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Wenbin Tian, Yang Liu, Yibo Zhang, Enhancing learning to solve multicomponent fractional viscoelastic equations with U-net Fourier neural operators, Journal of Geophysics and Engineering, Volume 22, Issue 1, February 2025, Pages 16–35, https://doi.org/10.1093/jge/gxae110
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Abstract
The research of viscoelastic media is currently a hot topic in the interpretation and processing of seismic data. To accurately simulate the propagation of seismic waves in viscoelastic media, the fractional viscoelastic equation has emerged as an indispensable method. However, solving this equation numerically has proven to be challenging due to the complexity introduced by its fractional Laplacian operators. Recently, deep learning, especially Fourier neural operators (FNO), has shown excellent performance in learning to fast solve partial differential equations. Traditional FNO methods may face crosstalk problems and this make it difficult to achieve satisfactory accuracy when solving the multicomponent fractional order viscoelastic equation. To solve this problem, we introduce a novel approach based on U-net Fourier neural operator (U-FNO). As an enhanced learning method to the traditional FNO-based method, the U-FNO-based method integrates a U-Fourier layer following the standard Fourier layer as a form of regularization, thereby achieving superior prediction accuracy for multicomponent equations. Specifically, both the Fourier layers and U-Fourier layers in U-FNO are trained with the solutions of the equation from previous time steps as inputs. This training process enables the U-FNO to efficiently produce more accurate solutions for subsequent wavefield. Numerical simulations reveal that the U-FNO-based method efficiently learns to solve the fractional viscoelastic wave equation independent of fractional Laplacian operators. Additionally, U-FNO-based method offers superior prediction accuracy in comparison with the traditional FNO-based method.
1. Introduction
Seismic wave forward modelling is essential in the process of seismic data processing (Carcione 1993; Carcione et al. 2002; Liu & Sen 2009a,b; Ren & Liu 2013; Xu et al. 2023). During their propagation through media, seismic waves are affected by attenuation and elasticity: inherent properties of the earth (Robertsson et al. 1994; Alkhalifah 2000; Duveneck & Bakker 2011). The viscoelastic equation, widely recognized for describing the characteristics of viscoelastic media, has garnered considerable attention. Numerous scholars have contributed to the advancement of the fractional viscoelastic wave equation, utilizing constant-Q models to simulate these properties during the propagation of the seismic wavefield (Kalyani et al. 2014). This equation can describe the attenuating effects in an attenuating medium (Li et al. 2016; Zhu 2017; Zhu et al. 2019). However, the solution to the complex equation needs several Fourier transforms because of the fractional Laplacian operators, which are associated with the quality factor Q. Considering the constraints of computing hardware, the necessity for multiple Fourier transforms constitutes a major problem (Yao et al. 2017). Furthermore, to accurately represent the complex propagation of the wavefield, the solution to the viscoelastic equation comprises multiple components, adding to the challenge of solving the equation. Recently, a range of techniques including low-rank decomposition (Sun et al. 2015; Chen et al. 2019), Taylor series expansion (Guo et al. 2016; Zhang et al. 2020), and independent fractional operators (Patnaik et al. 2020; Wang et al. 2022) are introduced into improve the simulation of viscoelastic media. To obtain seismic wavefields of higher frequency and resolution, these methods necessitate finer grid discretization, leading to increased computational costs in forward modelling (Liu & Sen 2011). This limitation often presents a significant challenge in applications such as seismic migration and inversion. Selecting the appropriate solution methods to enhance both accuracy and computational efficiency has emerged as a primary research focus in recent years (Xing & Zhu 2019; Xiong & Guo 2022; Zhou et al. 2023).
The successful implementation of deep learning, particularly through data-driven methods, has inspired numerous scholars to explore suitable approaches for solving partial differential equations (PDEs). A data-driven method for solving PDEs involves learning the mathematical and physical behaviour of these equations from data in a supervised learning framework without relying on complex traditional equations (LeCun et al. 2015; Qu et al. 2022). Recent progress in data-driven-based techniques has dramatically improved computational efficiency, often achieving speeds several orders of magnitude faster than those of traditional solvers. In the realm of data-driven strategies for seismic simulation, two popular methods are widely used. The initial method is the spatial mapping technique, which frequently utilizes convolutional neural networks (CNNs) to apply finite-dimensional operators within its category. These methods have demonstrated remarkable success in quickly and accurately predicting outputs for high-dimensional PDEs (Jiang et al. 2021; Zhong et al. 2023). However, the lack of physical information results in reduced prediction accuracy for this technique (Wang et al. 2017; Smith et al. 2020; Bhattacharya et al. 2021; Wen et al. 2021). The second technique employs the physical information constraint approach, integrating these constraints into a complex network to achieve solutions with high accuracy (Raissi et al. 2019; Zhang et al. 2023). During training, this method (such as physical information neural network (PINN)) automatically satisfies specific physical information using automatic differentiation, leading to enhanced accuracy and improved generalization. Unlike data-driven CNN-based methods, the PINN-based approach falls into the category of wave-equation methods and shares strong resemblances to classical numerical solvers (Moseley et al. 2020; Alkhalifah et al. 2021; Karniadakis et al. 2021; Song et al. 2021; Chen & Ge 2023). The PINN-based method is more suitable for equations that are amenable to solution by finite-difference methods. However, solving this fractional equation with the finite-difference method poses challenges, rendering the PINN-based method unsuitable for equation solving (Wang et al. 2018).
