Novel physical information neural network for solving seismic wave equation

Abstract

The invention relates to the technical field of seismic engineering, in particular to a novel physical information neural network (gs-PINN) for solving a seismic wave equation, which comprises the following steps of: 1, establishing the novel physical information neural network; step 2, dividing configuration points required by network training; 3, determining the form of incident waves; 4, network training is carried out; 5, a solution of a one-dimensional seismic wave equation is obtained; the method is composed of two sub-networks, parameters of a first hidden layer of each sub-network are non-trainable parameters, values are determined by seismic wave propagation directions, other parameters are set as trainable parameters, the network is ensured to automatically meet a one-dimensional seismic wave equation, the network is effectively prevented from falling into local optimum, training difficulty is greatly reduced, and training efficiency is improved. And a powerful technical support is provided for solving a one-dimensional seismic wave equation.

Description

Technical Field

[0001] The invention relates to the technical field of earthquake engineering, and in particular to a novel physical information neural network for solving earthquake wave equations. Background Art

[0002] The propagation of seismic waves is a fundamental problem in geophysics and is crucial for understanding earthquake dynamics, underground imaging, and resource exploration. Seismic wave propagation follows the seismic wave equation, which is a type of partial differential equation (PDE). Seismic wave modeling through this equation can simulate the wave propagation process. In earthquake engineering, seismic wave modeling is used to predict ground motion and assess the potential impact of earthquakes on buildings and infrastructure. In recent years, machine learning, especially deep learning, has gradually become a promising method for solving complex mathematical problems, especially in the field of PDE. Physically-Informed Neural Networks (PINNs) are an innovative framework that combines neural networks with basic physical laws and provides a new paradigm for solving partial differential equations. Using PINNs to solve PDEs has shown good prospects in many fields, including fluid dynamics, heat conduction, and electromagnetism.

[0003] However, the application of PINN in the field of seismic wave propagation is relatively rare. Scholars initially applied PINN to solve seismic wave propagation problems in infinite domains and proved that PINN can predict high-precision solutions even with less labeled data. Subsequently, a continuous training strategy for time domain decomposition was developed, which enabled PINN to successfully simulate seismic wave propagation in infinite and semi-infinite domains under free surface conditions and complex terrain. In addition, a PINN-based proxy model was developed to predict seismic waves excited by seismic loads at different locations. Although these PINN-based methods have shown impressive results in solving seismic wave equations, their performance in long-lasting problems and scenarios involving complex wave inputs has not been effectively verified, which are usually difficult for PINN to solve effectively. Summary of the invention

[0004] The purpose of this application is to provide a new physical information neural network for solving seismic wave equations, aiming to address the challenges faced by existing PINNs in dealing with long-term seismic wave propagation and complex wave input problems.

[0005] The embodiment of the present application provides a new physical information neural network for solving the seismic wave equation. The specific operation steps are as follows:

[0006] Step 1: Establish a new physical information neural network;

[0007] The new physical information neural network consists of two sub-networks, both of which are fully connected neural networks with the same network architecture. The first hidden layer parameter of each sub-network is set to be non-trainable, and the specific value is determined by the propagation direction of the seismic wave. The remaining network parameters are trainable parameters. The input of each sub-network is a space-time coordinate point, and the displacement wave field solved by the network is the superposition value of the output of the two sub-networks.

[0008] Step 2: Divide the configuration points required for network training;

[0009] Determine the size of the two-dimensional space-time computational domain, and divide the three edges corresponding to t=0, z=0, and z=L to obtain the configuration points required for training;

[0010] Step 3: Determine the form of the incident wave;

[0011] Collect incident waves with different duration and frequency components and use them as lower boundary conditions;

[0012] Step 4: Conduct network training:

[0013] In order to solve the solution of the wave equation under specific initial boundary conditions and make the network training results meet the initial boundary conditions, it is necessary to construct a loss function consistent with the initial boundary conditions, which is as follows:

[0014] L all =λ i L i +λ bl L bl +λ bu L bu (20)

[0015] Where L i , L bl and L bu They represent the loss function components corresponding to the initial value condition, the lower boundary condition, and the upper boundary condition, respectively, and the coefficient λ i ,λ l and λ bu are the weight factors of each loss function respectively;

[0016] Step 5, obtaining the solution of the one-dimensional seismic wave equation;

[0017] Input the coordinates of any point in the solution domain into the trained network, and the output value of the network is the solution to the initial boundary value problem at that point.

