https://orcid.org/0000-0001-6955-7323
Abstract
Accurate temporal resolution of the eikonal equation forms the cornerstone of seismological studies, including microseismic source localization, and travel-time tomography. Physics-informed neural networks (PINNs) have gained significant attention as an efficient approximation technique for numerical computations. In this study, we put forth a novel model named Eiko-PIGCNet, a graph convolutional neural network that incorporates physical constraints. We demonstrate the effectiveness of our proposed model in solving the 3D eikonal equation for travel-time estimation. In our approach, the discretized grid points are converted into a graph data structure, where every grid point is regarded as a node, and the neighboring nodes are interconnected via edges. The node characteristics are defined by incorporating the velocity and spatial coordinates of the respective grid points. Ultimately, the efficacy of the Eiko-PIGCNet and PINNs is evaluated and compared under various velocity models. The results reveal that Eiko-PIGCNet outshines PINNs in terms of solution accuracy and computational efficiency.
1. Introduction
The eikonal equation, a nonlinear partial differential equation (PDE), describes the time required for a seismic wave to propagate from a source point to any location within a given spatial domain. The accurate estimation of the initial seismic wave arrival times is crucial for various scientific investigations, including travel-time tomography (Schuster & Quintus-Bosz 1993) and the microseismic localization (Grechka et al.2015). Traditional methods to solve this problem primarily use the fast-marching method (FMM) (Hassouna & Farag 2007) and the fast-sweeping method (Qian et al.2007). However, accurately computing 3D travel times demands a higher level of precision and increased computational time, leading to increased computational costs (Jeong & Whitaker 2008). Given these challenges, recent advances in computational techniques and deep learning have the potential to improve the efficiency and accuracy of travel-time estimation, thereby addressing the limitations of traditional approaches.
Deep learning has emerged as a promising approach for solving PDEs, leveraging its powerful function-fitting capabilities to advance PDE-solving techniques (Lu et al.2019; Beck et al.2020). Since 2019, various network models based on different architectures have been introduced to tackle complex PDEs. These operators are extensions of neural networks designed to learn intricate mapping relationships in PDEs through supervised training. Li et al. (2020) introduced Fourier neural operators (FNO) as tools for learning multiple PDEs, while Wei & Fu (2022) developed physically informed Fourier neural operators for approximating the mathematical physical behavior of fluctuation equations. Konuk & Shragge (2021) propose a physics-based deep learning methodology to solve the AWE in the frequency domain. Song & Wang (2022) employed the framework of FNO to establish a mapping relationship between low-frequency and high-frequency wavefields, enabling the prediction of high-frequency wavefields based on a limited number of low-frequency components. Introduced by Raissi et al. (2019), Physics-informed neural networks (PINNs) present a groundbreaking approach that unites data-driven and physics-guided techniques for the construction and resolution of PDEs. Moseley et al. (2020) extended the original PINNs with favorable results in solving fluctuation equations. Akhalifah et al. (2020) established fundamental physical equations as a lattice of loss functions to solve the Helmholtz equation, while Song et al. (2021) used PINNs to solve the acoustic wave equation for a transversely isotropic medium with a vertical symmetry axis. Rasht-Behesht et al. (2022) demonstrated significant progress in solving wave propagation and full waveform inversion using PINNs. In the realm of solving eikonal equations, Smith et al. (2020) introduced EikoNet, a deep learning network for eikonal equation solutions. Bin Waheed et al. (2021) applied PINNs to solving the eikonal equation, achieving better accuracy than first-order fast scanning methods. Meanwhile, Bin Waheed et al. (2022) used PINNs to compute the travel-time solutions of the eikonal equations corresponding to anisotropic media, thereby offering new possibilities for solving the complex forms eikonal equations. However, traditional PINNs have limitations, such as the need to store differential computation maps for computing higher-order differentials, consuming significant storage space and training time. Moreover, the structure of traditional PINNs is typically simplistic, primarily utilizing fully connected layers. To mitigate these drawbacks, convolutional neural networks (CNNs) have been employed in PINNs. Li et al. (2022) proposed a convolutional neural network (Res-Unet) and a hybrid loss function for stratigraphic seismic wave velocity inversion. The primary distinction between feedforward neural networks and CNNs lies in the localized spatial nature of convolutional operations, allowing CNNs to learn spatially localized evolutions that are consistent with physical processes and exhibit enhanced interpretability. In addition to CNNs, graph neural networks (GNNs) also exhibit spatial localization. GNNs operate on graph data, learning computations locally based on the information transfer mechanism of neighboring nodes or connected edges. Research (Sanchez-Gonzalez et al.2020; He et al.2023) demonstrates that GNNs possess desirable fitting and generalization capabilities in the evolutionary modeling of physical processes.
