A new physical information neural network method for solving seismic wave equations

Abstract

The present invention relates to the field of earthquake engineering technology, and in particular to a novel physical information neural network (gs-PINN) for solving seismic wave equations. The gs-PINN comprises the following steps: first, establishing the novel physical information neural network; second, dividing the configuration points required for network training; third, determining the form of the incident wave; fourth, conducting network training; and fifth, obtaining a solution to the one-dimensional seismic wave equation. The gs-PINN comprises two sub-networks, wherein the parameters of the first hidden layer of each sub-network are non-trainable parameters whose values ​​are determined by the propagation direction of the seismic wave, and the remaining parameters are set as trainable parameters, thereby ensuring that the network itself automatically satisfies the one-dimensional seismic wave equation, effectively preventing the network from falling into a local optimum, significantly reducing the training difficulty, and providing strong technical support for solving the one-dimensional seismic wave equation.

Description

Technical Field

[0001] The present invention relates to the field of earthquake engineering technology, in particular to a method for solving earthquake wave equations. Background Art

[0002] Seismic wave propagation is a fundamental problem in geophysics, crucial for understanding earthquake dynamics, subsurface imaging, and resource exploration. Seismic wave propagation follows the seismic wave equation, a type of partial differential equation (PDE). Using this equation to model seismic waves can simulate wave propagation. 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, particularly deep learning, has emerged as a promising approach for solving complex mathematical problems, particularly in the field of PDEs. Physically Informed Neural Networks (PINNs), an innovative framework that combines neural networks with fundamental physical laws, offer a new paradigm for solving partial differential equations. Using PINNs to solve PDEs has shown promise in a variety of 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 problems that PINN finds difficult to solve effectively. Summary of the Invention

[0004] The purpose of this application is to provide a new physical information neural network method 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 present embodiment provides a novel physical information neural network method for solving seismic wave equations. The specific 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 non-trainable, with the specific value determined by the propagation direction of the seismic wave, while 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 of the outputs 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 wave equation under specific initial and boundary conditions and make the network training results meet the initial and boundary conditions, it is necessary to construct a loss function consistent with the initial and boundary conditions, which is as follows:

[0014] (20)

[0015] Among them L i 、L bl and L bu Respectively represent the loss function components corresponding to the initial value condition, lower boundary condition and upper boundary condition, and the coefficients λ i , λ l and λ bu are the weight factors of each loss function respectively;

[0016] Step 5: Obtain 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, the gs-PINN in step 1 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 within the Earth. For a one-dimensional isotropic homogeneous medium, the equation can be expressed as follows:

[0020] (1)

[0021] in u ( z,t ) represents the spatial coordinate z and time coordinates t The displacement at

[0022] The initial value conditions are as follows:

[0023] (2)

[0024] (3)

[0025] The boundary conditions are as follows:

[0026] (4)

[0027] (5)

[0028] in V is the wave propagation speed, u g (t) represents the incident motion, L Indicates the thickness of the medium, subscript t and z Respectively represent the time coordinate t and spatial coordinates z The partial derivative of

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

[0030] (6)

[0031] in f (×) and g (×) is any twice continuously differentiable function;

[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] (7)

[0034] in:

[0035] (8)

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

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

[0038] (9)

[0039] Where N1( z , t ) and N2( z , t ) represent the 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] (10)

[0041] (11)

[0042] 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 sub-network 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;

[0043] Set the activation function to a linear activation function, and let the weights and bias values ​​satisfy the following relationship:

[0044] (12)

[0045] (13)

[0046] At this time, formula (9) can be written as:

[0047] (14)

[0048] The above formula is correct t and z The first-order partial derivative and the second-order partial derivative can be expressed as:

[0049] (15)

[0050] (16)

[0051] (17)

[0052] (18)

[0053] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0054] (19)

[0055] 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 as in traditional PINN.

[0056] Furthermore, in step 1, a Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0057] (twenty four)

[0058] in s 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:

[0059] (25)

[0060] in fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

[0061] Furthermore, each subnetwork of the gs-PINN in step 1 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.

[0062] 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 20 m and the time interval is set to 0.02 s.

[0063] 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, respectively, and a lateral seismic motion with a duration of 37.76 s were collected.