[0018] Furthermore, in step 1, the gs-PINN includes a one-dimensional seismic wave equation and a gs-PINN construction process; wherein gs-PINN represents a new physical information neural network in the present invention, and PINN is an existing physical information neural network;

[0019] The one-dimensional seismic wave equation describes the propagation of waves generated by seismic events in the interior of the earth; for a one-dimensional isotropic homogeneous medium, the equation can be expressed as follows:

[0020]

[0021] Where u(z,t) represents the displacement at the spatial coordinate z and the time coordinate t;

[0022] The initial value conditions are as follows:

[0023] u(z,0)=0 (2)

[0024] u t (z,0)=0 (3)

[0025] The boundary conditions are as follows:

[0026] u(0,t)=u g (t) (4)

[0027] u z (L,t)=0 (5)

[0028] Where V is the wave propagation speed, u g (t) represents the incident motion, L represents the thickness of the medium, and the subscripts t and z represent the partial derivatives with respect to the time coordinate t and the space coordinate z, respectively;

[0029] By reducing equation (1) to its second standard form, the general solution to equation (1) can be easily determined, and consists of the sum of two functions:

[0030]

[0031] Where f(·) and g(·) are any twice continuously differentiable functions;

[0032] Based on the D'Alembert formula, the solution of the initial boundary value problem defined in equation (1-5) can be expressed as follows:

[0033]

[0034] in:

[0035]

[0036] The construction of gs-PINN is as follows:

[0037] gs-PINN can be expressed as follows:

[0038] u(z,t)=N1(z,t)+N2(z,t) (9)

[0039] Where N1(z, t) and N2(z, t) represent two sub-networks respectively; in order to make gs-PINN automatically satisfy the general solution of the one-dimensional wave equation, assuming that the number of neurons in the first hidden layer of each sub-network is 1, the outputs of the first hidden layers of the two sub-networks can be expressed as follows:

[0040]

[0041] in represents the activation function of the first hidden layer of the j-th sub-network, represents the output of the first hidden layer of the j-th sub-network, represents the weight parameter from the i-th input neuron in the j-th subnetwork to the first neuron in the first hidden layer, Represents the bias value of the first neuron in the first hidden layer of the j-th sub-network;

[0042] Set the activation function to a linear activation function, and let the weights and bias values ​​satisfy the following relationship:

[0043]

[0044] At this time, formula (9) can be written as:

[0045]

[0046] The first-order partial derivative and second-order partial derivative of the above formula with respect to t and z can be expressed as:

[0047]

[0048] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0049]

[0050] Through the above process, equation (19) is transformed into equation (1), which shows that the network itself satisfies the general solution of the wave equation. Therefore, it is not necessary to consider the constraints of the control equation in the loss function during training like the traditional PINN.

[0051] Furthermore, in step 1, a Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0052] w~N(0,σ 2 )(twenty four)

[0053] where σ is the bandwidth parameter, which is set to e -0.5 ; For the remaining layers, the Xavier weight initialization scheme is used, which can be expressed as:

[0054]

[0055] Among them fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

[0056] Furthermore, each subnetwork of the gs-PINN in step 1 is composed of an input layer, five hidden layers and an output layer, the number of neurons in the input layer is 2, the number of neurons in the output layer is 1, the number of neurons in the first hidden layer is 1, and the number of neurons in the remaining hidden layers is set to 64.

[0057] Furthermore, in the step 2, when the equidistant grid partitioning strategy is used to divide the configuration points required for the training network, the spatial interval is set to 20m and the time interval is set to 0.02s.

[0058] Furthermore, in step three, simple harmonic waves with a frequency of 0.5 Hz and durations of 10 s, 20 s and 37.76 s and a lateral seismic motion with a duration of 37.76 s were collected.

[0059] Furthermore, each loss function in step 4 can be calculated as follows:

[0060]

[0061] Where N i 、N bl and N bu They represent the number of sampling points associated with the initial value condition, the lower boundary value loss term, and the upper boundary value loss term, respectively.

[0062] Furthermore, in the step 4, an automatic differentiation module is introduced after the output layer of the network to calculate the derivative, thereby obtaining the loss required for training.

