Numerical simulation method of near-field wave based on physical driving deep learning

Abstract

The present application relates to a kind of physical driving deep learning-based near-field wave numerical simulation method, including sampling step, training step, simulation step;In the sampling step, space-time sample collection is realized using Sobol sequence algorithm, in the training step, artificial neural network model and loss function are constructed, and optimizer mode is optimized, and in the simulation step, the wave equation solution and the solution of each order partial derivative of the near-field wave of any spatial point under any resolution is predicted using the trained artificial neural network.The present application has the advantages of meshless, refinement, wave field side boundary full transmission, etc., for typical working conditions, the neural network formed by training has the generalization ability under different initial conditions, combined with transfer learning, the training efficiency of network can be efficiently improved.

Description

Technical Field

[0001] This invention belongs to the field of artificial intelligence technology, specifically relating to a numerical simulation method for near-field fluctuations based on physics-driven deep learning. Background Technology

[0002] To simulate the propagation characteristics of seismic waves in an infinite domain, various numerical methods place high demands on the spatial discretization methods of the computational model and the handling of artificial boundaries. In the past few decades, wave simulation has primarily employed methods such as the finite difference method, the finite element method, and the spectral element method, transforming analytical problems involving continuous variables into numerical problems involving discrete variables. The quality of the solution depends heavily on the mesh generation; for complex wave problems, this can lead to enormous computational and storage requirements. Furthermore, the accuracy and stability of the introduced artificial boundaries have always been a crucial issue in the numerical simulation of near-field waves. Summary of the Invention

[0003] To address the above-mentioned technical problems in this field, the present invention proposes a near-field wave numerical simulation method based on physics-driven deep learning, the method comprising the following steps:

[0004] Sampling steps: The Sobol sequence algorithm is used to collect several spatiotemporal sampling points in the global domain Ω, Dirichlet boundary, and Neumann boundary of the wave equation; the wave equation includes a one-dimensional wave equation and a two-dimensional SH wave equation; the wave equation satisfies the differential equation and constraint form shown in Equation I:

[0005]

[0006] In formula I, Let f(x,t) be a differential operator with parameter λ, u(x,t) be a solution to the differential equation f(x,t), and vector x be a spatial variable in the global domain Ω. Indicates the Dirichlet boundary. Denotes the Neumann boundary, with the global domain Ω∈R. d (d×1,2,…,n), where t∈[0,T] is the time variable, D(t) and N(t) represent the Dirichlet boundary and Neumann boundary conditions in the initial-boundary value problem of the partial differential equation, respectively, and 0(x) and 1(x) represent the initial conditions, respectively.

[0007] Training steps: Calculate the initial conditions using numerical methods. The initial conditions include two early displacement fields u1(x1,z1,t1) and u2(x2,z2,t2) at times t1 and t2.

[0008] The initial conditions and the spatiotemporal sampling points are input into an artificial neural network, which outputs an approximate solution.

[0009] The approximate solution is calculated using a deep learning-based automatic differentiation method. Second-order partial derivative

[0010] The physical total loss L(θ) at each spatiotemporal sampling point is calculated according to the loss function, and the physical total loss L(θ) is fed back to the artificial neural network to adjust the trainable parameter θ for the next training, until the physical total loss L(θ) is reduced to a preset value after several training cycles.

[0011] Simulation steps: The artificial neural network trained by the above training steps is used to predict the solution of the wave equation and the partial derivatives of the solution for near-field waves at any spatiotemporal point at any resolution.