[0064] Furthermore, each loss function in step 4 can be calculated as follows:

[0065] (twenty one)

[0066] (twenty two)

[0067] (twenty three)

[0068] where N i 、N bl and N bu They represent the number of sampling points related to the initial value condition, the lower boundary loss term, and the upper boundary loss term, respectively.

[0069] Furthermore, 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.

[0070] Furthermore, in step 4, the full batch strategy was used for training, the Adam algorithm was used as the optimizer, the training rounds were set to 200,000, and the learning rate decay strategy was introduced. The learning rate was 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 .

[0071] Furthermore, the weight factors of the loss function components corresponding to the initial value condition, lower boundary condition and upper boundary condition in step 4 are λ i , λ l and λ bu Set to 10 respectively 0 , 10 0 and 10 5 .

[0072] The beneficial effects of the present invention are 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. This ensures that the network itself automatically satisfies the one-dimensional seismic wave equation, effectively avoids the network from falling into local optimality, greatly reduces the difficulty of training, and provides strong technical support for solving the one-dimensional seismic wave equation.

[0073] Compared to existing PINN-based methods for solving one-dimensional seismic wave equations, the network proposed in this paper exhibits superior convergence performance and avoids the penalties associated with the governing equations, which require the most configuration points, during training. This significantly improves convergence efficiency and solution accuracy, resulting in improved usability. This method not only demonstrates excellent performance under simple harmonic wave incident conditions but has also been validated under measured seismic motion incident conditions. Compared to PINN, it has faster convergence speed and prediction accuracy, helping to improve the accuracy of site response analysis, assess earthquake risk, and ensure the safety of buildings and foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 Schematic diagram of one-dimensional wave propagation problem.

[0075] Figure 2 The architecture of the gs-PINN for solving one-dimensional seismic wave propagation problems, including the network structure and physical constraint loss function. The black boxes in the figure represent the two sub-networks, and the different colors within the dashed boxes indicate the different activation functions used: red corresponds to the sine function, orange represents the hyperbolic tangent function, and green represents the linear function.

[0076] Figure 3 Loss curves of the two models considering incident simple harmonic waves with durations of 10s, 20s, and 37.76s: (ac) is PINN, and (df) is gs-PINN.

[0077] Figure 4 Displacement fields predicted by two models and 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 exact solutions.

[0078] Figure 5 The absolute errors of the displacement fields predicted by PINN and gs-PINN relative to the exact solution for incident harmonics with durations of 10s, 20s, and 37.76s: (ac) is PINN, and (df) is gs-PINN.

[0079] Figure 6 Ground motion 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.

[0080] Figure 7 Loss curves of the network when using selected ground motions as incident waves: (a) for PINN and (b) for gs-PINN.

[0081] 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 for: (a) PINN, (b) gs-PINN, (c) exact solution; absolute errors for: (d) PINN, (e) gs-PINN. DETAILED DESCRIPTION

[0082] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. 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 making creative efforts are within the scope of protection of the present invention.

[0083] A novel physical information neural network method for solving seismic wave equations, referred to as gs-PINN, is characterized by the following specific operating steps:

[0084] Step 1: Establish gs-PINN;

[0085] 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 non-trainable, with the specific value 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 of the outputs of the two sub-networks. The network includes a one-dimensional seismic wave equation and the construction of gs-PINN.

[0086] The one-dimensional seismic wave equation describes the propagation of waves generated by seismic events within the Earth. For a one-dimensional isotropic homogeneous medium, the equation can be expressed as follows:

[0087] (1)

[0088] in u ( z,t ) represents the spatial coordinate z and time coordinates t The displacement at .

[0089] The initial value conditions are as follows:

[0090] (2)

[0091] (3)

[0092] The boundary conditions are as follows:

[0093] (4)

[0094] (5)

[0095] in V is the wave propagation speed, u g (t) represents the incident motion, L Indicates the thickness of the medium, subscript t and z Respectively represent the time coordinate t and spatial coordinates z The partial derivative of .

[0096] The general solution to Equation (1) can be easily determined by reducing it to its second normal form, which consists of the sum of two functions:

[0097] (6)

[0098] in f (×) and g (×) is any twice continuously differentiable function.