[0063] Furthermore, in step 4, the full batch strategy is used for training, the optimizer is the Adam algorithm, the training rounds are set to 200,000, and the learning rate decay strategy is introduced. The learning rate is kept at 10 in the first 100,000 training rounds. -3 , and the learning rate decays to 10 in the next 50,000 training rounds. -4 , in the last 50,000 training rounds, the learning rate decays to 10 -5 .

[0064] Furthermore, the weight factor λ of the loss function component corresponding to the initial value condition, the lower boundary condition and the upper boundary condition in step 4 is i ,λ l and λ bu Set to 10 respectively0 , 10 0 and 10 5 .

[0065] The beneficial effect of the present invention is as follows: based on the general solution form of the one-dimensional seismic wave equation, the present invention designs a new type of PINN (i.e., gs-PINN), which consists of two sub-networks. The parameters of the first hidden layer of each sub-network are non-trainable parameters, and the values ​​are determined by the propagation direction of the seismic wave. The remaining parameters are set as trainable parameters, which ensures that the network itself automatically satisfies the one-dimensional seismic wave equation, effectively avoids the network from falling into the local optimum, greatly reduces the difficulty of training, and provides strong technical support for solving the one-dimensional seismic wave equation.

[0066] Compared with the current one-dimensional seismic wave equation solution method based on PINN, the network proposed in the present invention has better convergence performance, and does not need to impose the loss associated with the control equation with the most required configuration points during the training process, which greatly improves the convergence efficiency and solution accuracy, and thus has better usability. This method not only demonstrates excellent performance under simple harmonic wave incident conditions, but also has been verified under measured seismic incident conditions. Compared with PINN, it has faster convergence speed and prediction accuracy, which helps to improve the accuracy of site response analysis, assess earthquake risks, and ensure the safety of buildings and foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 Schematic diagram of one-dimensional wave propagation problem.

[0068] Figure 2 The architecture of gs-PINN for solving one-dimensional seismic wave propagation problems, covering the network structure and physical constraint loss function. The black boxes in the figure represent two sub-networks, and the different colors in the dotted boxes represent the use of different activation functions: red corresponds to the sine function, orange represents the hyperbolic tangent function, and green represents the linear function.

[0069] Figure 3 Loss curves of two models considering incident simple harmonic waves with durations of 10s, 20s and 37.76s: (ac) is PINN, and (df) is gs-PINN.

[0070] Figure 4 The displacement fields predicted by the two models and the exact solutions considering incident harmonics with durations of 10 s, 20 s, and 37.76 s: (a, d, g) are PINN, (b, e, h) are gs-PINN, and (c, f, i) are the exact solutions.

[0071] Figure 5The absolute errors of the displacement fields predicted by PINN and gs-PINN relative to the exact solution under incident harmonics with durations of 10s, 20s, and 37.76s are considered: (ac) is PINN, and (df) is gs-PINN.

[0072] Figure 6 Earthquake records recorded at El Centro Array Station 8 during the 1979 Imperial Valley earthquake: (a) acceleration time history, (b) velocity time history, (c) displacement time history, and (d) Fourier amplitude spectrum of the displacement time history.

[0073] Figure 7 Loss curves of the network when using selected ground motions as incident waves: (a) for PINN and (b) for gs-PINN.

[0074] Figure 8 Comparison of the displacement fields and absolute errors predicted by the two models when using selected ground motions as incident waves. Displacement fields: (a) PINN, (b) gs-PINN, (c) exact solution; absolute errors: (d) PINN, (e) gs-PINN; DETAILED DESCRIPTION

[0075] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0076] A new physical information neural network for solving seismic wave equations, referred to as gs-PINN, is characterized by the following specific operation steps:

[0077] Step 1: Establish gs-PINN;

[0078] gs-PINN consists of two sub-networks, both of which are fully connected neural networks with the same network architecture; the first hidden layer parameter of each sub-network is set to be non-trainable, and the specific value is determined by the propagation direction of the seismic wave, and the remaining network parameters are trainable parameters; the input of each sub-network is a time-space coordinate point, and the displacement wave field solved by the network is the superposition value of the output of the two sub-networks; the network includes a one-dimensional seismic wave equation and the construction of gs-PINN;

[0079] The one-dimensional seismic wave equation describes the propagation of waves generated by seismic events in the interior of the earth; for a one-dimensional isotropic homogeneous medium, the equation can be expressed as follows:

[0080]

[0081] Where u(z,t) represents the displacement at the spatial coordinate z and the time coordinate t.