[0012] This method can simulate waves and achieve high-precision generalization by combining sparse initial wavefield data. It has advantages such as being meshless, fine-grained, and having full transmission of wavefield side boundaries. For typical working conditions, the trained neural network has the ability to generalize under different initial conditions. Combined with transfer learning, it can efficiently improve the training efficiency of the network. This method can also embed different boundary conditions such as stress type and displacement type. For specific technical effects, please refer to the embodiments and verification examples. Attached Figure Description

[0013] Figure 1 : A diagram of the physical information neural network architecture for solving the two-dimensional SH wave equation;

[0014] Figure 2 One-dimensional fluctuation problem without passive terms: spatiotemporal distribution of sampling points for different loss terms;

[0015] Figure 3 One-dimensional fluctuation problem without passive terms: Evolution of different loss terms during neural network training;

[0016] Figure 4 One-dimensional fluctuation problem without passive terms: Comparison of analytical solutions and PINN predictions

[0017] Figure 5 : Calculation model of intrinsic SH wave in an infinite homogeneous medium;

[0018] Figure 6 The spatiotemporal sampling point distribution of different loss terms in the wave problem of an infinitely uniform medium;

[0019] Figure 7 Wave field problems in infinite homogeneous media: Snapshots and absolute errors of wave fields in spectral element method and PINN simulation;

[0020] Figure 8 Wavefield snapshots and absolute errors obtained through transfer learning using the spectral element method and the PINN method;

[0021] Figure 9 Physical wave velocity distribution in spatially inhomogeneous media;

[0022] Figure 10 Wave field problems in spatially inhomogeneous media: Wave field snapshots and absolute errors in spectral element method and PINN simulation;

[0023] Figure 11 Computational model for endogenous wave problems in undulating terrain;

[0024] Figure 12 Spatiotemporal sampling point distribution of different loss terms in the problem of endogenous fluctuations in undulating terrain;

[0025] Figure 13 : The problem of intrinsic waves in undulating surfaces: wave field snapshots and absolute errors in spectral element method and PINN simulation;

[0026] Figure 14 : Near-field wave numerical simulation device. Detailed Implementation

[0027] The following embodiments further illustrate the content of the present invention, but should not be construed as limiting the present invention. Any modifications or substitutions made to the methods, steps, or conditions of the present invention without departing from the spirit and essence of the invention are within the scope of the present invention.

[0028] Some implementation methods include the following steps:

[0029] Sampling steps: The Sobol sequence algorithm is used to collect several spatiotemporal sampling points in the global domain Ω, Dirichlet boundary, and Neumann boundary of the wave equation; the wave equation includes a one-dimensional wave equation and a two-dimensional SH wave equation; the wave equation satisfies the differential equation and constraint form shown in Equation I:

[0030]

[0031] In formula I, Let f(x,t) be a differential operator with parameter λ, u(x,t) be a solution to the differential equation f(x,t), and vector x be a spatial variable in the global domain Ω. Indicates the Dirichlet boundary. Denotes the Neumann boundary, with the global domain Ω∈R. d (d = 1, 2, ..., n), where t ∈ [0, T] is the time variable. D (t), N (t) represent the Dirichlet boundary and Neumann boundary conditions in the initial-boundary value problem of the partial differential equation, respectively, and 0(x) and 1(x) represent the initial conditions, respectively.

[0032] Training steps: Calculate the initial conditions using numerical methods. The initial conditions include two early displacement fields u1(x1,z1,t1) and u2(x2,z2,t2) at times t1 and t2.

[0033] The initial conditions and the spatiotemporal sampling points are input into an artificial neural network, which outputs an approximate solution.

[0034] The approximate solution is calculated using a deep learning-based automatic differentiation method. Second-order partial derivative

[0035] The physical total loss L(θ) at each spatiotemporal sampling point is calculated according to the loss function, and the physical total loss L(θ) is fed back to the artificial neural network to adjust the trainable parameter θ for the next training, until the physical total loss L(θ) is reduced to a preset value after several training cycles.

[0036] Simulation steps: The artificial neural network trained by the above training steps is used to predict the solution of the wave equation and the partial derivatives of the solution for near-field waves at any spatiotemporal point at any resolution.

[0037] The one-dimensional wave equation has the following partial differential equation and constraints:

[0038]

[0039] Initial conditions:

[0040] u(x,0)=u t (x,0)=sin(πx)

[0041] Boundary conditions:

[0042] u(0,t)=0

[0043] u(L,t)=0

[0044] Wherein, wave speed c = 1.