[0099] Based on the D'Alembert formula, the solution of the initial boundary value problem defined in Equation (1-5) can be expressed as follows:

[0100] (7)

[0101] in:

[0102] (8)

[0103] The construction of the gs-PINN is as follows:

[0104] gs-PINN can be expressed as follows:

[0105] (9)

[0106] Where N1( z , t ) and N2( z , t ) represent the 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:

[0107] (10)

[0108] (11)

[0109] in Representative j The activation function of the first hidden layer of the sub-network is Representative j The output of the first hidden layer of the sub-network, Representative j In the sub-network i The weight parameters from the first input neuron to the first neuron in the first hidden layer, Representative j The bias value of the first neuron in the first hidden layer of the subnetwork;

[0110] Set the activation function to a linear activation function, and let the weights and bias values ​​satisfy the following relationship:

[0111] (12)

[0112] (13)

[0113] At this time, formula (9) can be written as:

[0114] (14)

[0115] The above formula is correct t and z The first-order partial derivative and the second-order partial derivative can be expressed as:

[0116] (15)

[0117] (16)

[0118] (17)

[0119] (18)

[0120] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0121] (19)

[0122] 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 as in traditional PINN.

[0123] Step 2: Divide the configuration points required for network training;

[0124] 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.

[0125] Step 3: Determine the form of the incident wave;

[0126] Collect incident waves with different duration and frequency components and use them as lower boundary conditions;

[0127] Step 4: Conduct network training:

[0128] 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:

[0129] (20)

[0130] Among them L i 、L bl and L bu Respectively represent the loss function components corresponding to the initial value condition, lower boundary condition and upper boundary condition, and the coefficients λ i , λ l and λ bu are the weight factors of each loss function. Each loss function can be calculated as follows:

[0131] (twenty one)

[0132] (twenty two)

[0133] (twenty three)

[0134] where N i 、N bl and N bu They represent the number of sampling points related to the initial value condition, the lower boundary loss term, and the upper boundary loss term respectively;

[0135] Step 5: Obtain the solution of the one-dimensional seismic wave equation;

[0136] 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.

[0137] In step 1, the Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0138] (twenty four)

[0139] in s 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:

[0140] (25)

[0141] in fan in and fan out are the number of input neurons and output neurons in the fully connected layer, respectively.

[0142] Each subnetwork of the gs-PINN in step 1 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.

[0143] 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 20 m and the time interval is set to 0.02 s.

[0144] In step 3, simple harmonic waves with a frequency of 0.5 Hz and durations of 10 s, 20 s and 37.76 s, and a lateral earthquake motion with a duration of 37.76 s were collected. Figure 6 Shown is a lateral ground motion recorded at El Centro Array Station 8 during the 1979 Imperial Valley earthquake with a duration of 37.76 seconds.

[0145] 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.

[0146] 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 .

[0147] The weight factors of the loss function components corresponding to the initial value conditions, lower boundary conditions and upper boundary conditions in step 4 are λ i , λ l and λ bu Set to 10 respectively 0 , 10 0 and 10 5 .

[0148] The specific operation steps of the present invention are described below with reference to specific embodiments;

[0149] Step 1: Create gs-PINN:

[0150] 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 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; each sub-network 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.

[0151] 1. One-dimensional seismic wave equation

[0152] 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:

[0153] (1)

[0154] where u(z, t) represents the displacement at the spatial coordinate z and the time coordinate t.

[0155] The initial value conditions are as follows:

[0156] (2)

[0157] (3)

[0158] The boundary conditions are as follows:

[0159] (4)

[0160] (5)

[0161] Where V is the propagation speed of the wave, 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.

[0162] The general solution to Equation (1) can be easily determined by reducing it to its second normal form, which consists of the sum of two functions:

[0163] (6)

[0164] in f (×) and g (×) is any twice continuously differentiable function.

[0165] Based on the D'Alembert formula, the solution of the initial boundary value problem defined in Equation (1-5) can be expressed as follows:

[0166] (7)

[0167] in:

[0168] (8)

[0169] 2. Construction of gs-PINN

[0170] like Figure 2 As shown, the proposed gs-PINN can be expressed as follows:

[0171] (9)

[0172] Where N1( z , t ) and N2( z , t ) represent two sub-networks respectively.