In this study, we introduce a novel deep learning model, Eiko-GCNet, for solving eikonal equations. This model adapts PINNs by retaining their physical constraints on initial and boundary conditions and transforms the fully connected structure into a graph convolutional neural grid structure. The computational domain of eikonal equations is divided into a regular grid, treating the grid as a graph, and the model is trained using a physically constrained loss function. Similar to PINNs, Eiko-PIGCNet does not require a dataset for training. We trained and tested Eiko-PIGCNet alongside conventional PINNs on various velocity models, including the homogenous, layered, block, and 3D Marmousi velocity models. Travel times were computed for each model on an 11th Gen Intel® Core™ i5-11260H @ 2.60 GHz CPU. The results were then compared with those obtained using the FMM. The findings indicate that Eiko-PIGCNet is capable of accurately modeling eikonal equations in 3D and demonstrates significantly superior performance compared to traditional PINNs, particularly in complex velocity models.
2. Methods
2.1. The eikonal equation
The eikonal equation is a first-order nonlinear PDE. In the context of isotropic velocity models with heterogeneous properties, as shown in Equation (1).
where |$\nabla $| is the gradient operator, |$T( x )$| denotes the travel time from the source point |${x}_s$| to any point in the domain, and |$v( x )$| is the velocity defined by |${{\Omega }}$| in the domain.
The eikonal equation is commonly used in seismic wave analysis to estimate travel time with high-frequency approximations. However, this approach is known to suffer from significant numerical errors and a strong singularity at the source origin (Zhao 2005). To mitigate the impact of the singularity, researchers have proposed factorizing the eikonal equation, resulting in a modified Equation (2). This modification aims to improve the accuracy of the calculation while addressing the challenge posed by the singularity at the source location.
where |${T}_0( x )\ $|denotes the known operator and |$\tau ( x )$| denotes the unknown operator with a value of |$\tau \ $|of 1 at the source. In this study, the solution of the eikonal equation function is calculated by fitting |$\tau $|.
2.2. Graph structure data with GCN
A graph comprises vertices and edges, with each vertex characterized by a feature vector, and inter-vertex relationships delineated by edges. Owing to the universal approximation properties of deep neural networks, input feature vectors for a graph convolutional network can be any discretized spatially varying field on a lattice. It processes an input graph in which each vertex corresponds to a spatial coordinate within the lattice. In the realm of computational PDEs, the use of graph theory is a commonly employed technique for representing a discretized domain as an interconnected network of nodes and edges (Jiang et al.2022). Specifically, the grid structure of the discretized domain is used to construct the graph, with each node representing a discrete point in the domain and each edge connecting nodes that are adjacent in space. The weight of each edge is typically determined by the Euclidean distance between the nodes it connects, reflecting the underlying spatial relationships between the discrete points in the domain. In this study, the spatial domain of the eikonal equation is discretized into grid nodes. For a 2D space, this is a grid with x and y elements, while for a 3D space, the grid has x, y, and z elements representing the corresponding spatial dimensions. This paper considers the spatial domain as an undirected graph for training purposes, where the grid points are represented as nodes in the graph. The nodes in the vicinity of the |$m-th$| node located in the space domain are identified as |$( {m - 1,l,n} )$|, |$( {m + 1,l,n} )$|, |$( {m,l - 1,n} ),( {m,l + 1,n} ),( {m,l,k - 1} ),( {m,l,n + 1} )\ $| and are connected to it through edges. In Fig. 1, the spatial domain is presented through a construction diagram where each node represents a distinct location within the domain. In the framework of GCNs, these nodes function as the spatial coordinates in the discretized grid while the edges encode the inherent relationships between these nodes. Especially, each node possesses a feature vector, including relevant properties and variables such as coordinates, velocity, and the time taken to travel from the node to a source point. The formulation of the node information is represented by
where |$\vec{x}$| is the coordinate of the grid node |$( {x,y,z} )$|, v is the velocity at the node position, and |${T}_0$| is the time of propagation from the node position to the source position in an isotropic medium.