Currently, the Fourier neural operator (FNO) is introduced to solve the problem of PDEs in the supervised learning framework (Li et al. 2020a,b). Differing from the methods previously mentioned, FNO adopts an inductive bias strategy by integrating physical information within the structure of the convolutional blocks (Lu et al. 2019; Grady et al. 2023). The FNO-based method uses a framework that accurately captures the configuration of the wave propagator in the Fourier domain. It is achieved through the use of two fully connected layers, specifically designed for the spatial and wavenumber domains (Kosloff & Baysal 1982; Stoffa et al. 1990). This strategy estimates the mathematical–physical properties of PDEs by establishing the mapping with solutions of the equations at different times and spatial locations (Rashid et al. 2022). Subsequently, the FNO-based method has been widely applied to learning solutions for seismic wave equations (Song & Wang 2022; Zhang et al. 2023). The commonly used training strategy in the time domain is to use the wavefield snapshot at this time step to predict the wavefield snapshot at next time step and establish an appropriate loss function to update the network (Wei & Fu 2022). Recently, by introducing model parameters to modify the FNO network structure, the FNO method has been successfully applied to the rapid inversion of model parameters (Yang et al. 2023; Yin et al. 2023). The FNO-based method has shown great potential for development in the fields of forward modelling and inversion in geophysics. However, its prediction accuracy for complex equations falls short because of the intrinsic regularization effect present in the FNO framework. Meanwhile, solving the viscoelastic equation, especially with multiple component solutions, poses a significant challenge. Recently, U-net has become increasingly popular as a deep learning architecture for data analysis, especially in image processing (Long et al. 2015). It uses convolutional layers to extract features and has succeeded in tasks like image classification and object detection (Krizhevsky et al. 2012; He et al. 2016). Combining the benefits of both FNO and U-net, researchers have proposed the U-net Fourier neural operator (U-FNO) and solved the problems with complex equations, such as multiphase flow equations (Wen et al. 2022). Their research declares that the U-FNO-based method outperforms the traditional FNO in improving the accuracy of complex PDE solutions. Currently, U-FNO has yet to be applied in the field of geophysics.
We examine the performance of FNOs in learning viscoelastic wave equations and compare the predictive performance between the standard FNO-based method and the U-FNO-based method. The U-FNO method combines the advantages of both FNO- and U-net-based methods. The U-net, employed as a regularization term, effectively addresses the crosstalk problem in traditional FNO methods for multicomponent equations and achieves high-accuracy solutions for multicomponent fractional viscoelastic equations. This provides a foundation for achieving high-precision inversion by modifying the network.
This paper first introduces the fractional viscoelastic equation. Subsequently, we offer an overview of the FNO framework used for seismic modelling. Next, the theory of the U-FNO architecture for seismic modelling is introduced. Finally, we assess the precision and performance of our proposed method by a homogeneous model and a partial Hess model.
2. Methodology
In this section, we detail the fractional viscoelastic equation along with its solving techniques, discuss the structure and problem of the FNO, and introduce the U-FNO framework as an innovative solution to these problems.
2.1. The introduction of fractional viscoelastic equation
The 2D fractional viscoelastic equation, derived from the constant-Q model, is often formulated as follows (Zhu & Carcione 2014):
where
where
where
Owing to the involvement of two fractional Laplacian operators
where t and
where G is a function representing Equation (4). From Equation (5), to predict the wavefield snapshot at this time step, we only need the wavefield at previous time step. To simplify the subsequent use of symbols we let
2.2. The review of FNO architecture for seismic modelling
Currently, an innovative method known as the FNO is emerged, which entails the direct parameterization of the integral kernel within Fourier domain. Its main objective is to build a mapping in an infinite-dimensional space with a limited set of an input–output dataset, a critical task in geophysics for modelling complex wave propagation. This can be accomplished by employing convolution with functions related to low wavenumbers to capture global features, and then applying an activation function to these features to transition back to the high wavenumber model. This process is crucial for modelling seismic wave propagation. The exceptional capability of the FNO allows it to approximate functions characterized by Fourier modes. Its architecture is comprised of multiple Fourier layers, with each layer hosting two fully connected layers in both the spatial and wavenumber domains, as shown in Fig. 1b. The fully connected layers in the spatial domain, which includes a local transform W, plays a pivotal role as the trainable phase-screen compensator, enabling a model to accommodate local variations. Meanwhile, the fully connected layer within the wavenumber domain integrates a sequence of transformations: a Fourier transform F, followed by a linear transformation R, and concludes with an inverse Fourier transform
where
where
where O stands for a nonlinear map operator.

