A seismic wave travel time calculation method and system based on a semi-discrete physical information neural network

Abstract

This invention discloses a method and system for calculating seismic wave travel time based on a semi-discrete physical information neural network, belonging to the field of geophysical imaging technology. It acquires velocity field data and source locations, and generates a spatial collocation set. Then, it constructs a neural network with alternating layers of smooth and non-smooth activation functions to capture gradient discontinuities in the travel time field. The output is processed using a decomposition method and an exponential activation function to eliminate source singularities and ensure non-negativity of the results. A semi-discrete difference scheme based on virtual step size is introduced, and a Hamiltonian loss function with upwind characteristics is constructed. Numerical dissipation is explicitly injected into the training process, forcing the network to converge to a physically correct viscous solution. Finally, the loss function is minimized to achieve high-precision and robust travel time field reconstruction under complex geological models. This invention solves the problems of standard physical information neural networks easily getting trapped in non-physical solutions and failing to characterize singular interfaces, and has the advantages of being meshless, highly flexible, and physically rigorous.

Description

Technical Field

[0001] This invention relates to the field of geophysical imaging technology, and more specifically to a method and system for calculating seismic wave travel time based on a semi-discrete physical information neural network. Background Technology

[0002] Seismic wave travel time calculation is fundamental to geophysical research, primarily involving solving the Eikonal equations to simulate the propagation of seismic waves on geological models. However, efficiently and accurately obtaining seismic wave travel time fields under complex geological models remains a significant challenge.

[0003] First, existing seismic wave travel time imaging techniques have limitations. Traditional ray tracing methods are prone to ray shadowing and multipath problems when dealing with complex velocity interfaces, and struggle to handle first-wave travel times. While grid-based numerical methods (such as finite difference solvers) are stable, they are extremely dependent on high-quality computational grids and face enormous computational overhead and the "curse of dimensionality" in large-scale 3D imaging.

[0004] Against this backdrop, emerging deep learning methods (especially physically-informed neural network methods) have shown certain advantages over traditional methods, but they still have inherent limitations. Physically-informed neural network methods approximate solutions to partial differential equations through neural networks, offering advantages such as meshless operation and flexible parameterization, enabling direct travel-time tomography in continuous space. However, standard physically-informed neural network methods lack an "upwind" mechanism when solving the Eikonal equation. Since the Eikonal equation is a typical Hamilton-Jacobi equation, its solutions (viscous solutions) are physically strictly causal and prone to causation under complex velocity fields. Standard physically-informed neural network methods typically use automatic differentiation to calculate gradients; however, the network does not adhere to causality and is prone to converging to physically incorrect solutions or producing unrealistic oversmoothing effects.

[0005] To address the shortcomings of the aforementioned physical information neural network methods in travel-time tomography, a more physically constrained improvement method is urgently needed. Existing improvement schemes often struggle to simultaneously achieve both computational accuracy and singularity capture capability.

[0006] Therefore, how to propose a seismic wave travel time calculation method and system based on a semi-discrete physical information neural network to achieve high-precision and robust reconstruction of the first arrival travel time field of seismic waves under complex geological models is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0007] In view of this, the present invention provides a method and system for calculating seismic wave travel time based on a semi-discrete physical information neural network. By introducing an alternating activation function network structure, non-smooth operators (such as ReLU) can be used to capture cusps in the travel time field gradient discontinuity. Simultaneously, by combining a semi-discrete fusion upwind scheme, classical numerical upwind operators (such as the Godunov scheme) are embedded into the loss function of the neural network, explicitly injecting numerical dissipation and physical causal constraints into the neural network. This fusion method effectively solves the problem of physical information neural network methods getting trapped in inviscid solutions when dealing with the Eikonal equation, thereby significantly improving the imaging accuracy under complex geological conditions.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: On one hand, this invention discloses a method for calculating seismic wave travel time based on a semi-discrete physical information neural network, comprising the following steps: S1. Obtain the solution domain Velocity field data The travel time field model is defined by combining the Eikonal equation. Based on the location of the earthquake source and the solution domain Sampling is performed to generate a set of spatial collocation points; where ; S2. Construct a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions; S3. Input the spatial collocation points into the semi-discrete physical information neural network, and use the decomposition method to preprocess the output of the neural network to obtain the approximate solution of the travel time field; S4. A semi-discrete difference operator is constructed by introducing a virtual step size. The gradient magnitude approximation of the travel-time field approximation solution is calculated using the semi-discrete difference operator, and combined with the velocity field data. Obtain the Hamiltonian operator residuals, and then construct the loss function; S5. Use the gradient descent algorithm to minimize the loss function and train a semi-discrete physical information neural network; S6. Solve the domain The semi-discrete physical information neural network is trained by inputting any spatial coordinate point within the network and outputs the first arrival travel time field of the seismic wave.