Graph structure data: (a) with 2D spatial nodes and (b) 3D spatial nodes.
The GCN is a convolutional network variant that processes graphs efficiently. It operates directly on the graph structure, propagating information layer by layer through a multi-layer graph convolutional network. This is achieved by using specific rules:
In Equation (4), |$\tilde{A}$| is obtained as the summation of adjacency matrix of the undirected graph and the identity matrix to include self-connections. The diagonal degree matrix is represented by |$\tilde{D}$|. The matrix of activations in the lth layer is denoted by |${H}^{( l )}$|, with|$\ {H}^{( 0 )\ }$|representing the feature representation of the nodes. The convolution weights for the lth layer are represented by |${W}^{( l )}$|, while σ represents the activation function.
2.3. Eiko-PIGCNet based 3D travel-time solution of the eikonal equation
The Eiko-PIGCNet is a neural network model featuring a residual connected structure composed of several GCN layers. The model presented in this study employs N convolutional layers, each with a filter size of 1 × 1 and implementing the Tanh activation function. The GCN is designed to accept inputs from both the initial conditions and all nodes present within the system. To achieve this, a partial differential operator, as defined in Equation (5), is utilized, and all relevant terms in Equation (2) are reorganized to the left side, ensuring the right side of the expression equals zero. These techniques aim to optimize the model's performance for complex, dynamic systems:
In this study, the input format for the model is N × P, where N denotes the number of vertices, and P represents the count of vertex attributes. For the 3D eikonal equation, P = 3, corresponding to the attributes|$\ [ {\vec{x},v,{T}_0} ]$|. Moreover, the graph structure, specifically edge information, is provided to the model through the adjacency matrix. The architecture of Eiko-PIGCNet is shown in Fig. 2, where the red nodes represent the source points, and the blue nodes represent the receiver points. For a given velocity model, Eiko-PIGCNet fits the mapping between the values of |$\vec{x}$| and |$\tau ( {\vec{x}} )$| of the eikonal equation. Subsequently, the 3D travel time is obtained by multiplying |$\tau ( {\vec{x}} )$| by the known factor |${T}_0( {\vec{x}} )$|.
Architecture of Eiko-PIGCN.
The Eiko-PIGCNet employs a loss function that combines two components based on the PINNs. The first component enforces the differential operator constraint within the specified domain, while the second component minimizes the difference between the actual initial conditions and the model predictions. The equation function consists of two main terms, namely |$\tau ( x )$|, with positive value and |$\tau ( {{x}_s} )$|, with a value of 1 at the source. The loss function is formulated using the following equation:
Here, |${N}_I$| denotes the number of nodes involved in training, and H is the Heaviside function. Given the absence of a validation dataset for Eiko-PIGCNet and PINNs, the FMM is employed to compute the reference travel time, |${T}_{ref}$|, for a 3D velocity model. Subsequently, |${T}_{pred}$| is derived using the Eiko-PIGCNet, and the difference between |${T}_{pred}$| and |${T}_{ref}$|, i.e. the relative travel-time error, |${T}_{error}$|, is used to assess the accuracy of the Eiko-PIGCNet. The same procedure is applied to the results obtained from the PINNs, and the relative travel-time error of the two models is compared to evaluate the precision of the model solutions:
3. Results
In this study, we valuate the computational accuracy of the Eiko-PIGCNet in solving the 3D eikonal equation using various velocity models, including the homogeneous, block, and layered, and 3D Marmousi. The performance of the Eiko-PIGCNet is compared against two other methods, the FMM and the PINNs. The results demonstrate the efficacy of the Eiko-PIGCNet in accurately solving the eikonal equation for the tested velocity models.