[0045] The two-dimensional SH wave equation has the following partial differential equation form:

[0046]

[0047] u(x,z,t) is the out-of-plane wave displacement, and c is the physical wave velocity of the medium. f(x,z,t) is the external load, which is set to ≡ 0 and is applied by means of an equivalent external force through a given initial wave field.

[0048] The loss function includes a PDE residual loss term, an initial condition loss term, and a boundary condition loss term; the loss function has an expression as shown in Equation II-V:

[0049]

[0050]

[0051]

[0052]

[0053] in, These are the PDE residual loss term, boundary condition loss term, and initial condition loss term, respectively; in the global domain Ω, Dirichlet boundary... and Neumann boundary The number of selected spatiotemporal sampling points is N. p N d N n where time t∈[0,T]; This represents the approximate solution. D (t), N 0(t) and 1(t) represent the Dirichlet and Neumann boundary conditions in the initial-boundary value problem of the partial differential equation, respectively, and 0(t) and 1(t) represent the initial condition functions, respectively.

[0054] Deep learning has now implemented strong-form automatic differentiation techniques, which are integrated into development platforms including PyTorch and Tensorflow. These techniques obtain the partial derivatives of each order of the equation solution approximated by the neural network, and incorporate the PDE and initial and boundary value residuals as regularization terms into the loss function.

[0055] In such Figure 1 In the PINN framework for solving the two-dimensional SH wave equation, the input to the network is specified as the spatiotemporal coordinates X = (x, z, t), and the output is the PDE solution approximated by the neural network, i.e., the displacement field Y = u(x, z, t). Using a deep learning automatic differentiation method, the following is calculated: And construct the loss function.

[0056] In some implementations, the training steps involve using the spectral element method to calculate the early displacement fields u1(x1,z1,t1) and u2(x2,z2,t2), which employs a second-order explicit Newmark time step scheme. The first displacement field u1(x1,z1,t1) is used to constrain the position and shape of the earthquake source, and the second displacement field u2(x2,z2,t2) is used to constrain the direction of wave propagation.

[0057] In some implementations, the Tanh activation function is used as the nonlinear activation function of the artificial neural network during the training steps.

[0058] In some implementations, the artificial neural network is trained using the Adam and / or L-BFGS optimizers during the training steps.

[0059] In some implementations, Xavier is used to initialize the parameters of the artificial neural network during the training steps.

[0060] In some implementations, the training steps of the artificial neural network include a fully connected artificial neural network with 3-5 hidden layers and 30-100 neurons per layer.

[0061] Some implementations include simulation steps that predict the intrinsic SH wave equation in an infinitely homogeneous medium and the SH wave equation in a spatially inhomogeneous medium.

[0062] Some implementations include simulation steps that involve using a perfectly matched layer (PML) at the spatial boundary of the wave equation to simulate the propagation of a point source in an infinite medium.

[0063] The method of the present invention will be explained in detail through the following embodiments.

[0064] Example 1

[0065] Taking a passive-term one-dimensional wave (one-dimensional wave equation) as an example, this paper introduces the implementation method of physics-driven deep learning. The one-dimensional wave equation has the following partial differential equation and constraints:

[0066]

[0067] Initial conditions:

[0068] u(x,0)=u t (x,0)=sin(πx)

[0069] Boundary conditions:

[0070] u(0,t)=0

[0071] u(L,t)=0

[0072] Wherein, wave speed c = 1.

[0073] The Sobol sequence algorithm was used to generate spatiotemporal sampling points for the PDE residual loss term, initial condition loss term, and boundary condition loss term, respectively. The spatial distribution of the sampling points is as follows: Figure 1 As shown, a fully connected neural network with 3 hidden layers and 100 neurons per layer is used. The relative error between the prediction result and the analytical solution of the near-field wave numerical simulation method based on physics-driven deep learning PINN is defined as:

[0074]

[0075] In the above formula, The displacement field at time T1 predicted by the PINN method, u ref (x, t = T1) represents the displacement field at time T1 in the analytical solution. The number of sampling points for prediction is N.