Training and prediction processes by FNO (a) and U-FNO (c) and the architectures of Fourier layers (b) and U-Fourier layers (d).
By comparing Equation (5) with Equation (8), it becomes clear that the method based on the FNO seeks to refine the nonlinear mapping operator O to capture the mathematical–physical characteristics of the function G. This objective can be attained with optimizing a loss function that evaluates the difference between the predicted
2.3. The theory of U-FNO architecture for seismic modelling
U-FNO is introduced to enhance the optimization of FNO. It consists of the U-Fourier layer (Fig. 1d) and the Fourier layer (Fig. 1b). By incorporating a U-net architecture, the U-Fourier layer improves its capability for parameterizing high-frequency information in the wavefield through local convolutional kernels, distinguishing it from the traditional Fourier layer. This addition increases the flexibility and robustness of FNO, making it more effective in solving the fractional viscoelastic wave equation and in handling complex seismic wavefield modelling. The training process in wavefield modelling can be revised by employing the U-FNO-based method, as shown in Fig. 1c. Specifically, the two-component wavefield with partial time steps is sequentially processed through the FNO layer followed by the U-FNO layer. The process of U-FNO, which is aimed at obtaining the solution of the fractional viscoelastic equation, is as follows:
The U-FNO employs the wavefield snapshots from the previous several time steps (
) as input, which is then elevated to a higher-dimensional channel domain through a fully connected layer ( ), leading to a promoted snapshots of wavefield (V).This promoted snapshots of wavefield (V) is further refined with some Fourier layers. In each Fourier layer, the promoted wavefield is transformed into the wavenumber domain by a sequence of operations: Fourier transform F, linear transform R, and inverse Fourier transform
. Meanwhile, this promoted wavefield is linearly transformed by local transform (W) in the spatial domain. Moreover, the outputs of these two parts are combined by activation function to obtain the output from this Fourier layer. Subsequently, this output is input into the next Fourier layer.The final output from these Fourier layer (
) serves as the input for several U-Fourier layers. In each U-Fourier layer, the input is processed to obtain ( ) not only through two components of the conventional FNO but also via the U-net ( ).The final output, which is obtained by the last U-Fourier layer (
), is then mapped back to the target dimension using a fully connected layer ( ). This network includes two fully connected layers and a function, which enables it to predict the wavefield at the current time.
Similar to Equation (6), the learning processes of FNO and U-FNO can be mathematically represented as follows:
The final predicted wavefield at this time step (i) is.
Through training U-FNO with snapshots of solutions to this complex equation, the resulting U-FNO becomes a PDE solver capable of generating accurate solutions for Equation (1). By comparing Equations (9) and (6), it is demonstrated that the features of the wavefield snapshots extracted by the U-net in the U-FNO method can be used as regularization terms to mitigate the crosstalk problem between the two component datasets. The layer number of Fourier (L) and U-Fourier (N) is often tuned based on the characteristics and requirements of the specific problem being solved. After many experiments, we find that employing two Fourier layers followed by two U-Fourier layers can obtain the optimal performance for solving a fractional viscoelastic equation. The input and training approach utilized in this method play a critical role in the training process. Figure 2 offers a schematic depiction of the input and output within the U-FNO (taking snapshots