In the implementation of the PINNs, TensorFlow and Sciann (Haghighat & Juanes 2021) frameworks were employed. The model utilized a [20] × 10 network architecture, consisting of 10 hidden layers with 20 neurons in each layer. For the homogeneous, block, and layered velocity models, 50 000 points were randomly selected to constitute the training dataset. The model was trained using the Adam optimizer, undergoing 5000 iterations to optimize its performance. Similarly, for the Eiko-PIGCNet, the homogeneous, layered, and block velocity models used N = 41 × 41 × 41 nodes, with 70% of these nodes randomly selected for training. This was achieved by selecting nodes in the x- and y-axis directions and all nodes in the z-axis, while excluding the source location, and then superimposing the nodes at the source. The selected nodes and node attributes were combined to form the training dataset. For the homogeneous velocity model, a [12] × 3 network architecture was employed, which consists of three hidden layers with 12 neurons per layer. In contrast, for the layered and block velocity models, a [20] × 10 network architecture was used, featuring 10 hidden layers with 20 neurons in each layer. The Eiko-PIGCNet was trained using L-BFGS with a learning rate of 0.0001 and a maximum of 50 000 iterations.
3.1. The homogeneous velocity
In this study, we considered a homogeneous velocity model of size (40, 40, 40 km), with a grid spacing of 1 km and a source location of (20, 20, 20 km). The travel-time distribution at the source point is illustrated in Fig. 3. Red circles are used to indicate the positions of seismic stations. Black solid lines represent the travel times obtained from the FMM method, while white dashed lines denote those calculated using the Eiko-PIGCNet, and red dashed lines signify the travel times computed by the PINNs. The travel-time distance on the three sections was measured to be 1 s.
Distribution of the 3D travel-time field at the source slices for homogeneous velocity model.
Figure 4 depicts the relative travel-time errors and the slices at the source point obtained using Eiko-PIGCNet. It was used to calculate the relative travel-time errors of the three cuts, which were found to be 0.00732, 0.00776, and 0.00686%. On the other hand, the PINNs yielded relative timing errors of 0.00357, 0.00438, and 0.00343%, respectively. These results indicate that the PINNs provides slightly better accuracy in calculating travel time for the homogeneous velocity model, compared to the Eiko-PIGCNet.
The diagram for the relative travel-time error of the Eiko-PIGCNet for the 3D homogeneous velocity model.
3.2. The block velocity
This study examined the computational accuracy of two models, Eiko-PIGCNet and PINNs, in a block velocity model with dimensions of (40, 40, 40 km) and a grid spacing of 1 km. A high-speed anomaly of 7 km s-1 was located between (15, 15, 15 km) and (25, 25, 25 km), with a source location (20, 20, 20 km). The spatial distribution of travel-time field slices at the source location obtained using three different methods are shown in Fig. 5. The travel-time distance on the three sections was measured to be 0.5223s.
Distribution of the 3D travel-time field at the source slices for block velocity model.
A slicing procedure was performed at source, the relative error in outcomes as depicted in Fig. 6. The Eiko-PIGCNet exhibited remarkable performance, as underscored by its average relative time errors of 0.55743, 0.58923, and 0.55615% across the three sections. This is in stark contrast to the PINNs, which recorded relative time errors of 1.01486, 1.07846, and 1.01123%, respectively, thus highlighting the superior accuracy of the Eiko-PIGCNet. However, it is noteworthy that Eiko-PIGCNet displayed an increase in relative time errors in the context of the velocity abrupt change plane, as depicted in Fig. 6. This observation indicates a potential limitation of Eiko-PIGCNet when navigating scenarios characterized by abrupt velocity changes, necessitating further research to enhance its robustness under such conditions.