[0076] The Adam optimizer was trained for 40,000 steps, and the convergence process of each loss term and relative error is as follows: Figure 2 As shown. A comparison of PINN's predicted solution with its analytical solution and the numerical solution calculated using the finite difference method is presented, for example... Figure 3 As shown in the figure. In the finite difference method, both spatial and temporal discretization employ the standard second-order central difference scheme. The maximum absolute errors between the predicted solution of PINN and the analytical solution of the finite difference method are 0.008m and 0.007m, respectively. Figure 3 This indicates that the trained neural network has a strong ability to approximate the true solution of the wave equation.

[0077] Example 2

[0078] This paper uses the two-dimensional SH wave problem to illustrate the implementation process of solving the wave equation using a physics-driven deep learning method. The two-dimensional SH wave equation has the following partial differential equation form:

[0079]

[0080] u(x,z,t) is the out-of-plane wave displacement, and c is the physical wave velocity of the medium. f(x,z,t) is the external load, which is set to ≡ 0 and is applied by means of an equivalent external force through a given initial wave field.

[0081] In such Figure 1 In the PINN framework for solving the two-dimensional SH wave equation, the input to the network is specified as the spatiotemporal coordinates X = (x, z, t), and the output is the PDE solution approximated by the neural network, i.e., the displacement field Y = u(x, z, t). Using a deep learning automatic differentiation method, the following is calculated: And construct the loss function.

[0082] This embodiment uses the two early displacement fields at times t1 and t2 calculated by the spectral element method as initial conditions, and the first wavefield snapshot. The location and shape of the seismic source were constrained; a second wavefield snapshot. Constrain the direction of wave propagation. The boundary condition of zero stress perpendicular to the ground surface can be achieved using the Neumann boundary condition in the loss function. This indicates that it includes the PDE residual loss term. Initial condition loss term Free boundary condition loss term loss function It can be represented as:

[0083]

[0084]

[0085] The solution to the partial differential equation approximated by the neural network is, i.e. In the example of the SH wave problem in an infinite medium, the loss function consists only of the PDE residual loss term and the initial condition loss term, i.e., (θ) = p (θ)+ ic (θ). The physical information neural network architecture for solving the two-dimensional SH wave equation is as follows: Figure 4 As shown.

[0086] This embodiment selects the Tanh activation function as the nonlinear activation function of the neural network to ensure that the network has a non-zero second derivative and achieves efficient convergence. The optimization strategies employed are Adam and L-BFGS optimizers. The Adam optimization algorithm achieves fast convergence across the entire network, followed by L-BFGS optimization to obtain a high-precision solution. Xavier initialization is used to initialize the trainable parameters in the neural network.

[0087] Verification Example 1

[0088] Taking the problem of intrinsic SH waves in an infinitely homogeneous medium as an example, the feasibility of the method is verified. The region where the model is established in this embodiment is as follows: Figure 5 The target is a rectangle 600m wide and 600m high, with a medium physical wave velocity c = 400m / s. Displacement fields calculated using the spectral element method at t1 = 0.18s and t2 = 0.2s were used as initial conditions for training. Results at other times were used as true solutions for accuracy comparison with PINN calculations. The spectral element method uses a second-order explicit Newmark time-step scheme with a time step of 1e. -4 The total computation time was 0.9s. According to this invention, the computation time starts from the initial condition time t = 0.18s, and the total training time is 0.72s. The model was spatially discretized on a 100×100 grid using fourth-order Legendre spectral units, with only 50×50 sampling points (2500 points) used for the initial condition, to verify the effectiveness of the physical information neural network under sparse observation data.

[0089] Perfectly matched layers (PMLs) were used at the boundary of regions with a thickness of 5 elements on each of the four sides of the model to simulate the propagation of a point source in an infinite medium. The input wave displacement time history used a Gaussian pulse with a dominant frequency of 20Hz, and the source location was at the center of the model at x = 300m and z = 300m. The Sobol sequence algorithm was used to generate sampling points and initial condition sampling points throughout the entire domain for calculating the PDE residual loss term, with the spatial distribution as shown below. Figure 6 As shown. Number of sampling points N p =10000, Nic1 =2500, N ic2 =2500.