[0009] Preferably, the travel time field model in S1 ; The point source conditions are as follows: ; In the formula, gradient operator

[0010] Preferably, in S2, the smooth activation function is the tanh function, and the non-smooth activation function is the ReLU function.

[0011] Preferably, in S3, the neural network output is preprocessed using a decomposition method to obtain an approximate solution for the travel time field, as shown in the following formula: ; In the formula, This is an approximate solution for the travel time field; For neural network output; For distance function term, .

[0012] Preferably, S4 includes: Based on virtual step size , for the set of points in space Each spatial point in Constructing a neighbor set ,in , for Euler space along the first Unit vectors along the coordinate axes; Based on the nearest point set, the gradient magnitude approximation of the travel time field approximation solution is calculated using a semi-discrete difference operator; Approximate value of gradient magnitude using the travel-time field approximation solution With velocity field data The Hamiltonian operator residual is obtained. as follows:

[0013] Constructing a loss function using Hamiltonian operator residuals for: ;in, This represents the number of sampling points.

[0014] Preferably, semi-discrete difference operators include, but are not limited to, the Godunov scheme and the Lax-Friedrichs scheme.

[0015] Preferably, the Hamiltonian operator residuals obtained based on the Godunov scheme. as follows: .

[0016] Preferably, the Hamiltonian operator residuals obtained based on the Lax-Friedrichs scheme as follows: .

[0017] Preferably, in S5, the AdamW optimizer is used to execute the gradient descent algorithm.

[0018] On the other hand, the present invention also discloses a seismic wave travel time calculation system based on a semi-discrete physical information neural network, comprising: The information acquisition module is used to obtain the solution domain. Velocity field data The travel time field model is defined by combining the Eikonal equation. and the solution domain in ; The model building module is used to build a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions. The forward modeling module is used to input spatial collocation points into a semi-discrete physical information neural network, and to preprocess the output of the neural network using a decomposition method to obtain an approximate solution for the travel time field. The physical constraint module is used to introduce a virtual step size to construct a semi-discrete difference operator. This semi-discrete difference operator is used to calculate the approximate gradient magnitude of the travel-time field solution, and then combined with velocity field data. Obtain the Hamiltonian operator residuals, and then construct the loss function; The training module is optimized to minimize the loss function using the gradient descent algorithm and train a semi-discrete physical information neural network. The travel-time field output module is used to output the solution domain. The semi-discrete physical information neural network is trained by inputting any spatial coordinate point within the network and outputs the first arrival travel time field of the seismic wave.

[0019] As can be seen from the above technical solution, the present invention discloses a method and system for calculating seismic wave travel time based on a semi-discrete physical information neural network, which has the following advantages compared with the prior art: 1. Overcoming the challenge of capturing caustics in complex media: Standard physical information neural networks typically use smooth activation functions such as hyperbolic tangent (tanh), which can lead to oversmoothing errors at gradient discontinuities (caustics) where the wavefront meets. This invention constructs a piecewise smooth function approximator by alternately using smooth and non-smooth activation functions. This approximator can accurately characterize sharp singular features in the travel time field while maintaining global solution smoothness, significantly improving computational accuracy in complex geological models (such as the Marmousi model).

[0020] 2. Introducing an upwind mechanism ensures a sticky solution to the physical causality law: Standard physical information neural networks rely on automatic differentiation to calculate gradients, and their central symmetry leads to a lack of directionality when dealing with the Eikonal equation. This invention calculates the neural network gradient in a semi-discrete manner, incorporating upwind discrete schemes (such as Godunov or Lax-Friedrichs schemes), explicitly injecting numerical dissipation and upwind characteristics into the neural network. This forces the network to converge to a unique sticky solution that conforms to physical laws, avoiding convergence to an erroneous non-sticky solution.