[0082] The initial value conditions are as follows:

[0083] u(z,0)=0 (2)

[0084] u t (z,0)=0 (3)

[0085] The boundary conditions are as follows:

[0086] u(0,t)=u g (t) (4)

[0087] u z (L,t)=0 (5)

[0088] Where V is the wave propagation speed, u g (t) represents the incident motion, L represents the thickness of the medium, and the subscripts t and z represent the partial derivatives with respect to the time coordinate t and the space coordinate z, respectively.

[0089] By reducing equation (1) to its second standard form, the general solution to equation (1) can be easily determined, which consists of the sum of two functions:

[0090]

[0091] where f(·) and g(·) are any twice continuously differentiable functions.

[0092] Based on the D'Alembert formula, the solution of the initial boundary value problem defined in equation (1-5) can be expressed as follows:

[0093]

[0094] in:

[0095]

[0096] The construction of gs-PINN is as follows:

[0097] gs-PINN can be expressed as follows:

[0098] u(z,t)=N1(z,t)+N2(z,t) (9)

[0099] Where N1(z, t) and N2(z, t) represent two sub-networks respectively; in order to make gs-PINN automatically satisfy the general solution of the one-dimensional wave equation, assuming that the number of neurons in the first hidden layer of each sub-network is 1, the outputs of the first hidden layers of the two sub-networks can be expressed as follows:

[0100]

[0101] in represents the activation function of the first hidden layer of the j-th sub-network, represents the output of the first hidden layer of the j-th sub-network, represents the weight parameter from the i-th input neuron in the j-th subnetwork to the first neuron in the first hidden layer, Represents the bias value of the first neuron in the first hidden layer of the j-th sub-network;

[0102] Set the activation function to a linear activation function, and let the weights and bias values ​​satisfy the following relationship:

[0103]

[0104] At this time, formula (9) can be written as:

[0105]

[0106] The first-order partial derivative and second-order partial derivative of the above formula with respect to t and z can be expressed as:

[0107]

[0108] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0109]

[0110] Through the above process, equation (19) is transformed into equation (1), which shows that the network itself satisfies the general solution of the wave equation. Therefore, it is not necessary to consider the constraints of the control equation in the loss function during training like the traditional PINN.

[0111] Step 2: Divide the configuration points required for network training;

[0112] Determine the size of the two-dimensional space-time computational domain. Unlike traditional PINN, which requires partitioning the entire computational domain, gs-PINN only needs to partition the three edges corresponding to t=0, z=0, and z=L to obtain the configuration points required for training.

[0113] Step 3: Determine the form of the incident wave;

[0114] Collect incident waves with different duration and frequency components and use them as lower boundary conditions;

[0115] Step 4: Conduct network training:

[0116] Since gs-PINN itself satisfies the wave equation, in order to solve the wave equation under specific initial and boundary conditions, it is also necessary to make the network training results satisfy the initial and boundary conditions. To achieve this goal, it is necessary to construct a loss function consistent with the initial and boundary conditions, which is as follows:

[0117] L all =λ i L i +λ bl L bl +λ bu L bu (20)

[0118] Where L i , L bl and L bu They represent the loss function components corresponding to the initial value condition, the lower boundary condition, and the upper boundary condition, respectively, and the coefficient λ i ,λ l and λ bu are the weight factors of each loss function. Each loss function can be calculated as follows:

[0119]

[0120]

[0121] Where N i 、N bl and N bu They represent the number of sampling points related to the initial value condition, the lower boundary value loss term, and the upper boundary value loss term respectively;

[0122] Step 5, obtaining the solution of the one-dimensional seismic wave equation;

[0123] Input the coordinates of any point in the solution domain into the trained network, and the output value of the network is the solution to the initial boundary value problem at that point.

[0124] In step 1, a Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0125] w~N(0,σ 2 )(twenty four)

[0126] where σ is the bandwidth parameter, which is set to e -0.5 For the remaining layers, the Xavier weight initialization scheme is used, which can be expressed as:

[0127]

[0128] Among them fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

[0129] In the step 1, each subnetwork of the gs-PINN consists of an input layer, five hidden layers and an output layer. The number of neurons in the input layer is 2, the number of neurons in the output layer is 1, the number of neurons in the first hidden layer is 1, and the number of neurons in the remaining hidden layers is set to 64.

[0130] When the equidistant grid partitioning strategy is used in step 2 to divide the configuration points required for the training network, the spatial interval is set to 20m and the time interval is set to 0.02s.

