Method and system for solving seismic wave propagation equation based on physical information neural network

Abstract

The invention relates to the technical field of seismic engineering, and provides a method and system for solving a seismic wave propagation equation based on a physical information neural network, and the method comprises the steps: determining a space domain, a time domain, an initial condition and a boundary condition of a one-dimensional seismic wave equation; initial boundary value coordinate points are randomly extracted to serve as a training data set constraint network, and Latin hypercube sampling is utilized to generate space-time domain configuration points to meet partial differential equation constraint; a physical information neural network model is established, after space-time coordinates are input, equation residual errors are calculated through automatic differential, and initial boundary value loss and physical residual errors are combined to construct a composite loss function for training; and proposing a self-adaptive region sampling strategy based on residual errors, resampling a high residual error region according to an average residual error value of the configuration points in a training period, and iteratively optimizing network parameters after a newly added point and an initial configuration point are fused. According to the method, by dynamically enhancing the sampling density of a high-error region, the L2 relative error of a prediction solution is reduced by one order of magnitude, the network is effectively prevented from falling into local optimum, the convergence efficiency is remarkably improved, grid discretization or prior data support is not needed, and a high-precision meshless solution scheme is provided for a one-dimensional seismic wave equation.

Description

Technical Field

[0001] The present invention belongs to the field of integrable systems, and in particular relates to a method and system for solving seismic wave propagation equations based on physical information neural networks. Background Art

[0002] Nonlinear science is an interdisciplinary discipline that studies the common laws of nonlinear phenomena in complex systems. Its core research objects include typical nonlinear behaviors such as solitons, chaos, and fractals. Since the establishment of the theoretical system in the 1960s, this discipline has made breakthrough progress in both basic theory and engineering application. The propagation of seismic waves is a fundamental problem in geophysics and is crucial for understanding earthquake dynamics, underground imaging, and resource exploration. The governing equation for seismic wave propagation is essentially a wave equation, a nonlinear partial differential equation (NPDE). Modeling seismic waves through this equation can simulate the wave propagation process and better study the potential impact of seismic waves on buildings and infrastructure.

[0003] In terms of the system of methods for solving nonlinear partial differential equations, analytical methods such as the backscattering transform, the Hirota bilinear method, and the Darboux transform have formed a relatively complete theoretical framework. These methods provide an important path to obtaining accurate analytical solutions by revealing the integrability characteristics of nonlinear systems. In the field of seismic exploration, the inhomogeneity of underground media and the complexity of exploration targets make analytical solutions to wave equations difficult. At the same time, numerical methods have been proposed for many complex systems that are difficult to obtain explicit analytical solutions. Through the strategy of space-time discretization, these methods successfully transform infinite-dimensional continuous problems into finite-dimensional computational problems and have been widely used in engineering practice. However, existing numerical algorithms still have significant limitations: first, there is an inherent contradiction between the universality of the algorithms and the computational accuracy and efficiency; second, when dealing with high-dimensional nonlinear partial differential equations, they encounter the curse of dimensionality, and the computational complexity increases exponentially with the increase in dimensionality.

[0004] With the explosive growth of computing resources, deep learning technology has developed rapidly and has been applied to a wide range of fields. Physics-Informed Neural Networks (PINN), an innovative approach that integrates deep learning with the laws of physics, have demonstrated significant advantages in solving partial differential equations (PDEs) in recent years, providing innovative solutions for scientific computing and engineering. The core concept is to encode the governing equations and initial / boundary conditions as loss functions, leveraging neural networks to achieve generalized solutions. In the field of seismic wave propagation, the PINN method has also been applied to solving one-dimensional seismic wave equations.