[0021] 3. Completely resolves the physical conflict between source singularity and negative travel time: By combining the decomposition method and the exponential activation function of the last layer of the neural network, this invention mathematically ensures the validity of the obtained travel time field. The value is strictly greater than zero, effectively eliminating the numerical instability caused by the absence of gradients near the earthquake source. This allows the algorithm to maintain high robustness in both the near-field region close to the earthquake source and the far-field region far from the earthquake source.

[0022] 4. Strong adaptability and stability to complex velocity models: Experiments in this paper demonstrate that, when dealing with extremely non-uniform velocity fields such as normal distributions, this method exhibits stronger anti-interference capabilities and convergence stability compared to existing algorithms such as PINNeik and NES. Even in extreme scenarios with multiple caustics, this invention maintains physical accuracy.

[0023] 5. Combining the flexibility of meshless computing with the rigor of numerical computation: This invention inherits the meshless characteristics of deep learning, handling irregular solution regions without the need for complex mesh generation. Simultaneously, due to the introduction of a semi-discrete scheme, it also possesses the mathematical rigor of traditional numerical methods when dealing with nonlinear partial differential equations. This hybrid approach of "neural network approximation + semi-discrete scheme constraints" provides an efficient and reliable tool for large-scale travel-time tomography in geophysics.

[0024] 6. The semi-discrete method framework design has good scalability: the semi-discrete framework is not limited to the two classic schemes, Godunov and Lax-Fridrichs; higher-order discrete schemes such as WENO can also be directly integrated into the algorithm framework. Compared with traditional methods, this not only improves the adaptability of the algorithm but also provides innovative ideas for handling larger-scale and more diverse geological problems. Attached Figure Description

[0025] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0026] Figure 1 A flowchart of the method provided by the present invention; Figure 2 This is a schematic diagram of the calculation results of the Marmousi model; Figure 3 The system architecture diagram provided for this invention. Detailed Implementation

[0027] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] On one hand, embodiments of the present invention disclose a method for calculating seismic wave travel time based on a semi-discrete physical information neural network, such as... Figure 1 As shown, it includes the following steps: S1. Obtain the solution domain Velocity field data and the location of the epicenter The travel time field model is defined by combining the Eikonal equation. ; and based on the location of the earthquake source and the solution domain Sampling is performed to generate a set of spatial collocation points.

[0029] Among them, the travel time field model The formula is as follows: ; The point source conditions are as follows: ; In the formula, For gradient operators, The solution domain The coordinates of the point in the diagram.

[0030] In this embodiment, the target region (i.e., the solution region) Sampling is performed using a uniform grid with a grid step size of [value missing]. The set of collocations is obtained. .

[0031] Determine the seismic source within the target area. In this embodiment, the point seismic source is placed at the center of the top of the model, i.e., at coordinates... The location of the earthquake source will serve as a reference point for calculating the analytical terms in the subsequent decomposition method.

[0032] S2. Construct a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions.

[0033] This embodiment uses a fully connected deep neural network as its basic architecture. For the complex Marmousi model, the input layer dimension is set to 2 (corresponding to spatial coordinates). The output layer dimension is 1 (corresponding to the function value). ).

[0034] The network consists of one input layer, four hidden layers, and one output layer. Each hidden layer has 100 neurons to ensure sufficient nonlinear expressive power to simulate travel-time evolution in complex velocity fields. The odd-numbered hidden layers (layers 1 and 3) are activated using a smooth activation function; in this example, the hyperbolic tangent function is used. Even-numbered hidden layers (layers 2 and 4) are activated using non-smooth activation functions; in this example, the ReLU activation function is used. To ensure that the output value of the neural network is positive, the output layer needs to use an exponential function. Activate.

[0035] The parameters of the neural network are initialized using Xavier to alleviate the problems of vanishing or exploding gradients. At the same time, the bias term of the linear transformation parameter of the output layer can be initialized to a large positive value (e.g., 2.0 in this embodiment) to ensure that the model is far from local optima in the initial stage of training.