[0131] In step 3, simple harmonic waves with a frequency of 0.5 Hz and durations of 10 s, 20 s and 37.76 s, respectively, and a lateral earthquake motion with a duration of 37.76 s are collected. Figure 6 Shown is a lateral ground motion with a duration of 37.76 s recorded at El Centro Array Station 8 during the 1979 Imperial Valley earthquake.

[0132] In step 4, an automatic differentiation module is introduced after the output layer of the network to calculate the derivative, thereby obtaining the loss required for training.

[0133] In step 4, the full batch strategy is used for training, the optimizer is the Adam algorithm, the training rounds are set to 200,000, and the learning rate decay strategy is introduced. The learning rate is kept at 10 in the first 100,000 training rounds. -3 , and the learning rate decays to 10 in the next 50,000 training rounds. -4 , in the last 50,000 training rounds, the learning rate decays to 10 -5 .

[0134] The weight factor λ of the loss function component corresponding to the initial value condition, the lower boundary condition and the upper boundary condition in step 4 i ,λ l and λ bu Set to 10 respectively 0 , 10 0 and 10 5 .

[0135] The specific operation steps of the present invention are described below in conjunction with specific embodiments;

[0136] Step 1: Create gs-PINN:

[0137] gs-PINN consists of two subnetworks, both of which are fully connected neural networks with the same network architecture; the first hidden layer parameter of each subnetwork is set to be non-trainable, and the specific value is determined by the propagation direction of the seismic wave, and the remaining network parameters are trainable parameters; the input of each subnetwork is a time-space coordinate point, and the displacement wave field solved by the network is the superposition value of the output of the two subnetworks; each subnetwork of gs-PINN consists of an input layer, five hidden layers and an output layer, the number of neurons in the input layer is 2, the number of neurons in the output layer is 1, the number of neurons in the first hidden layer is 1, and the number of neurons in the remaining hidden layers is set to 64.

[0138] 1. One-dimensional seismic wave equation

[0139] The seismic wave equation describes the propagation of waves generated by seismic events (such as earthquakes or artificial seismic sources) in the Earth's interior. For a one-dimensional isotropic homogeneous medium such as Figure 1 As shown, the equation can be expressed as follows:

[0140]

[0141] Where u(z,t) represents the displacement at the spatial coordinate z and the time coordinate t.

[0142] The initial value conditions are as follows:

[0143] u(z,0)=0 (2)

[0144] u t (z,0)=0 (3)

[0145] The boundary conditions are as follows:

[0146] u(0,t)=u g (t) (4)

[0147] u z (L,t)=0 (5)

[0148] Where V is the wave propagation speed, u g (t) represents the incident motion, L represents the thickness of the medium, and the subscripts t and z represent the partial derivatives with respect to the time coordinate t and the space coordinate z, respectively.

[0149] By reducing equation (1) to its second standard form, the general solution to equation (1) can be easily determined, which consists of the sum of two functions:

[0150]

[0151] where f(·) and g(·) are any twice continuously differentiable functions.

[0152] Based on the D'Alembert formula, the solution of the initial boundary value problem defined in equation (1-5) can be expressed as follows:

[0153]

[0154] in:

[0155]

[0156] 2. Construction of gs-PINN

[0157] like Figure 2 As shown, the proposed gs-PINN can be expressed as follows:

[0158] u(z,t)=N1(z,t)+N2(z,t) (9)

[0159] Where N1(z,t) and N2(z,t) represent two sub-networks respectively.

[0160] In order to make the proposed gs-PINN automatically satisfy the general solution of the one-dimensional wave equation, it is assumed that the number of neurons in the first hidden layer of each sub-network is 1, then the outputs of the first hidden layers of the two sub-networks can be expressed as follows:

[0161]

[0162] in represents the activation function of the first hidden layer of the j-th sub-network, represents the output of the first hidden layer of the j-th sub-network, represents the weight parameter from the i-th input neuron in the j-th subnetwork to the first neuron in the first hidden layer, Represents the bias value of the first neuron in the first hidden layer of the j-th subnetwork.