[0090] A fully connected network with 5 hidden layers and 30 neurons per layer approximates the solution to Equation VIII. A learning rate of 6e is used. -3 The Adam optimizer and L-BFGS optimizer were each trained for 10,000 steps. This optimized combination enhanced PINN's global search and local tuning capabilities, resulting in high-precision results. After training, the model can predict the solutions to the wave equation and their partial derivatives at any spatiotemporal point with any resolution. Figure 7 A comparison of the prediction results from the PINN method and the simulation results from the spectral element method at four time points is presented. The coefficients of determination R0 between the wavefields simulated by the PINN method and the spectral element method at these four time points are calculated. 2 At t=0s, R 2 =0.999830; at t=0.02s, R 2 =0.999738; at t=0.32s, R 2 =0.999280; at t=0.72s, R 2 =0.999203.

[0091] Figure 7 This indicates that the PINN method's prediction results for the wave field are basically consistent with those of the spectral elements, accurately capturing the propagation characteristics of waves in an infinite medium. No constraints were imposed on the boundaries during training, and waves within the inner domain can also achieve boundary transmission.

[0092] After multi-condition testing, it was found that the trained neural network has the ability to generalize under different initial conditions. By simply changing the source positions in the previous example to x = 200m and z = 200m, displacement fields generated using the spectral element method at t = 0.18s and t = 0.2s were input into the network as new initial conditions to constitute the loss function. The neural network trained at source positions x = 300m and z = 300m can directly generalize to the new initial conditions, obtaining high-precision prediction results, such as... Figure 8 As shown, wavefield boundary transmission can be achieved across the entire domain. The coefficients of determination R0 between the wavefield simulated by the PINN method and the spectral element method at four time points were calculated. 2 At t=0s, R 2 =0.999569; at t=0.17s, R 2 =0.999105; at t=0.52s, R 2 =0.998754; at t=1.02s, R 2 =0.998382. The simulation results verify the feasibility of applying transfer learning to the near-field fluctuation problem.

[0093] Furthermore, based on the prediction comparison results at t=1.02s, it can be found that the neural network with a total training time of 0.72s has the ability to extrapolate time under new initial conditions, achieving high-precision prediction of wave fields exceeding the original training time by at least 42.8%. Transfer learning can improve the generalization ability of the PINN method in near-field wave problems. It can also be connected to the LBFGS optimizer for further training, thereby obtaining even more accurate results.

[0094] Verification Example 2

[0095] Consider the two-dimensional SH wave problem in a spatially inhomogeneous medium. The model is established in a rectangular region 1000m wide and 1000m high. Based on a wave velocity c = 1000m / s, four complex two-dimensional Gaussian mixture wave velocity distributions are added. The computational model and the physical wave velocity distribution of the medium are as follows: Figure 9 As shown, the wave velocity distribution ranges from 500 to 1000 m / s.

[0096] The displacement fields at t=0.15s and t=0.175s (with 50×50 sampling points) were calculated using the spectral element method as initial conditions for training. Results at other time points were compared with those obtained using the PINN method. The time step in the second-order explicit Newmark time step format of the spectral element method is 5e. -5 The total computation time is 0.7s. In PINN, the computation time starts from the time of the first initial condition t = 0.15s, and the total training time is 0.55s.

[0097] The training consists of two parts, first using a learning rate of 6e -4 The Adam optimizer was trained for 10,000 epochs to achieve global convergence, and then the L-BFGS optimizer was called to achieve optimization convergence, with a training step count of 40,000 steps. Figure 10 The results of PINN predictions and spectral element method simulations at different time points were compared. The coefficients of determination R0 between the wavefields simulated by the PINN method and the spectral element method at four time points were calculated. 2 At t=0s, R 2 =0.999985; at t=0.025s, R 2 =0.999984; at t=0.45s, R 2 =0.999757; at t=0.55s, R 2 =0.998933. The results show that the PINN method can achieve high-precision simulation of fluctuations in non-uniform media.