Schematic description of input and output in training and prediction (number of snapshots is three).
3. Numerical examples
In this section, the capability of U-FNO to predict seismic wavefield propagation in viscoelastic media is demonstrated by a homogeneous model and a partial Hess model. The SGPS-based method, in conjunction with PML boundary conditions, is employed to solve the fractional viscoelastic wave equation for making dataset. These methodologies allow us to generate datasets for training, validation, and prediction. Importantly, informed by insights from previous research and extensive experiments, we set the dimensions of the Fourier and U-Fourier layers to 12 × 20. Additionally, we use structural similarity (SSIM) and signal-to-noise ratio (SNR) (see Appendix A) to evaluate the quality of the predicted wavefield snapshots derived from both the homogeneous and partial Hess models.
3.1. The test on a homogeneous model
For the first test, a 2D homogeneous model is employed. This model is designed with a grid size of 100 × 100 and a grid spacing of 10 m. In Table 1, the parameters for velocity and the quality factor Q associated with P- and S-waves are detailed. We use a Ricker wavelet at 35 Hz with an interval of 1 ms, and a maximum sampling time of 0.24 s as the source. Figure 3 provides a visual representation of the distribution of training, validation, and prediction sets. This figure depicts the even distribution of 625 sources across the velocity model, each spaced at 40 m. From shots 100 to 500 (highlighted in yellow), the 314th shot, located at the model's centre and marked by a red star, is selected to evaluate the prediction performance. Here, 80% of the yellow region is randomly allocated to the training dataset, while the remaining 20% is allocated to the validation dataset. These datasets are compiled with a maximum time interval limited to 0.18 s, and particle velocity snapshots are resampled every 10 ms.