[0036] S3. Input the spatial collocation points into the semi-discrete physical information neural network, and preprocess the neural network output using the decomposition method to obtain the approximate solution of the simulated travel time field, as shown in the following formula: ; In the formula, This is an approximate solution for the travel time field; For neural network output; For distance function term, .

[0037] Based on the decomposition method, singularities at the wave source are eliminated by explicitly multiplying by the distance function term, and the positive value of the travel time calculation results is ensured.

[0038] S4. A semi-discrete difference operator is constructed by introducing a virtual step size. The gradient magnitude approximation of the travel-time field approximation solution is calculated using the semi-discrete difference operator, and combined with the velocity field data. Calculate the Hamiltonian operator residuals, and then construct the loss function, including: S41. Based on virtual step size , for the set of points in space Each spatial point in Constructing a neighbor set ,in , for Euler space along the first A unit vector along the coordinate axes.

[0039] Set a virtual step size Used to compute discrete difference operators, where These are hyperparameters that are set manually. yes Euler space along the first A unit vector along the coordinate axes.

[0040] In this example, a virtual step size is used. For each sampling point Locally introduce neighboring points ,in, It is a unit vector along the coordinate axes in a 2D plane, and then the simulated travel time field data is calculated through neural network forward modeling. .

[0041] S42. Based on the nearest point set, calculate the approximate gradient magnitude of the travel time field approximation solution using a semi-discrete difference operator.

[0042] The semi-discrete architecture is applicable to common differential equation discretization schemes. Any finite difference discretization scheme with wind resistance can be selected. In this embodiment, the Godunov scheme and the Lax-Friedrichs scheme are used as examples.

[0043] S43. Approximate value of gradient magnitude through the solution obtained by the travel-time field approximation. With velocity field data The Hamiltonian operator residual is obtained. as follows: .

[0044] Specifically, the Hamiltonian operator residuals obtained based on the Godunov scheme as follows: .

[0045] Hamiltonian operator residuals obtained based on the Lax-Friedrichs scheme as follows: .

[0046] S44. Constructing a loss function using Hamiltonian operator residuals for: ;in, This represents the number of sampling points.

[0047] S5. Use the gradient descent algorithm to minimize the loss function and train a semi-discrete physical information neural network.

[0048] After initializing the neural network parameters and setting the calculation logic of the loss function, a suitable optimizer can be selected and its hyperparameters set. Gradient descent is then used to wait for the network to complete training. This embodiment uses the AdamW optimizer, setting the maximum number of iterations to 5000 and the initial learning rate... Set as The optimizer's weight_decay parameter is set to In this example, the minibatch training strategy is not used; the complete training set is used for each iteration of network parameters.

[0049] S6. Input any spatial coordinate point within the solution area into the trained semi-discrete physical information neural network, and output the first arrival travel time field of the seismic wave.

[0050] After the neural network is trained, input any coordinates from which the travel time function value needs to be calculated. The time-travel function value simulated by the neural network can then be obtained. This example selects uniform grid points (grid step size) on the training area. Using this as the test set, we obtain the set of collocations. The solution results for the mainstream PINNeik algorithm, the traditional fast scan algorithm, the semi-discrete physical information neural network algorithm using the Godunov discretization scheme, and the semi-discrete physical information neural network algorithm using the Lax-Friedrichs discretization scheme are presented on the test set. (Reference) Figure 2 .

[0051] On the other hand, such as Figure 3 As shown, this invention also proposes a seismic wave travel time calculation system based on a semi-discrete physical information neural network, comprising: The information acquisition module is used to obtain the solution domain. Velocity field data The travel time field model is defined by combining the Eikonal equation. and the solution domain in ; The model building module is used to build a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions. The forward modeling module is used to input spatial collocation points into a semi-discrete physical information neural network, and to preprocess the output of the neural network using a decomposition method to obtain an approximate solution for the travel time field. The physical constraint module is used to introduce a virtual step size to construct a semi-discrete difference operator. This semi-discrete difference operator is used to calculate the approximate gradient magnitude of the travel-time field solution, and then combined with velocity field data. Obtain the Hamiltonian operator residuals, and then construct the loss function; The training module is optimized to minimize the loss function using the gradient descent algorithm and train a semi-discrete physical information neural network. The travel-time field output module is used to output the solution domain. The semi-discrete physical information neural network is trained by inputting any spatial coordinate point within the network and outputs the first arrival travel time field of the seismic wave.