[0005] However, traditional PINN methods generally use a fixed sampling strategy during training, pre-arranging collocation points randomly or uniformly within the computational domain. This static sampling approach prevents the network from capturing the complex characteristics of the PDE solution in the early stages of training, and the uniformly distributed sampling points make it difficult to effectively represent high-gradient regions. In the later stages of training, when the residual distribution becomes non-uniform, the fixed sampling points are unable to dynamically focus on high-error regions. Furthermore, training neural networks based on fixed sampling points often achieves locally optimal results, failing to fully utilize the neural network's powerful expressive power. Summary of the Invention

[0006] In order to solve the problems existing in the prior art, the present invention provides a method for solving the seismic wave propagation equation based on a physical information neural network. The method aims to avoid the network from falling into local optimality as much as possible by periodically resampling configuration points in areas with large residuals when training the neural network. At the same time, the method makes full use of the expressive power of the neural network and applies it to solving the seismic wave wave equation, thereby achieving a more accurate solution to the seismic wave propagation problem.

[0007] To achieve the above objectives, in a first aspect, the present invention provides a method for solving seismic wave propagation equations based on a physical information neural network, comprising the following steps: S1. Randomly train a number of data points based on the initial conditions and boundary conditions of the seismic wave equation; use the Latin hypercube sampling method to configure points in the time and space domain of the seismic wave; S2. Build a fully connected physical information neural network model; S3. Initialize adaptive region sampling hyperparameters, including adaptive sampling interval , number of regional sampling points , regional step ratio and the residual threshold , and perform adaptive sampling; S4, iteratively train the physical information neural network several times, the training data uses the initial boundary condition data and internal configuration point data, wherein each training is performed Second, perform the S3 adaptive sampling step, merge the resampled internal configuration points with the internal configuration points in S1 to train the physical information neural network; S5. Calculate the data in the dataset to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation.

[0008] Furthermore, the spatial domain of the seismic wave propagation equation for , time domain for , the equation and its Dirichlet boundary form are:

[0009]

[0010]

[0011] in, is the propagation velocity of the seismic wave, and the seismic wave satisfies the equation at the initial moment , the seismic wave is 0 at the boundary of the spatial domain.

[0012] Furthermore, building a fully connected physical information neural network model includes: S21. Set the number of hidden layers and neurons in each layer, and the weight of the network. and bias Initialize and set the activation function; S22. Design a loss function, add physical information constraints, and obtain a fully connected physical information neural network model.

[0013] Furthermore, the loss function consists of three parts, namely initial condition loss , boundary condition loss and residual loss of equation , as follows:

[0014] in,

[0015]

[0016]

[0017] in, and Represent the output of the physical information neural network and the true solution of the equation respectively ; The seismic wave equation describes the real physical world, They represent the number of initial data points, boundary data points and residual collocation points, respectively, and describe the size of the seismic wave data.

[0018] Furthermore, when designing the loss function and adding physical information constraints, the initial boundary condition loss is the output of the neural network. The mean square error between the true value and the physical information neural network is used to constrain the physical information neural network to meet the given initial boundary conditions. The residual loss of the equation is obtained by applying physical information , Right now , the mean square error of the residual value is calculated to constrain the physical information neural network to satisfy the wave equation, where the partial derivatives are obtained by automatic differentiation.

[0019] Furthermore, during the training of the physical information neural network, adaptive regional sampling is performed periodically to update the internal configuration point data involved in the training, including: S31, the number of training times reaches the adaptive sampling interval , let the adaptive sampling point set If it is empty, adaptive sampling will start; S32, taking the absolute sum of the internal configuration point residuals during training within the adaptive sampling interval and calculating the average value during the iteration, that is: ,in, For the After iterations, at the adaptive sampling interval The residuals of the collocation points during the period are summed, and ; S33, what you will get Perform min-max normalization; S34. Use the nearest neighbor algorithm to calculate the average distance between the internal configuration points involved in the training ; S35, loop through the internal configuration points involved in training, if , then at point Centered on Sampling within the region at the upper and lower bounds Point is a set , will be collected Merge into adaptive sampling point set ,Right now: ,in, for No. Residual values, for The corresponding Internal configuration points, The calculation method is as follows:

[0020] in, Specify a dimension. is the space-time domain of the one-dimensional seismic wave equation.

[0021] Furthermore, the calculation formula is as follows:

[0022] in, is the input spatial coordinate data vector, is the input time coordinate vector, is the output of the neural network, It is the solution of partial differential equations approximated by physical information neural network.