A visual depiction of the distribution of the training, validation, and prediction sets.
Superscript . | ||||
---|---|---|---|---|
Parameter | 3500 | 2500 | 40 | 30 |
Superscript . | ||||
---|---|---|---|---|
Parameter | 3500 | 2500 | 40 | 30 |
Superscript . | ||||
---|---|---|---|---|
Parameter | 3500 | 2500 | 40 | 30 |
Superscript . | ||||
---|---|---|---|---|
Parameter | 3500 | 2500 | 40 | 30 |
Respectively training by FNO-based and U-FNO-based approaches, we present their predictions for the

Snapshots of particle velocity

Snapshots of particle velocity
To provide further validation for the effectiveness of our method, we enhance our analysis by vertical profiles of particle velocity snapshots. Figure 6 illustrates the particle velocity

Comparison of the vertical profiles in particle velocity

Comparison of the vertical profiles in particle velocity
Furthermore, Table 2 presents the computed SSIM and SNR values, providing additional support for the aforementioned observation. Simultaneously, a clear trend emerges from the information in Table 2 is that the prediction of

Training losses and validation losses by inputting one snapshot of particle velocity (a) and three snapshots of particle velocity (b).
SSIM and SNR of the prediction by FNO and U-FNO with the homogeneous model.
. | FNO . | U-FNO . | |||
---|---|---|---|---|---|
Superscript . | One snapshot . | Three snapshots . | One snapshot . | Three snapshots . | |
SSIM | 0.3075 | 0.6504 | 0.6944 | 0.8805 | |
0.2924 | 0.6113 | 0.6373 | 0.8773 | ||
SNR | 13.75 | 18.96 | 19.68 | 28.57 | |
9.71 | 16.19 | 16.62 | 25.10 |
. | FNO . | U-FNO . | |||
---|---|---|---|---|---|
Superscript . | One snapshot . | Three snapshots . | One snapshot . | Three snapshots . | |
SSIM | 0.3075 | 0.6504 | 0.6944 | 0.8805 | |
0.2924 | 0.6113 | 0.6373 | 0.8773 | ||
SNR | 13.75 | 18.96 | 19.68 | 28.57 | |
9.71 | 16.19 | 16.62 | 25.10 |
SSIM and SNR of the prediction by FNO and U-FNO with the homogeneous model.
. | FNO . | U-FNO . | |||
---|---|---|---|---|---|
Superscript . | One snapshot . | Three snapshots . | One snapshot . | Three snapshots . | |
SSIM | 0.3075 | 0.6504 | 0.6944 | 0.8805 | |
0.2924 | 0.6113 | 0.6373 | 0.8773 | ||
SNR | 13.75 | 18.96 | 19.68 | 28.57 | |
9.71 | 16.19 | 16.62 | 25.10 |
. | FNO . | U-FNO . | |||
---|---|---|---|---|---|
Superscript . | One snapshot . | Three snapshots . | One snapshot . | Three snapshots . | |
SSIM | 0.3075 | 0.6504 | 0.6944 | 0.8805 | |
0.2924 | 0.6113 | 0.6373 | 0.8773 | ||
SNR | 13.75 | 18.96 | 19.68 | 28.57 | |
9.71 | 16.19 | 16.62 | 25.10 |
3.2. The test on a partial Hess model
To evaluate the adaptability of U-FNO to complex models, we select a partial Hess model for assessing the accuracy of the predicted particle velocities (

Figures 10 and 11 provide comparisons of predicted particle velocity snapshots at 0.15 s, a time step that extends beyond the training period respectively. In Figure 10b and d, both methods reconstruct the particle velocity snapshots of the complex model. However, on comparing Fig. 10c with e, it becomes evident that the difference between the predicted particle velocity snapshots

Snapshots of particle velocity

Snapshots of particle velocity

Comparison of the vertical profiles in true and predicted particle velocity

Training and validation losses of epochs for partial Hess model.
The differences observed in Figs. 14c and f and 15c and f indicate that as seismic waves propagate, both methods experience errors after a long period of propagation. However, compared to traditional FNO-based methods, our proposed method maintains commendable predictive performance. Simultaneously, we further evaluate the training performance of our proposed method on wavefields with source locations distant from the training set. We select the wavefield snapshots from the 614th source location as the prediction set. As shown in Fig. 16, the wavefield of the 614th source is already distant from the training set, and its prediction performance can well illustrate the generalization ability of our method. From the wavefield snapshot predicted at 0.17 s for the 614th shot (Fig. 17), it can be observed that even when far from the training region, our proposed method demonstrates good prediction performance. This indicates that our method has better generalization. U-FNO-based method exhibits better predictive performance in component (

Snapshots of particle velocity in 314th shot at 0.17 s: true particle velocity

Snapshots of particle velocity in 314th shot at 0.19 s: true particle velocity

A visual depiction of the distribution of 614th shot prediction dataset.

