θ) T N(x2;θ) T …N(x n ;θ) T ] T is the global node response vector, is the global node load vector; thus, we express the energy as a quadratic form with respect to the output of the neural network function:

[0085]

[0086] Assuming the spatial dimension is d, K is the nd×nd stiffness matrix and P is the nd-dimensional load vector.

[0087] The loss function of the proposed method consists of physical term loss and data term loss, and its general form is shown in the following formula:

[0088] L=L phys. (θ)+L dat. (θ) (12)

[0089] The data item loss is as follows:

[0090]

[0091] The data loss can be used to introduce observation data and boundary conditions; the overall structure of the proposed method is as follows: Figure 1 shown.

[0092] In addition, boundary conditions can also be introduced through other methods, such as modifying the neural network structure, as shown below:

[0093] u(x)=D(x)N(x; θ)+G(x) (14)

[0094] Where D(x) represents the distance function, whose value on the boundary is 0, N(x;θ) represents a trainable neural network, G(x) represents the extension of the Dirichlet boundary condition on the boundary, and u(x) represents the final output; through the above modifications, the neural network automatically satisfies the Dirichlet boundary condition.

[0095] The traditional finite element method of introducing boundary conditions is to modify the stiffness matrix and node load vector; Figure 1 As shown in (a), when the given displacement is zero displacement, the main diagonal can be changed to 1 and other elements can be changed to 0 in the rows and columns corresponding to the zero displacement nodes in the stiffness matrix K, and the corresponding elements in the node load vector can be changed to 0; when the given node displacement is a constant value When the diagonal element K in the jth row of the stiffness matrix K is jj Multiply by a large number α, and use Replace p j ,like Figure 1 (b) shown.

[0096] The proposed method is more stable than PINN. The following is an analysis of its gradient. Assuming that the boundary conditions or observation data are satisfied, only the convergence of the physical loss term is considered. Many existing optimization methods promote the convergence of the loss function, that is, the derivative of the loss function with respect to the neural network parameters is close to 0. Therefore, we assume that the model is in a converged state. The derivative of the physical loss term with respect to the neural network parameters of the proposed method is as follows:

[0097]

[0098] In principle, when the deep finite element model converges to an extreme value, the gradient shown in Equation (41) should be equal to zero; at this time, it can be inferred from the right side of the equation that K·N(θ)-P should also be equal to zero; from the finite element principle, it can be seen that K·N(θ)-P=0 is the target solution of the formal partial differential equation; this shows that when the Deep FEM model converges to an extreme value, the solution obtained is the correct solution that satisfies the partial differential equation.

[0099] The second-order derivative of the loss function with respect to the parameters is:

[0100]

[0101] When the model converges, the first term in the above formula is greater than 0, which is due to the positive definiteness of K, and the second term is close to 0. Therefore, the second-order derivative of the model loss is greater than 0, that is, it is locally convex, making it easy to obtain a stable optimal solution.

[0102] The present invention inherits the advantages of PINN and provides a flexible learning framework that can realize the fusion of physical knowledge and observation data; the trained model can be used to quickly predict the response performance of any position; it has a unified modeling process and a concise loss function; it can utilize the powerful pre- and post-processing technology of existing finite elements, and is applicable to large-scale complex geometric elastic mechanics problems with large degrees of freedom, and the loss function has a unified form for simple and complex problems; the model training is stable and simple; the loss has extreme values ​​near the true solution, and the converged model is not easy to jump out, with better convergence and training efficiency; the same hyperparameters and training strategies are applicable to problems of different complexities, and there is no need for tedious parameter adjustment for different specific problems; compared with PINN, it has higher computational efficiency and accuracy; the proposed method inherits the mesh convergence of finite elements, and gradually approaches the exact solution as the mesh is refined; general PINN does not have this property, and the accuracy of the model cannot be increased indefinitely with the increase in the number of configuration points.

[0103] The present invention will be further described in detail below with reference to the accompanying drawings and three specific embodiments. The three specific embodiments are referred to as Examples 1 to 3. The experimental parameters of each example are shown in Tables 1 to 5 below:

[0104] Table 1 Relative errors of DFEM and PINN at selected points in Example 1

[0105]

[0106] Table 2 Relative errors of DFEM and PINN at selected points in Example 2

[0107]

[0108] Table 3 Relative error of DFEM in selected points of example 3

[0109]

[0110] Table 4. Transfer learning training time and finite element calculation time for example 1

[0111]

[0112]

[0113] Table 5 Transfer learning training time and finite element calculation time for example 3

[0114]

[0115] Step 1: Meshing

[0116] The proposed method is illustrated by taking three specific problems of partial differential equations of elasticity as examples. The three problems are plate tension with a hole, asymmetric tension, and crane boom. The solution domain is discretized into units, and then the stiffness matrix and node load array are calculated according to formula (8). In order to show the advantages of the proposed method, the PINN is compared with the plate tension with a hole and the asymmetric tension problem, and the configuration points are selected for them respectively. However, due to the high degree of freedom, it is difficult for PINN to obtain a convergent solution for the crane boom problem. The problem description and mesh division are as follows: Figure 2 、 Figure 3 、 Figure 4 shown.

[0117] Step 2: Model training

[0118] The physical loss function of deep finite element is shown in formula (11), while the physical loss function of PINN is shown in formula (17).

[0119]

[0120] In order to introduce the Dirichlet boundary condition, the neural network is transformed as follows for the first two examples:

[0121]

[0122] After the transformation, the neural network can automatically meet the boundary conditions. For the third example, due to the complexity of its boundary surface, the boundary conditions are introduced by modifying the stiffness matrix.