[0052] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0053] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for calculating seismic wave travel time based on a semi-discrete physical information neural network, characterized in that, Includes the following steps: S1. Obtain the solution domain Velocity field data and the location of the epicenter The travel time field model is defined by combining the Eikonal equation. ; and based on the location of the earthquake source and the solution domain Sampling is performed to generate a set of spatial collocation points; where ; S2. Construct a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions; S3. Input the spatial collocation points into the semi-discrete physical information neural network, and use the decomposition method to preprocess the output of the neural network to obtain the approximate solution of the travel time field; S4. A virtual step length is introduced to construct a semi-discrete difference operator, and the gradient module length approximation value of the travel time field approximate solution is calculated by the semi-discrete difference operator, and the velocity field data is combined The Hamiltonian operator residual is obtained, and then a loss function is constructed, including: Based on virtual step size , for the set of points in space Each spatial point in Constructing a neighbor set ,in , for Euler space along the first Unit vectors along the coordinate axes; Based on the nearest point set, the gradient magnitude approximation of the travel time field approximation solution is calculated using a semi-discrete difference operator; Traveltime field gradients approximated by neural networks and velocity field data in space resulting in hamiltonian operator residuals as follows: ； Constructing a loss function using hamiltonian operator residuals is: ; wherein, is the number of sampling points; S5. Use the gradient descent algorithm to minimize the loss function and train a semi-discrete physical information neural network; S6. Input any spatial coordinate point in the solution region into the trained semi-dispersive physical information neural network, and output the first arrival travel time field of the seismic wave.

2. The method of claim 1, wherein, Travel time field model in S1 The formula is as follows: ； The point source conditions are as follows: ； In the formula, is the gradient operator.

3. The method of claim 1, wherein, In S2, the smooth activation function is the tanh function, and the non-smooth activation function is the ReLU function.

4. The method of claim 2, wherein, In S3, the neural network output is preprocessed using a decomposition method to obtain an approximate solution for the travel time field, as shown in the following formula: ； wherein is a travel time field approximation solution; is a neural network output; is a distance function term, .

5. The method of claim 1, wherein, Semi-discrete difference operators include, but are not limited to, the Godunov scheme and the Lax-Friedrichs scheme.

6. The method of claim 5, wherein, Hamiltonian operator residual based on Godunov scheme As follows: 。 7. The method of claim 5, wherein the method is characterized by, Hamiltonian operator residual based on the Lax-Friedrichs scheme as follows: 。 8. The method of claim 1, wherein, In S5, the AdamW optimizer is used to execute the gradient descent algorithm.

9. A seismic wave travel time calculation system based on a semi-discrete physical information neural network, used to implement the seismic wave travel time calculation method based on a semi-discrete physical information neural network as described in any one of claims 1-8, characterized in that, include: The information acquisition module is used to obtain the solution domain. Velocity field data and the location of the epicenter The travel time field model is defined by combining the Eikonal equation. ; and based on the location of the earthquake source and the solution domain Sampling is performed to generate a set of spatial collocation points; where ; The model building module is used to build a semi-discrete physical information neural network, which includes alternating layers of smooth activation functions and layers of non-smooth activation functions. The forward modeling module is used to input spatial collocation points into a semi-discrete physical information neural network, and to preprocess the output of the neural network using a decomposition method to obtain an approximate solution for the travel time field. The physical constraint module is used for introducing a virtual step length to construct a semi-discrete difference operator, calculating a gradient module length approximation value of the travel time field approximate solution through the semi-discrete difference operator, and combining velocity field data A Hamiltonian operator residual is obtained, and then a loss function is constructed. The training module is optimized to minimize the loss function using the gradient descent algorithm and train a semi-discrete physical information neural network. The travel-time field output module is used to output the solution domain. The semi-discrete physical information neural network is trained by inputting any spatial coordinate point within the network and outputs the first arrival travel time field of the seismic wave.

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