The diagram for the relative travel-time error of the Eiko-PIGCNet for the 3D block velocity model.
3.3. The layered velocity
In this study, we assume a layered velocity model with dimensions of (40, 40, 40 km) and a grid spacing of 1 km. The source is located at coordinates (20, 20, 20 km). The velocity increases along the z-axis, with an increment of 0.5 km s-1 every 5 km, as shown in Fig. 7a. The travel-time field slice at the source location is depicted in Fig. 7b. The travel-time contours on the three cross-sections were generated with a uniform interval of 1 s.
layered: (a) 3D Velocity model; (b) Distribution of the 3D travel-time field at the source slices.
The relative error and source location slice of Eiko-PIGCNet are shown in Fig. 8. The Eiko-PIGCNet exhibited relative travel-time errors of 3.72041, 1.23751, and 1.21784% across three slices, while the PINNs demonstrated lower errors of 2.2235, 0.83356, and 0.83344%, respectively. Notably, the Eiko-PIGCNet demonstrated stratification in the xz and zy sections, with larger relative time errors at the velocity abrupt change interface due to the source location being located at the partitioning interface. The results indicate that the PINNs outperformed the Eiko-PIGCNet in terms of 3D travel-time accuracy in the layered velocity model.
The diagram for the relative travel-time error of the Eiko-PIGCNet for the 3D layered velocity. model.
3.4. 3D Marmousi velocity
To assess the computational accuracy of Eiko-PIGCNet in intricate geological models, we constructed a 3D Marmousi velocity model characterized by dimensions of 35 × 35 × 35 km, source coordinates at (17.5, 17.5, 17.5 km), a grid spacing of 0.35 km, and a total node count of N = 101 × 101 × 101, as shown in Fig. 9a. PINNs were employed for training at all grid points, iterating 5000 times, and Eiko-PIGCNet was also selected for training across all nodes. Slicing was conducted at the source, resulting in travel-time field distributions as displayed in Fig. 9b. The travel-time contours on the three cross-sections were generated with a uniform interval of 1 s.
Marmousi: (a) 3D velocity model and (b) distribution of the 3D travel-time field at the source slices.
Figure 10 illustrates the relative travel-time error of the Eiko-PIGCNet and the source slice. The Eiko-PIGCNet demonstrated average relative timing errors of 2.7205, 2.29723, and 1.42265% across three sections. By comparison, the PINN exhibited higher average relative timing errors of 9.01563, 11.0111, and 8.9% for the same sections. These results indicate that the Eiko-PIGCNet outperforms the PINNs in terms of timing accuracy and constraint capability when applied to complex geological models.
The diagram for the relative travel-time error of the Eiko-PIGCNet for the 3D Marmousi velocity model.
4. Discussion
The Eiko-PIGCNet utilizes discretized grid and graphical inputs to effectively constrain the residuals of the equations and values at the source. Unlike the PINNs, which relies on fully connected layers, the Eiko-PIGCNet employs graph convolutional layers, which have better scalability and faster convergence due to their physically based discrete learning method (Zhu & Zabaras 2018; Gao et al.2021). A comparative analysis was conducted between the Eiko-PIGCNet and PINNs, focusing on their convergence characteristics during training. Additionally, a performance evaluation of these models was carried out, assessing their computational precision and efficiency across four distinct models.