[0173] 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:

[0174] (10)

[0175] (11)

[0176] 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 sub-network 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.

[0177] Let's set the activation function to a linear activation function and let the weights and bias values ​​satisfy the following relationship:

[0178] (12)

[0179] (13)

[0180] At this time, formula (9) can be written as:

[0181] (14)

[0182] The first-order partial derivative and second-order partial derivative of the above formula with respect to t and z can be expressed as:

[0183] (15)

[0184] (16)

[0185] (17)

[0186] (18)

[0187] Substituting formula (16) into formula (18), the latter can be rewritten as follows:

[0188] (19)

[0189] 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 as in traditional PINN.

[0190] The Gaussian weight initialization scheme is used for the second hidden layer of the network, which can be expressed as:

[0191] (twenty four)

[0192] in s is the bandwidth parameter, which is set to For the remaining layers, the Xavier weight initialization scheme is used, which can be expressed as:

[0193] (25)

[0194] in fan in and fanout are the number of input neurons and output neurons in the fully connected layer, respectively.

[0195] Step 2: Divide the configuration points required for network training:

[0196] Determine the size of the two-dimensional space-time computational domain. Unlike the traditional PINN, which requires partitioning the entire computational domain, the proposed 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. An equidistant grid partitioning strategy is used to obtain the configuration points, with the spatial interval set to 20 m and the time interval set to 0.02 s.

[0197] Step 3: Determine the form of the incident wave:

[0198] The collection frequency is 0.5 Hz, and the durations are 10 s, 20 s, and 37.76 s respectively. The collection frequency is 0.5 Hz, and the durations are 10 s, 20 s, and 37.76 s respectively. The collection frequency is 0.5 Hz, and the collection frequency is 0.5 Hz, and the collection frequency is 0.5 s ... Figure 6 As shown, and it is used as the lower boundary condition. The form of simple harmonic wave is as follows:

[0199]

[0200] where A is set to 0.417 m so that the peak of the incident harmonic matches the peak of the selected earthquake motion.

[0201] Step 4: Conduct network training:

[0202] 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:

[0203] (20)

[0204] Among them L i 、L bl and L bu Respectively represent the loss function components corresponding to the initial value condition, lower boundary condition and upper boundary condition, and the coefficients 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 5Each loss function can be calculated as follows:

[0205] (twenty one)

[0206] (twenty two)

[0207] (twenty three)

[0208] 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 .

[0209] Step 5: Obtain the solution of the one-dimensional seismic wave equation:

[0210] 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.

[0211] Figure 3 、 4 Figures 5 and 6 show the results of solving the one-dimensional wave equation using the method proposed in the present invention under simple harmonic wave incident conditions. It can be clearly seen that as the duration increases, the final convergence loss value of PINN gradually increases, while the final convergence loss value of the present invention remains relatively stable. At 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 perspective of the predicted wave field, gs-PINN has better overall consistency with the exact solution than PINN. Error analysis also shows that the accuracy of PINN decreases significantly with increasing duration, especially when the duration is 37.76s. In contrast, gs-PINN maintains high accuracy under different duration conditions. Table 1 provides a quantitative comparison of PINN and gs-PINN based on two performance indicators: relative L2 error and training time. The results show that the prediction accuracy of gs-PINN is improved by at least 11.8 times, and the training speed is increased by at least 6.5 times. Moreover, the improvements in prediction accuracy and training speed are more significant with increasing incident wave duration.

[0212] Figure 7and 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 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 rate of PINN decreases significantly, while the performance of gs-PINN remains relatively stable. From the predicted wave field, when processing incident waves with complex frequency components, the predicted wave field of PINN is significantly distorted, especially in the later time period. However, gs-PINN is highly consistent with the exact solution, accurately capturing 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.