[0098] Verification Example 3

[0099] A key challenge in using physics-driven neural networks to simulate wave motion is the method for applying free boundary conditions. While the concept of free surface conditions is theoretically clear and can be automatically satisfied by the finite element method and spectral element method, surface undulations increase the difficulty of solving and modeling. The method of this invention is used to verify relevant working conditions.

[0100] Build as Figure 11 The computational model has a left boundary length of 450m and a bottom boundary length of 1200m. The physical wave velocity of the medium is c = 1000m / s. The displacement fields in the infinite medium at t = 0.09s and t = 0.1s were calculated as initial conditions using the spectral element method, and the displacement field at t = 0.3s under the undulating surface was used for accuracy verification. Both computational models are spatially discretized on a 100×100 grid using fourth-order Legendre spectral elements, with 2500 sampling points in a 50×50 grid for each initial condition. The absorbing boundary still uses a perfectly matched layer (PML) with a thickness of 5 elements. The input wave displacement time history uses a Gaussian pulse with a dominant frequency of 20Hz, and the source location is x = 250m and z = 750m. The Sobol sequence algorithm is used to generate sampling points in the entire domain for calculating the PDE residual loss term and the free boundary condition loss term, with the number of sampling points being N. p =10000,N bc =2500. The spatial distribution of sampling points is as follows: Figure 12 As shown.

[0101] A fully connected network with 5 hidden layers and 30 neurons per layer. The learning rate is 6e. -3 The Adam optimizer was trained for 10,000 epochs, and then the L-BFGS optimizer was trained for 40,000 epochs. Figure 13 The results of PINN predictions and spectral element method simulations at three time points are presented. The coefficients of determination R0 between the wavefields simulated by the PINN method and the spectral element method at four time points are calculated. 2 At t=0s, R 2 =0.999880; at t=0.01s, R 2 =0.999894; at t=0.085s, R 2 =0.999271; at t=0.21s, R 2 =0.998152.

[0102] Some implementation methods propose, such as Figure 14 A near-field wave numerical simulation device, comprising a calculator and a memory, wherein the calculator sends near-field wave numerical simulation data based on physics-driven deep learning to the memory.

[0103] The embodiments of the subject matter described in this specification can be implemented as one or more computer programs, that is, one or more modules of computer program instructions encoded on one or more tangible, non-transitory program carriers, for execution by a data processing device or to control the operation of a data processing device. The computer program instructions can be encoded on artificially generated propagated signals. The computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) can be written in any form of programming language, including compiled languages, interpreted languages, declarative languages, or procedural languages, and the computer program can be expanded in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment.

[0104] The processing and logic flows described in this specification can be executed by one or more programmable computers that execute one or more computer programs by processing input data and generating output to run functions. Computers suitable for implementing computer programs include, and exemplarily may be based on general-purpose microprocessors, special-purpose microprocessors, or both, or any other type of central processing unit. Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media, and memory devices.

[0105] While this specification contains numerous specific implementation details, these should not be construed as limiting the scope of any invention or the scope of the claims, but rather as descriptions of features that can embody specific embodiments of a particular invention. Specific features described in this specification within the context of an independent embodiment may also be implemented in combination with a single embodiment. Conversely, various features described within the context of a single embodiment may also be implemented independently in multiple embodiments, or in any suitable sub-combination. Furthermore, while features may be described for combination and even initially claimed in this way, one or more features from a claimed combination may be removed from that combination in some cases, and the claimed combination may be redirected to a sub-combination or a variation thereof.

[0106] Similarly, although operations are described in the accompanying drawings in a specific order, it should not be construed as requiring that such operations be performed in the specific order shown or in sequential order, or that all illustrated operations be performed, in order to achieve the desired result. In certain cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above embodiments should not be construed as requiring such separation in all embodiments, and it should be understood that program components and systems can generally be integrated into a single software product or packaged into multiple software products.

[0107] Although the present invention has been described in detail above with general descriptions, specific embodiments, and experiments, modifications or improvements can be made to it, which will be obvious to those skilled in the art. Therefore, all such modifications or improvements made without departing from the spirit of the present invention fall within the scope of protection claimed by the present invention.