The study focuses on a simple velocity model, using a block model as an example, and illustrates the variation of the loss function with the number of iterations for the training process of the Eiko-PIGCNet and PINNs in Fig. 11. After 5000 iterations of the PINNs, the loss function value decreases slowly, while the Eiko-PIGCNet stops after 261 iterations, as indicated by the blue and red lines, respectively. Additionally, the loss value of the PINN essentially does not decrease with the increase of iterations after 100 iterations, whereas the loss function of the Eiko-PIGCNet gradually decreases with the increase of iterations. In the case of the complex geological model, as shown in Fig. 12, the loss function values exhibit similar characteristics with the increase of iterations. Overall, the results indicate that the Eiko-PIGCNet converges significantly better than the PINNs.
Variation of Eiko-PIGCNet and PINN loss functions for the block velocity model.
Variation of Eiko-PIGCNet and PINN loss functions for the Marmousi velocity model.
Table 1 illustrates a comparative analysis of relative travel-time errors in 3D travel-time calculations between the Eiko-PIGCNet and PINNs. In homogenous and stratified velocity models, the computational precision of PINNs marginally surpasses that of Eiko-PIGCNet. Conversely, within the context of complex velocity models, Eiko-PIGCNet exhibits a pronounced superiority over PINNs. As indicated by the data in Table 2, a significant disparity was observed in the training times of Eiko-PIGCNet and PINNs across four different velocity models. In the homogeneous velocity model, the runtime ratio of Eiko-PIGCNet to PINNs was 1:6.68, indicating that Eiko-PIGCNet exhibited a significantly faster training time. Similar trends were observed in the block velocity model (1:3.8), layered velocity model (1:3.75), and Marmousi model (1:3.23). The data suggest that across all velocity models, the Eiko-PIGCNet consistently outperforms the PINNs in terms of training efficiency, with the most significant improvement observed in the homogeneous velocity model. For the remaining models, the PINNs training time was ∼3.5 times longer than that of Eiko-PIGCNet, highlighting the superior efficiency of Eiko-PIGCNet over traditional PINNs. Furthermore, a comparative assessment was conducted between deep learning models, including Eiko-PIGCNet and PINNs, and the finite-difference method in terms of travel-time computation. The travel-time prediction performance of Eiko-PIGCNet has slightly better PINNs across different velocity models. Moreover, both models substantially outperformed the finite-difference method, indicating their potential superiority for applications requiring travel-time computations.
Relative error comparison for Eiko-PIGCNet and PINNs.
| Model | Homogeneous (%) | Block (%) | Layered (%) | Marmousi (%) |
|---|---|---|---|---|
| Eiko-PIGCNet | 0.0071 | 0.5412 | 1.2497 | 2.7166 |
| PINNs | 0.0056 | 0.9573 | 1.1468 | 11.6422 |
| Model | Homogeneous (%) | Block (%) | Layered (%) | Marmousi (%) |
|---|---|---|---|---|
| Eiko-PIGCNet | 0.0071 | 0.5412 | 1.2497 | 2.7166 |
| PINNs | 0.0056 | 0.9573 | 1.1468 | 11.6422 |
Relative error comparison for Eiko-PIGCNet and PINNs.
| Model | Homogeneous (%) | Block (%) | Layered (%) | Marmousi (%) |
|---|---|---|---|---|
| Eiko-PIGCNet | 0.0071 | 0.5412 | 1.2497 | 2.7166 |
| PINNs | 0.0056 | 0.9573 | 1.1468 | 11.6422 |
| Model | Homogeneous (%) | Block (%) | Layered (%) | Marmousi (%) |
|---|---|---|---|---|
| Eiko-PIGCNet | 0.0071 | 0.5412 | 1.2497 | 2.7166 |
| PINNs | 0.0056 | 0.9573 | 1.1468 | 11.6422 |
Comparison of training and prediction times for Eiko-PIGCNet, PINNs, and finite-difference (FD) methods.