[0213] This paper proposes a new PINN-based approach to solve the one-dimensional wave propagation problem. This approach redesigns the PINN architecture to form a new architecture, termed 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, responsible for coordinate transformation. This architectural design ensures that the network architecture is intrinsically aligned with the general solution of the one-dimensional wave equation, eliminating the equation residual loss typically required during PINN training, thereby improving the network's convergence speed and prediction accuracy. To validate the performance of the proposed gs-PINN, two test cases were conducted. In the first case, gs-PINN was used to solve the one-dimensional wave equation considering incident harmonic waves. Comparative experiments with PINN demonstrated that gs-PINN not only reduced training time but also achieved higher prediction accuracy. Furthermore, the impact of wave duration on network performance was analyzed. Results showed that the prediction error of gs-PINN remained constant regardless of wave duration, while the error of PINN increased significantly with increasing 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 demonstrates rapid convergence and high-precision predictions, while PINN exhibits significant distortion in predictions with complex input waves. These findings validate the superior reliability of gs-PINN.

[0214] In summary, the novel physics-informed neural network approach for solving the seismic wave equation is effective and reliable. This approach redesigns the PINN architecture, proposing a new architecture called gs-PINN. The gs-PINN architecture consists of two subnetworks. In each subnetwork, the first hidden layer consists of a single 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, eliminating the equation residual loss typically required during PINN training.

[0215] Table 1. Relative performance of PINN and gs-PINN for incident harmonics of different durations L 2. Comparison of error and training time.

[0216] Table 2. Comparison of relative L2 errors and training time of PINN and gs-PINN when using selected ground motions as incident waves.

[0217] The ratio is the relative L 2Comparison values ​​of error and training time.

[0218] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

Claims

1. A novel physical information neural network method 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 non-trainable, with the specific value determined by the propagation direction of the seismic wave, while 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 of the outputs 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 wave equation under specific initial and boundary conditions and make the network training results meet the initial and boundary conditions, it is necessary to construct a loss function consistent with the initial and boundary conditions, which is as follows: (20) Among them L i 、L bl and L bu Respectively represent the loss function components corresponding to the initial value condition, lower boundary condition and upper boundary condition, and the coefficients λ i , λ l and λ bu are the weight factors of each loss function respectively; Step 5: Obtain 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 method for solving seismic wave equations according to claim 1, 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, and PINN is an existing physical information neural network; The one-dimensional seismic wave equation describes the propagation of waves generated by seismic events within the Earth. For a one-dimensional isotropic homogeneous medium, the equation is expressed as follows: (1) in u ( z,t ) represents the spatial coordinate z and time coordinates t The displacement at The initial value conditions are as follows: (2) (3) The boundary conditions are as follows: (4) (5) in V is the wave propagation speed, u g (t) represents the incident motion, L Indicates the thickness of the medium, subscript t and z Respectively represent the time coordinate t and spatial coordinates z The partial derivative of By reducing Equation (1) to its second standard form, the general solution of Equation (1) is determined, which is the sum of two functions composition: (6) in f (×) and g (×) is any twice continuously differentiable function; Based on the D'Alembert formula, the solution of the initial boundary value problem defined in equation (1-5) is expressed as follows: (7) in: (8) The construction of gs-PINN is as follows: gs-PINN is represented by the following formula: (9) Where N1( z , t ) and N2( z , t ) represent the 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 are expressed as follows: (10) (11) 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 sub-network 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: (12) (13) At this time, formula (9) can be written as: (14) The above formula is correct t and z The first and second order partial derivatives of are expressed as: (15) (16) (17) (18) Substituting formula (16) into formula (18), the latter is rewritten as follows: (19) 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 as in traditional PINN.

3. The novel physical information neural network method for solving seismic wave equations according to claim 1 is characterized in that: In step 1, the Gaussian weight initialization scheme is used for the second hidden layer of the network, which is expressed as: (24) in s is the bandwidth parameter, which is set to e -0.5 ; For the remaining layers, the Xavier weight initialization scheme is used, which is expressed as: (25) in 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 method for solving seismic wave equations according to claim 1 is characterized in that: Each subnetwork of the gs-PINN in step 1 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 method 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 20 m and the time interval is set to 0.02 s.

6. The novel physical information neural network method for solving seismic wave equations according to claim 1, 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 method for solving seismic wave equations according to claim 1, characterized in that: Each loss function in step 4 is calculated as follows: (21) (22) (23) where N i 、N bl and N bu They represent the number of sampling points related to the initial value condition, the lower boundary loss term, and the upper boundary loss term, respectively.

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

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