[0023] In a second aspect, the present invention provides a system for solving seismic wave propagation equations based on a physical information neural network, comprising an initialization module, a model building module, an adaptive sampling module, a training module, and a solution module; The initialization module is used to randomly train a number of data points based on the initial conditions and boundary conditions of the seismic wave equation; Latin hypercube sampling method is used to configure points in the time and space domain of seismic waves; The model building module is used to build a fully connected physical information neural network model; The adaptive sampling module is used to initialize the adaptive region sampling hyperparameters, including the adaptive sampling interval. , number of regional sampling points , regional step ratio and the residual threshold , and perform adaptive sampling; The training module is used to iteratively train the physical information neural network several times. The training data uses the initial boundary condition data and internal configuration point data. Second, perform the adaptive sampling step and merge the resampled internal configuration points with the internal configuration points in S1 to train the physical information neural network; The solution module is used to calculate the data in the data set to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation.

[0024] At the same time, a computer device is provided, including a processor and a memory, the memory is used to store computer executable programs, the processor reads part or all of the computer executable programs from the memory and executes them, and when the processor executes part or all of the computer executable programs, it can implement the above-mentioned method of solving seismic wave propagation equations based on physical information neural networks.

[0025] A computer-readable storage medium may also be provided, in which a computer program is stored. When the computer program is executed by a processor, the method for solving the seismic wave propagation equation based on the physical information neural network can be implemented.

[0026] Compared with existing technologies, the present invention has at least the following beneficial effects: it dynamically avoids the local optimum trap. The static sampling of traditional PINNs leads to a rigid loss function topography, making the network prone to falling into error basins of attraction. This method adaptively samples internal configuration points, namely internal spatiotemporal coordinates, and dynamically adjusts the training dataset, preventing the network from falling into local optimums caused by a fixed training set. It also combines historical residuals with global residual information within the domain to achieve collaborative perception of spatiotemporal residual features, while reducing the computational overhead of resampling and training samples, and combining holistic information to achieve better optimization. It seamlessly integrates with traditional methods and can be applied to other network architectures, demonstrating the method's strong scalability and compatibility, providing strong technical support for more accurate solutions to one-dimensional seismic wave equations. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 Flowchart for training physical information neural network based on residual adaptive region sampling.

[0028] Figure 2 The architecture of a physical information neural network.

[0029] Figure 3 Wave equation solutions for one-dimensional seismic waves predicted by original PINN and RARS-PINN.

[0030] Figure 4 The absolute value diagram of the error between the original PINN and RARS-PINN predicted solutions and the true solutions.

[0031] Figure 5 The loss change curve of the original PINN and RARS-PINN during training. DETAILED DESCRIPTION

[0032] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0033] The present invention randomly extracts coordinate points of initial conditions and boundary conditions as constrained physical information neural networks to satisfy given initial and boundary value training data sets, performs Latin hypercube sampling on coordinate points within the spatial and time domains of the seismic wave propagation equation to constrain the physical information neural networks to satisfy given seismic wave partial differential equations; establishes a physical information neural network model and uses it for approximate solution after training to obtain a solution to the seismic wave propagation equation.

[0034] The present invention provides a method for solving seismic wave propagation equations based on physical information neural networks, referred to as RARS-PINN, which is also applied to solving seismic wave equations. The specific operation steps are as follows: The governing equation for seismic wave propagation is essentially a wave equation. To determine the time and space domain of the equation, assume that the spatial domain for , time domain for , the equation and its Dirichlet boundary form are:

[0035]

[0036]

[0037] in, is the propagation velocity of the seismic wave, and the seismic wave satisfies the equation at the initial moment , the seismic wave is 0 at the boundary of the spatial domain.

[0038] S1, randomly train 100 data points based on initial conditions and boundary conditions, and let the initial boundary training point sets be , and satisfy ,in ; Use Latin hypercube sampling to take 10,000 configuration points in the time and space domain, and let the configuration point set be .

[0039] S2, build a fully connected physical information neural network model, such as Figure 2 ; Further, comprising the following steps: S21, set the number of hidden layers of the network to 8, the number of neurons in each layer to 100, and the weight of the network and bias Perform Xavier initialization and use the hyperbolic tangent function as the activation function .