[0163] Let's set the activation function to a linear activation function and let the weights and bias values ​​satisfy the following relationship:

[0164]

[0165] At this time, formula (9) can be written as:

[0166]

[0167] The first-order partial derivative and second-order partial derivative of the above formula with respect to t and z can be expressed as:

[0168]

[0169] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0170]

[0171] Through the above process, equation (19) is transformed into equation (1), which shows that the network itself satisfies the general solution of the wave equation. Therefore, it is not necessary to consider the constraints of the control equation in the loss function during training like the traditional PINN.

[0172] The Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0173] w~N(0,σ 2 ) (twenty four)

[0174] where σ is the bandwidth parameter, which is set to e -0.5 For the remaining layers, the Xavier weight initialization scheme is used, which can be expressed as:

[0175]

[0176] Among them fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

[0177] Step 2: Divide the configuration points required for network training:

[0178] Determine the size of the two-dimensional space-time computational domain. Different from the traditional PINN which needs to divide the entire computational domain, the proposed gs-PINN only needs to divide the three edges corresponding to t=0, z=0 and z=L to obtain the configuration points required for training. The configuration points are obtained using an equidistant grid division strategy, with the spatial interval set to 20m and the time interval set to 0.02s.

[0179] Step 3: Determine the form of the incident wave:

[0180] The collection frequency is 0.5 Hz, and the durations are 10 s, 20 s, and 37.76 s, respectively. A lateral ground motion with a duration of 37.76 s was recorded at El Centro Array Station 8 during the 1979 Imperial Valley earthquake. Figure 6 As shown, and it is used as the lower boundary condition. The form of simple harmonic wave is as follows:

[0181]

[0182] where A is set to 0.417 m so that the peak of the incident harmonic matches the peak of the selected earthquake motion.

[0183] Step 4: Conduct network training:

[0184] Since the proposed gs-PINN itself satisfies the wave equation, in order to solve the wave equation under specific initial and boundary conditions, it is also necessary to make the network training results satisfy the initial and boundary conditions. To achieve this goal, it is necessary to construct a loss function consistent with the initial and boundary conditions, which is as follows:

[0185] L all =λ i L i +λ bl L bl +λ bu L bu (20)

[0186] Where L i , L bl and L bu They represent the loss function components corresponding to the initial value condition, the lower boundary condition, and the upper boundary condition, respectively, and the coefficient λ i ,λ l and λ bu are the weight factors of each loss function, which are set to 10 respectively. 0 , 10 0 and 10 5 Each loss function can be calculated as follows:

[0187]

[0188] Where N i 、N bl and N bu Respectively represent the number of sampling points related to the initial value condition, the lower boundary loss term and the upper boundary loss term. The derivative term in the loss function can be calculated by the automatic differentiation module. The present invention uses a full batch strategy for training, the optimizer is the Adam algorithm, the training round is set to 200,000, and the learning rate decay strategy is introduced. The learning rate is kept at 10 in the first 100,000 training rounds. -3 , and the learning rate decays to 10 in the next 50,000 training rounds. -4 , in the last 50,000 training rounds, the learning rate decays to 10 -5 .

[0189] Step 5: Obtain the solution of the one-dimensional seismic wave equation:

[0190] Input the coordinates of any point in the solution domain into the trained network, and the output value of the network is the solution to the initial boundary value problem at that point.

[0191] Figure 3 ,4 Figures 5 and 5 show the results of solving the one-dimensional wave equation using the method proposed in the present invention under the simple harmonic wave incident condition. It can be clearly seen that with the increase of duration, the final convergence loss value of PINN gradually increases, while the final convergence loss value of the present invention remains relatively stable. Under each duration, the final overall loss value of gs-PINN is always at least two orders of magnitude lower than that of PINN. From the predicted wave field, gs-PINN has better overall consistency with the exact solution than PINN. Error analysis also shows that with the increase of duration, the accuracy of PINN decreases significantly, especially when the duration is 37.76s. In contrast, gs-PINN can maintain a high accuracy under different duration conditions. Table 1 is a quantitative comparison of PINN and gs-PINN based on the two performance indicators of relative L2 error and training time. The results show that the prediction accuracy of gs-PINN is at least improved by 11.8 times, and the training speed is at least improved by 6.5 times. Moreover, with the increase of the duration of the incident wave, the improvements in prediction accuracy and training speed are more obvious.