Claims

1. A numerical simulation method for near-field waves based on physics-driven deep learning, characterized in that, The method includes the following steps: Sampling steps: The Sobol sequence algorithm is used to collect several spatiotemporal sampling points in the global domain Ω, Dirichlet boundary, and Neumann boundary of the wave equation; the wave equation includes a one-dimensional wave equation and a two-dimensional SH wave equation; the wave equation satisfies the differential equation and constraint form shown in Equation I: In formula I, Let f(x,t) be a differential operator with parameter λ, u(x,t) be a solution to the differential equation f(x,t), and vector x be a spatial variable in the global domain Ω. Indicates the Dirichlet boundary. Denotes the Neumann boundary, with the global domain Ω∈R. d (d×1,2,…,n), where t∈[0,T] is the time variable, D(t) and N(t) represent the Dirichlet and Neumann boundary conditions in the initial-boundary value problem of the partial differential equation, respectively, and 0(x) and 1(x) represent the initial condition functions, respectively. Training steps: Calculate the initial conditions using numerical methods. The initial conditions include two early displacement fields u1(x1,z1,t1) and u2(x2,z2,t2) at times t1 and t2. The initial conditions and the spatiotemporal sampling points are input into an artificial neural network, which outputs an approximate solution. The approximate solution is calculated using a deep learning-based automatic differentiation method. Second-order partial derivative The physical total loss L(θ) at each spatiotemporal sampling point is calculated according to the loss function, and the physical total loss L(θ) is fed back to the artificial neural network to adjust the trainable parameter θ for the next training, until the physical total loss L(θ) is reduced to a preset value after several training cycles. Simulation steps: The artificial neural network trained by the above training steps is used to predict the wave equation solution and the partial derivatives of the solution for near-field waves at any spatiotemporal point at any resolution.

2. The method as described in claim 1, characterized in that, The loss function includes a PDE residual loss term, an initial condition loss term, and a boundary condition loss term; the loss function has an expression as shown in equations II-V: in, These are the PDE residual loss term, boundary condition loss term, and initial condition loss term, respectively; in the global domain Ω, Dirichlet boundary... and Neumann boundary The number of selected spatiotemporal sampling points is N. p N d N n where time t∈[0,T]; Let D(t) and N(t) represent the approximate solution, where D(t) and N(t) represent the Dirichlet and Neumann boundary conditions in the initial-boundary value problem of the partial differential equation, respectively, and 0(t) and 1(t) represent the initial condition functions, respectively.

3. The method as described in claim 1, characterized in that, In the training step, the early displacement fields u1(x1,z1,t1) and u2(x2,z2,t2) are calculated using the spectral element method, which uses a second-order explicit Newmark time step format. The first displacement field u1(x1,z1,t1) is used to constrain the position and shape of the earthquake source, and the second displacement field u2(x2,z2,t2) is used to constrain the direction of wave propagation.

4. The method as described in claim 1, characterized in that, In the training step, the Tanh activation function is used as the nonlinear activation function of the artificial neural network.

5. The method as described in claim 1, characterized in that, In the training step, the artificial neural network is trained using the Adam and / or L-BFGS optimizers.

6. The method as described in claim 1, characterized in that, In the training step, Xavier is used to initialize the parameters of the artificial neural network.

7. The method as described in claim 1, characterized in that, The artificial neural network comprises a fully connected artificial neural network with 3-5 hidden layers and 30-100 neurons per layer.

8. The method as described in claim 1, characterized in that, The simulation steps include predicting the intrinsic SH wave equation in an infinitely homogeneous medium and the SH wave equation in a spatially inhomogeneous medium.

9. The method as described in claim 8, characterized in that, The simulation steps include simulating the propagation of a point source in an infinite medium using a perfectly matched layer PML at the spatial boundary of the wave equation.

10. A near-field wave numerical simulation device, comprising a calculator and a memory, characterized in that, The calculator sends near-field wave numerical simulation data based on any of the physical-driven deep learning methods described in claims 1-9 to the memory.

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