[0040] S22. Design loss function and add physical information constraints. The loss function consists of three parts, namely initial condition loss , boundary condition loss and residual loss of equation The initial boundary condition loss is the output of the neural network The mean square error between the true value and the physical information neural network is used to constrain the physical information neural network to meet the given initial boundary conditions. The residual loss of the equation is obtained by applying the physical information , which is the left-hand form of the equation, calculates the mean square error of the residual value. The physical information neural network satisfies the wave equation, where the partial derivatives are obtained by automatic differentiation. The loss function of this example can be expressed as follows:

[0041] in,

[0042]

[0043]

[0044] Here, and Represent the output of the physical information neural network and the true solution of the equation respectively ; For physical information; represent the number of initial data points, boundary data points, and residual collocation points, respectively.

[0045] S3. Initialize the hyperparameters of the adaptive region sampling method, including the adaptive sampling interval , number of regional sampling points , regional step ratio and the residual threshold The specific hyperparameters for this embodiment are set as follows: Furthermore, the specific process of the method is as follows: S31, the number of training times reaches the adaptive sampling interval , let the adaptive sampling point set If it is empty, adaptive sampling will start; S32, take the absolute value of each internal configuration point residual sum within the adaptive sampling interval and calculate the average value during the iteration, that is: ,in, is a vector and is the After iterations, The residuals of the collocation points during the period are summed, and .

[0046] S33, the above steps obtained Perform min-max normalization to shrink the mean residual of each collocation point to In order to facilitate the subsequent efficient selection of points with large residuals, the following formula is used to obtain :

[0047] S34. Use the nearest neighbor algorithm to calculate the average distance between the internal configuration points involved in the training .

[0048] S35, loop through the internal configuration points involved in training, if , then at point Centered on Sampling within the region at the upper and lower bounds Point is a set ,Will Merge into ,Right now: .in for The first Residual values, for The corresponding Internal configuration points, The calculation method is as follows:

[0049] in, Specify a dimension. is the space-time domain of the one-dimensional seismic wave equation.

[0050] S4. Establish an empty adaptive sampling point set The end condition of training the physical information neural network is set to 10,000 iterations or the training loss remains unchanged for a long time. The parameter optimizer of the physical information neural network uses L-BFGS. The training data uses the initial boundary condition data. and internal configuration point data , where each execution training, performing adaptive sampling updates , i.e., the specific execution steps of the method in S3 above. Execute physical information neural network training.

[0051] S5. Calculate the data set to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation. Specifically, the calculation formula is as follows:

[0052] in is the input spatial coordinate data vector, is the input time coordinate vector, is the output of the neural network, It is the solution of partial differential equations approximated by physical information neural network.

[0053] The original PINN and the RARS-PINN proposed in this patent were trained 10,000 times using the L-BFGS optimizer. The prediction solutions obtained by the two networks are as follows: Figure 3 The left and right sides of . The training loss of PINN is , the L2 relative error is ; The training loss of RARS-PINN is 5.1333e-06, and the L2 relative error is From the final results, we can see that RARS-PINN is one order of magnitude lower than the original PINN in terms of L2 relative error, achieving better results. At the same time, the absolute errors between the predicted solutions and the true solutions of the two networks are as follows: Figure 4 As shown, RARS-PINN ( Figure 4 The right side) is smoother in the domain, while the original PINN has two more obvious error peaks. The training loss iteration curve is as follows Figure 5 As shown in the figure, we can see that the original PINN (blue curve) falls into a local optimum after about 2000 iterations, and the subsequent training process cannot be further optimized (training is terminated after 5000 times); while RARS-PINN (orange curve) can continue to optimize before 8000 iterations. Although the convergence speed is slightly slower than the original PINN, it ultimately achieves a smaller loss.