[0192] Figure 7 and 8 The results of solving the one-dimensional wave equation using the proposed method in the measured earthquake incident conditions are given. The results show that gs-PINN still exhibits a higher convergence speed and lower final overall loss when dealing with complex incident waves. Figure 3 (c,f), compared with the simple harmonic wave incident condition of the same duration, the convergence speed of PINN decreases significantly, while the performance of gs-PINN remains relatively stable. From the predicted wave field, the predicted wave field of PINN is significantly distorted when processing incident waves with complex frequency components, especially in the later time period. However, gs-PINN is highly consistent with the exact solution and accurately captures the shape and amplitude of the wave field. Error analysis shows that the difference between the displacement field predicted by gs-PINN and the exact solution is significantly lower than that of PINN. Table 2 provides a quantitative comparison of the network performance of PINN and gs-PINN when processing incident waves with complex frequency components. gs-PINN consistently achieves higher prediction accuracy and shorter training time, further verifying its superior performance.

[0193] This paper proposes a new PINN-based approach to solve the one-dimensional wave propagation problem. The approach redesigns the PINN architecture to form a new architecture called gs-PINN. The gs-PINN architecture consists of two sub-networks. In each sub-network, the first hidden layer consists of a neuron with fixed parameters, which is responsible for coordinate transformation. This architectural design ensures that the network architecture is essentially aligned with the general solution of the one-dimensional wave equation, thereby eliminating the equation residual loss usually required during PINN training, thereby improving the convergence speed and prediction accuracy of the network. To verify the performance of the proposed gs-PINN, two test cases were conducted. In the first case, gs-PINN was used to solve the 1D wave equation considering incident harmonic waves. Comparative experiments with PINN show that gs-PINN not only reduces the training time but also achieves higher prediction accuracy. At the same time, the impact of the wave duration on the network performance is analyzed. The results show that the prediction error of gs-PINN remains constant regardless of the wave duration, while the error of PINN increases significantly with the increase of the wave duration. To examine the impact of the frequency content of the incident waves on the performance of the two models, ground motions recorded from the 1940 El Centro earthquake were selected as input. The results show that gs-PINN consistently exhibits fast convergence speed and high-precision predictions, while PINN shows significant distortion in predictions under complex input waves. These findings validate the superior reliability of gs-PINN.

[0194] In summary, the new physical information neural network for solving seismic wave equations is effective and realistic. This method redesigns the PINN architecture and proposes a new architecture called gs-PINN. The gs-PINN architecture consists of two sub-networks. In each sub-network, the first hidden layer consists of a neuron with fixed parameters, which performs coordinate transformation. This architectural design ensures that the network architecture is essentially equivalent to the general solution of the one-dimensional wave equation, thereby eliminating the equation residual loss usually required during PINN training. Table 1. Comparison of relative L2 error and training time of PINN and gs-PINN for incident harmonics of different durations.

[0195]

[0196] Table 2. Comparison of relative L2 errors and training time of PINN and gs-PINN when using selected ground motions as incident waves.

[0197]

[0198] The ratio is the comparison of the relative L2 error and training time of PINN and gs-PINN.

[0199] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A new physical information neural network for solving seismic wave equations, characterized in that: The specific steps are as follows: Step 1: Establish a new physical information neural network; The new physical information neural network consists of two sub-networks, both of which are fully connected neural networks with the same network architecture. The first hidden layer parameter of each sub-network is set to be non-trainable, and the specific value is determined by the propagation direction of the seismic wave. The remaining network parameters are trainable parameters. The input of each sub-network is a space-time coordinate point, and the displacement wave field solved by the network is the superposition value of the output of the two sub-networks. Step 2: Divide the configuration points required for network training; Determine the size of the two-dimensional space-time computational domain, and divide the three edges corresponding to t=0, z=0, and z=L to obtain the configuration points required for training; Step 3: Determine the form of the incident wave; Collect incident waves with different duration and frequency components and use them as lower boundary conditions; Step 4: Conduct network training: In order to solve the solution of the wave equation under specific initial boundary conditions and make the network training results meet the initial boundary conditions, it is necessary to construct a loss function consistent with the initial boundary conditions, which is as follows: L all =λ i L i +λ bl L bl +λ bu L bu (20) Where L i , L bl and L bu They represent the loss function components corresponding to the initial value condition, the lower boundary condition, and the upper boundary condition, respectively, and the coefficient λ i ,λ l and λ bu are the weight factors of each loss function respectively; Step 5, obtaining the solution of the one-dimensional seismic wave equation; Input the coordinates of any point in the solution domain into the trained network, and the output value of the network is the solution to the initial boundary value problem at that point.

2. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In step 1, the gs-PINN includes a one-dimensional seismic wave equation and a gs-PINN construction process; wherein gs-PINN represents a new physical information neural network in the present invention, and PINN is an existing physical information neural network; The one-dimensional seismic wave equation describes the propagation of waves generated by seismic events in the interior of the earth; for a one-dimensional isotropic homogeneous medium, the equation can be expressed as follows: Where u(z,t) represents the displacement at the spatial coordinate z and the time coordinate t; The initial value conditions are as follows: u(z,0)=0 (2) at t (of,0)=0 (3) The boundary conditions are as follows: u(0,t)=u g (t) (4) u z (L,t)=0 (5) Where V is the wave propagation speed, u g (t) represents the incident motion, L represents the thickness of the medium, and the subscripts t and z represent the partial derivatives with respect to the time coordinate t and the space coordinate z, respectively; By reducing equation (1) to its second standard form, the general solution of equation (1) can be easily determined, which consists of the sum of two functions composition: Where f(·) and g(·) are any twice continuously differentiable functions; Based on the D'Alembert formula, the solution of the initial boundary value problem defined in equation (1-5) can be expressed as follows: in: The construction of gs-PINN is as follows: gs-PINN can be expressed as follows: u(z,t)=N1(z,t)+N2(z,t) (9) Where N1(z, t) and N2(z, t) represent two sub-networks respectively; in order to make gs-PINN automatically satisfy the general solution of the one-dimensional wave equation, assuming that the number of neurons in the first hidden layer of each sub-network is 1, the outputs of the first hidden layers of the two sub-networks can be expressed as follows: in represents the activation function of the first hidden layer of the j-th sub-network, represents the output of the first hidden layer of the j-th sub-network, represents the weight parameter from the i-th input neuron in the j-th subnetwork to the first neuron in the first hidden layer, Represents the bias value of the first neuron in the first hidden layer of the j-th sub-network; Set the activation function to a linear activation function, and let the weights and bias values ​​satisfy the following relationship: At this time, formula (9) can be written as: The first-order partial derivative and second-order partial derivative of the above formula with respect to t and z can be expressed as: Substituting formula (16) into formula (18), the latter can be rewritten as follows: Through the above process, Equation (19) is transformed into Equation (1), which shows that the network itself satisfies the general solution of the wave equation. Therefore, during training, there is no need to consider the constraints of the control equation in the loss function like the traditional PINN.

3. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In step 1, a Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as: w~N(0,σ 2 (24) where σ is the bandwidth parameter, which is set to e -0.5 ; For the remaining layers, the Xavier weight initialization scheme is used, which can be expressed as: Among them fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

4. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In the step 1, each subnetwork of the gs-PINN consists of an input layer, five hidden layers and an output layer. The number of neurons in the input layer is 2, the number of neurons in the output layer is 1, the number of neurons in the first hidden layer is 1, and the number of neurons in the remaining hidden layers is set to 64.

5. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: When the equidistant grid partitioning strategy is used in step 2 to divide the configuration points required for the training network, the spatial interval is set to 20m and the time interval is set to 0.02s.

6. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In the step three, simple harmonic waves with a frequency of 0.5 Hz and durations of 10 s, 20 s and 37.76 s, respectively, and a lateral seismic motion with a duration of 37.76 s were collected.

7. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: Each loss function in step 4 can be calculated as follows: Where N i 、N bl and N bu They represent the number of sampling points associated with the initial value condition, the lower boundary value loss term, and the upper boundary value loss term, respectively.

8. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In step 4, an automatic differentiation module is introduced after the output layer of the network to calculate the derivative, thereby obtaining the loss required for training.

9. The novel physical information neural network for solving seismic wave equations according to claim 1 is characterized in that: In step 4, the full batch strategy is used for training, the optimizer is the Adam algorithm, the training rounds are set to 200,000, and the learning rate decay strategy is introduced. The learning rate is kept at 10 in the first 100,000 training rounds. -3 , and the learning rate decays to 10 in the next 50,000 training rounds. -4 , in the last 50,000 training rounds, the learning rate decays to 10 -5 .

10. The novel physical information neural network for solving seismic wave equations according to claim 1, characterized in that: The weight factor λ of the loss function component corresponding to the initial value condition, the lower boundary condition and the upper boundary condition in step 4 i ,λ l and λ bu Set to 10 respectively 0 , 10 0 and 10 5 .

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