| Model | Homogeneous | Block | Layered | Marmoursi | |
|---|---|---|---|---|---|
| Eiko-PIGCNet | Training time | 3.18 min | 6.16 min | 10.2 min | 452.13 min |
| Prediction time | 1.23 s | 1.43 s | 1.33 s | 2.78 s | |
| PINNs | Training time | 21.23 min | 23.45 min | 38.34 min | 1462.32 min |
| Prediction time | 1.34 s | 1.64 s | 1.74 s | 3.42 s | |
| FD | Training time | ||||
| Calculation time | 4.21 min | 4.74 min | 5.18 min | 517.46 min |
| Model | Homogeneous | Block | Layered | Marmoursi | |
|---|---|---|---|---|---|
| Eiko-PIGCNet | Training time | 3.18 min | 6.16 min | 10.2 min | 452.13 min |
| Prediction time | 1.23 s | 1.43 s | 1.33 s | 2.78 s | |
| PINNs | Training time | 21.23 min | 23.45 min | 38.34 min | 1462.32 min |
| Prediction time | 1.34 s | 1.64 s | 1.74 s | 3.42 s | |
| FD | Training time | ||||
| Calculation time | 4.21 min | 4.74 min | 5.18 min | 517.46 min |
Comparison of training and prediction times for Eiko-PIGCNet, PINNs, and finite-difference (FD) methods.
| Model | Homogeneous | Block | Layered | Marmoursi | |
|---|---|---|---|---|---|
| Eiko-PIGCNet | Training time | 3.18 min | 6.16 min | 10.2 min | 452.13 min |
| Prediction time | 1.23 s | 1.43 s | 1.33 s | 2.78 s | |
| PINNs | Training time | 21.23 min | 23.45 min | 38.34 min | 1462.32 min |
| Prediction time | 1.34 s | 1.64 s | 1.74 s | 3.42 s | |
| FD | Training time | ||||
| Calculation time | 4.21 min | 4.74 min | 5.18 min | 517.46 min |
| Model | Homogeneous | Block | Layered | Marmoursi | |
|---|---|---|---|---|---|
| Eiko-PIGCNet | Training time | 3.18 min | 6.16 min | 10.2 min | 452.13 min |
| Prediction time | 1.23 s | 1.43 s | 1.33 s | 2.78 s | |
| PINNs | Training time | 21.23 min | 23.45 min | 38.34 min | 1462.32 min |
| Prediction time | 1.34 s | 1.64 s | 1.74 s | 3.42 s | |
| FD | Training time | ||||
| Calculation time | 4.21 min | 4.74 min | 5.18 min | 517.46 min |
Finally, there also exists a limitation of our method. The current model framework is specifically tailored for the rule space grid of four velocity models, a limitation inherent in the design characteristics of the model. However, the flexibility of GCN presents a promising opportunity for future research to expand this model to 3D irregular grids. This is due to the capacity of GCN to handle complex graph structures, thereby enabling them to manage grids with irregular connections. It is noteworthy that the application of the Eiko-PIGCNet has primarily been confined to simulating the seismic wave travel-time field from a single fixed source. Its capability to simulate multiple random sources remains an area requiring further evaluation.
5. Conclusions
This research presents a novel, physically informed Eiko-PIGCNet, which leverages the potential of graph CNNs to compute first-arrival travel times with remarkable accuracy and efficiency within a 3D velocity model incorporating a depth grid. The proposed methodology demonstrates superior computational accuracy and efficiency compared to the traditional PINNs, particularly within the confines of complex geological models. Interestingly, despite these marked improvements, the Eiko-PIGCNet maintains a level of accuracy comparable to PINNs when applied to simpler geological models. The findings reveal its potential to outshine traditional methods like PINNs in terms of computational efficiency and accuracy, thereby opening new avenues for enhancing computational geophysics and related fields. Further work can explore refining this model to address its noted limitations in situations of abrupt velocity changes.
Acknowledgements
This research was funded by the National Natural Science Foundation of China (grant no. 41930112). The 3D figure showing tool new tools used for demonstration in the study are available via https://www.researchgate.net/publication/330778222_A_simple_code_Matlab_for_showing_3D_data_especially_for_geophysi-cal_inversion_results.
Conflict of interest statement. The authors declare that there are no conflicts of interest regarding the publication of this article.