[0054] In summary, the present invention relates to the field of earthquake engineering technology, and provides a method for solving seismic wave equations using a physical information neural network based on residual adaptive regional sampling, comprising the following steps: determining the spatial domain, time domain, initial conditions and boundary conditions of a one-dimensional seismic wave equation; randomly extracting coordinate points of the initial boundary conditions as a training data set constraining the physical information neural network to satisfy a given initial boundary value, performing Latin hypercube sampling on coordinate points within the spatial domain and time domain to constrain the physical information neural network to satisfy a given partial differential equation; establishing a physical information neural network model and using it for approximating the solution of the partial differential equation after training; and designing a basic The adaptive regional sampling strategy for residuals has the following basic idea: based on the average residual value of the configuration points during the adaptive sampling interval training period, the configuration points are re-sampled in the high residual area, and the resampled points are combined with the initial training configuration points to train the physical information neural network to obtain an optimized physical information neural network, which reduces the L2 relative error of the predicted solution by one order of magnitude, effectively preventing the network from falling into the local optimum, and providing a new solution method for the one-dimensional seismic wave equation. The RARS-PINN described in the present invention can get rid of the local optimum of the neural network during training, and continuously optimize the areas with higher residuals, ultimately achieving more accurate training results.

[0055] In Example 2, the present invention further provides a system for solving seismic wave propagation equations based on a physical information neural network, comprising an initialization module, a model building module, an adaptive sampling module, a training module, and a solving module; The initialization module is used to randomly train a number of data points based on the initial conditions and boundary conditions of the seismic wave equation; Latin hypercube sampling method is used to configure points in the time and space domain of seismic waves; The model building module is used to build a fully connected physical information neural network model; The adaptive sampling module is used to initialize the adaptive region sampling hyperparameters, including the adaptive sampling interval. , number of regional sampling points , regional step ratio and the residual threshold , and perform adaptive sampling; The training module is used to iteratively train the physical information neural network several times. The training data uses the initial boundary condition data and internal configuration point data. Second, perform the adaptive sampling step and merge the resampled internal configuration points with the internal configuration points in S1 to train the physical information neural network; The solution module is used to calculate the data in the data set to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation.

[0056] On the other hand, the present invention provides a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, it can implement the method of solving seismic wave propagation equations based on physical information neural network described in the present invention.

[0057] The present invention can also provide a computer device, including a processor and a memory, the memory is used to store a computer executable program, the processor reads the computer executable program from the memory and executes it, and when the processor executes the computer executable program, it can implement the method of solving seismic wave propagation equations based on physical information neural network described in the present invention.

[0058] The computer device may be a laptop computer, a desktop computer or a workstation.

[0059] The processor can be a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), or an off-the-shelf field-programmable gate array (FPGA).

[0060] The memory of the present invention may be an internal storage unit of a laptop computer, desktop computer or workstation, such as a memory or a hard disk; or an external storage unit, such as a mobile hard disk or a flash memory card.

[0061] Computer-readable storage media may include computer storage media and communication media. Computer storage media include volatile and non-volatile, removable and non-removable media implemented by any method or technology for storing information such as computer-readable instructions, data structures, program modules or other data. Computer-readable storage media may include: read-only memory (ROM), random access memory (RAM), solid-state drives (SSD) or optical disks, etc. Among them, random access memory may include resistance random access memory (ReRAM) and dynamic random access memory (DRAM).

Claims

1. A method for solving seismic wave propagation equations based on physical information neural networks, characterized in that: The following steps are involved: S1. Randomly train a number of data points based on the initial conditions and boundary conditions of the seismic wave equation; use the Latin hypercube sampling method to configure points in the time and space domain of the seismic wave; S2. Build a fully connected physical information neural network model; S3. Initialize adaptive region sampling hyperparameters, including adaptive sampling interval , number of regional sampling points , regional step ratio and the residual threshold , and perform adaptive sampling; S4, iteratively train the physical information neural network several times, the training data uses the initial boundary condition data and internal configuration point data, wherein each training is performed Second, perform the S3 adaptive sampling step, merge the resampled internal configuration points with the internal configuration points in S1 to train the physical information neural network; S5. Calculate the data in the dataset to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation.

2. The method for solving seismic wave propagation equations based on physical information neural network according to claim 1, characterized in that: Spatial domain of seismic wave propagation equations for , time domain for , the equation and its Dirichlet boundary form are: in, is the propagation velocity of the seismic wave, and the seismic wave satisfies the equation at the initial moment , the seismic wave is 0 at the boundary of the spatial domain.