[0123] In this patent, the neural network activation function used is tanh. The network structure of the first two examples is [2,200,200,200,2], and the network structure of the third example is [3,500,500,500,3]. The training loss curves of the three models are as follows: Figure 5 、 Figure 6 、 Figure 7As shown; for the first example, Figure 5 It can be seen that the proposed method is more stable in training and converges faster than PINN. The smaller energy at convergence indicates that it is closer to the true solution. Figure 6 It can be seen that the training of PINN is unstable and does not always converge to the correct solution. Even if it has converged, continued training may even lead to deviation from the optimal solution. However, the proposed deep finite element method is very stable and can quickly converge to the true solution. Figure 7 It is shown that the proposed method can still converge stably even for high degree of freedom problems, while PINN cannot converge on this example.

[0124] Step 3: Model testing

[0125] After training is completed, the model can be tested. Figure 8 、 Figure 9 、 Figure 10 as well as Figure 11 These are the model prediction results of three examples; Figure 8 The displacement field results of the first example are shown. It can be seen that the proposed deep finite element method is significantly more accurate than PINN, and the global error is extremely small. Figure 9 and Figure 10 The results of the second example are combined with Figure 6 It can be seen that when training for 2500 iterations, the results are less accurate than those of the deep finite element method, and when training for 5000 iterations, overfitting occurs and the model completely falls into an incorrect solution. This shows that the training of PINN is not stable enough and its convergence cannot be guaranteed. The proposed method is superior to PINN in terms of training speed, training stability, and prediction accuracy. The results of the third example are predicted as follows: Figure 11 As shown in Figure 1, PINN can no longer obtain a convergent solution for this three-dimensional large degree of freedom problem, while the deep finite element method can still obtain a convergent solution with a small error in the prediction results. The prediction errors of the three examples are shown in Tables 1 to 3.

[0126] Step 4: Model Migration

[0127] The trained model can be used as a pre-training model. If the load changes, transfer learning can be performed on this basis, while the finite element method can only be recalculated. Therefore, the proposed model has more advantages than the finite element method in terms of computational efficiency. Figure 12 and Figure 13 Tables 4 and 5 are the migration training curves of Example 1 and Example 3, respectively. Tables 4 and 5 are the comparisons of their training time and finite element calculation time.

[0128] The present invention can embed partial differential equations into neural networks, and by teaching the neural networks the physical laws, it can make up for the shortcomings of pure data-driven models and make the models more generalizable. Its basic idea is inspired by finite elements and PINN, and takes the system potential energy as the physical loss function. It uses grid technology and local simplification to calculate the potential energy, avoiding the derivative term in the loss function and improving the efficiency and stability of model training.

Claims

1. A deep learning method combining physical information neural network and finite element method, characterized by: Embedding partial differential equations into neural networks involves the following steps: S1. Divide the solution domain into grids and calculate the stiffness matrix and node load array; S2, collect observation data and boundary conditions; S3, using the energy represented by the stiffness matrix and the fitting error of the observed data as the loss function to train the neural network; S4, executing the optimization algorithm to optimize the neural network; S5, and repeat the training and optimization steps until convergence; The loss function consists of physical term loss and data term loss. The data term loss is used to introduce observation data or boundary conditions. The general form of the loss function is shown in the following formula: L=L phys. (θ)+L dat. (i) (12) The data item loss is as follows: The derivative of the physical loss term with respect to the neural network parameters is as follows: When the loss function converges, the solution obtained is the correct solution to the partial differential equation, and the second-order derivative of the loss function with respect to the parameter is greater than 0, which has local convexity.

2. The deep learning method combining physical information neural network and finite element analysis according to claim 1, characterized in that: The neural network is a multi-layer perceptron, comprising an input layer, a hidden layer and an output layer, wherein the hidden layer applies a nonlinear activation function.

3. The deep learning method combining physical information neural network and finite element method according to claim 1, characterized in that: The general form of the partial differential equation can be expressed as follows: Where L represents the differential operator, u(x) represents the vector field function defined on the domain Ω, and R n represents n-dimensional Euclidean space, q(x) represents an excitation term; in addition, B is the operator corresponding to the boundary condition, Γ is a part of the boundary, indicating the area of ​​action of B, represents the boundary of the domain Ω; Assuming that the order of operator L is 2m, the boundary conditions of order 0 to m-1 are called forced boundary conditions, and the boundary conditions of order m to 2m-1 are called natural boundary conditions. The natural variation method requires that the trial function u satisfy the forced boundary conditions in advance; the boundary conditions are decomposed into the following two types: where Γ u is the mandatory boundary, Γ σ is a natural boundary, and B u is a mandatory boundary condition operator with an order of 0 to m-1. σ It is a natural boundary condition operator with an order of m to 2m-1.

4. The deep learning method combining physical information neural network and finite element analysis according to claim 1, characterized in that: The optimization algorithm includes SGD, Adam or L-BFGS.

5. The deep learning method combining physical information neural network and finite element analysis according to claim 1, characterized in that: The stiffness matrix and the node load array provide a simple form for calculating energy, which is expressed as a quadratic function of the node displacement.

6. The deep learning method combining physical information neural network and finite element analysis according to claim 1, characterized in that: The method also includes decomposing the boundary conditions into mandatory boundary conditions and natural boundary conditions, and satisfying the mandatory boundary conditions; or introducing boundary conditions by modifying the neural network structure or modifying the stiffness matrix and the node load vector.

7. The deep learning method combining physical information neural network and finite element analysis according to claim 1, characterized in that: The neural network expresses the test function on the unit into an interpolation form through shape functions to handle complex structural problems.

8. The deep learning method combining physical information neural network and finite element method according to claim 1 or 3, characterized in that: When processing partial differential equations, the method performs conversion through the Galerkin form and obtains a global stiffness matrix and a global node response vector.

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