3. The method for solving seismic wave propagation equations based on physical information neural network according to claim 1, characterized in that: Building a fully connected physical information neural network model includes: S21. Set the number of hidden layers and neurons in each layer, and the weight of the network. and bias Initialize and set the activation function; S22. Design a loss function, add physical information constraints, and obtain a fully connected physical information neural network model.

4. The method for solving seismic wave propagation equations based on physical information neural network according to claim 3, characterized in that: The loss function consists of three parts, namely the initial condition loss , boundary condition loss and residual loss of equation , as follows: in, in, and Represent the output of the physical information neural network and the true solution of the equation respectively ; The seismic wave equation describes the real physical world, They represent the number of initial data points, boundary data points and residual collocation points, respectively, and describe the size of the seismic wave data.

5. The method for solving seismic wave propagation equations based on physical information neural network according to claim 3, characterized in that: When designing the loss function and adding physical information constraints, the initial boundary condition loss is the output of the neural network. The mean square error between the true value and the original value is used to constrain the physical information neural network to meet the given initial boundary conditions. The residual loss of the equation is obtained by applying physical information. , Right now , the mean square error of the residual value is calculated to constrain the physical information neural network to satisfy the wave equation, where the partial derivatives are obtained by automatic differentiation.

6. The method for solving seismic wave propagation equations based on physical information neural network according to claim 1, characterized in that: During the training of the physical information neural network, adaptive area sampling is performed periodically to update the internal configuration point data involved in the training, including: S31, the number of training times reaches the adaptive sampling interval , let the adaptive sampling point set If it is empty, adaptive sampling will start; S32, taking the absolute sum of the internal configuration point residuals during training within the adaptive sampling interval and calculating the average value during the iteration, that is: ,in, For the After iterations, at the adaptive sampling interval The residuals of the collocation points during the period are summed, and ; S33, what you will get Perform min-max normalization; S34. Use the nearest neighbor algorithm to calculate the average distance between the internal configuration points involved in the training ; S35, loop through the internal configuration points involved in training, if , then at point Centered on Sampling within the region at the upper and lower bounds Point is a set , will be collected Merge into adaptive sampling point set ,Right now: ,in, for No. Residual values, for The corresponding Internal configuration points, The calculation method is as follows: in, Specify a dimension. is the space-time domain of the one-dimensional seismic wave equation.

7. The method for solving seismic wave propagation equations based on physical information neural network according to claim 1, characterized in that: The calculation formula is as follows: in, is the input spatial coordinate data vector, is the input time coordinate vector, is the output of the neural network, It is the solution of partial differential equations approximated by physical information neural network.

8. A system for solving seismic wave propagation equations based on physical information neural networks, characterized in that: It includes initialization module, model building module, adaptive sampling module, training module and solution module; The initialization module is used to randomly train a number of data points based on the initial conditions and boundary conditions of the seismic wave equation; Latin hypercube sampling method is used to configure points in the time and space domain of seismic waves; The model building module is used to build a fully connected physical information neural network model; The adaptive sampling module is used to initialize the adaptive region sampling hyperparameters, including the adaptive sampling interval. , number of regional sampling points , regional step ratio and the residual threshold , and perform adaptive sampling; The training module is used to iteratively train the physical information neural network several times. The training data uses the initial boundary condition data and internal configuration point data. Second, perform the adaptive sampling step and merge the resampled internal configuration points with the internal configuration points in S1 to train the physical information neural network; The solution module is used to calculate the data in the data set to be solved through the trained physical information neural network model to solve the one-dimensional seismic wave equation.

9. A computer device, characterized in that: It includes a processor and a memory, the memory is used to store computer executable programs, the processor reads part or all of the computer executable programs from the memory and executes them, and when the processor executes part or all of the computer executable programs, it can implement the method for solving seismic wave propagation equations based on physical information neural networks as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that A computer program is stored in a computer-readable storage medium. When the computer program is executed by a processor, it can implement the method for solving seismic wave propagation equations based on a physical information neural network as described in any one of claims 1 to 7.

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