Snapshots of particle velocity in 614th shot at 0.17 s: True particle velocity
SSIM and SNR of the prediction by FNO and U-FNO with the partial Hess model.
. | . | . | ||||||
---|---|---|---|---|---|---|---|---|
. | SSIM . | SNR . | ||||||
. | . | . | ||||||
. | ||||||||
. | . | . | ||||||
Superscript . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . |
314th at 0.15 s | 0.8877 | 0.9366 | 0.8668 | 0.9145 | 32.81 | 37.37 | 29.1 | 34.06 |
314th at 0.17 s | 0.8841 | 0.9357 | 0.8652 | 0.9131 | 30.1 | 35.87 | 26.59 | 32.96 |
314th at 0.19 s | 0.8833 | 0.9342 | 0.8632 | 0.9127 | 28.13 | 34.91 | 24.59 | 31.33 |
614th at 0.17 s | 0.6463 | 0.7248 | 0.6063 | 0.7163 | 26.85 | 29.47 | 23.73 | 27.3 |
. | . | . | ||||||
---|---|---|---|---|---|---|---|---|
. | SSIM . | SNR . | ||||||
. | . | . | ||||||
. | ||||||||
. | . | . | ||||||
Superscript . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . |
314th at 0.15 s | 0.8877 | 0.9366 | 0.8668 | 0.9145 | 32.81 | 37.37 | 29.1 | 34.06 |
314th at 0.17 s | 0.8841 | 0.9357 | 0.8652 | 0.9131 | 30.1 | 35.87 | 26.59 | 32.96 |
314th at 0.19 s | 0.8833 | 0.9342 | 0.8632 | 0.9127 | 28.13 | 34.91 | 24.59 | 31.33 |
614th at 0.17 s | 0.6463 | 0.7248 | 0.6063 | 0.7163 | 26.85 | 29.47 | 23.73 | 27.3 |
SSIM and SNR of the prediction by FNO and U-FNO with the partial Hess model.
. | . | . | ||||||
---|---|---|---|---|---|---|---|---|
. | SSIM . | SNR . | ||||||
. | . | . | ||||||
. | ||||||||
. | . | . | ||||||
Superscript . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . |
314th at 0.15 s | 0.8877 | 0.9366 | 0.8668 | 0.9145 | 32.81 | 37.37 | 29.1 | 34.06 |
314th at 0.17 s | 0.8841 | 0.9357 | 0.8652 | 0.9131 | 30.1 | 35.87 | 26.59 | 32.96 |
314th at 0.19 s | 0.8833 | 0.9342 | 0.8632 | 0.9127 | 28.13 | 34.91 | 24.59 | 31.33 |
614th at 0.17 s | 0.6463 | 0.7248 | 0.6063 | 0.7163 | 26.85 | 29.47 | 23.73 | 27.3 |
. | . | . | ||||||
---|---|---|---|---|---|---|---|---|
. | SSIM . | SNR . | ||||||
. | . | . | ||||||
. | ||||||||
. | . | . | ||||||
Superscript . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . | FNO . | U-FNO . |
314th at 0.15 s | 0.8877 | 0.9366 | 0.8668 | 0.9145 | 32.81 | 37.37 | 29.1 | 34.06 |
314th at 0.17 s | 0.8841 | 0.9357 | 0.8652 | 0.9131 | 30.1 | 35.87 | 26.59 | 32.96 |
314th at 0.19 s | 0.8833 | 0.9342 | 0.8632 | 0.9127 | 28.13 | 34.91 | 24.59 | 31.33 |
614th at 0.17 s | 0.6463 | 0.7248 | 0.6063 | 0.7163 | 26.85 | 29.47 | 23.73 | 27.3 |
3.3. The test on a partial Marmousi model
To verify the applicability of the proposed method for complex models, we apply the partial Marmousi model in this section. Figure 18 presents the P- and S-wave velocities of the partial Marmousi model, along with the quality factor Q. All these models are divided into a 100 × 100 grid with grid intervals of 10 m. A Ricker wavelet, with a central frequency of 30 Hz, is used as the source wavelet. The maximum sampling duration and the time interval are 0.2 s and 0.5 ms, respectively. In this model test, we load the seismic source simultaneously on both the components (

Figures 19 and 20 compare the predicted particle velocity snapshots at 0.17 s, which is a time step beyond the training period. From Fig. 19b and d, it can be observed that both FNO and U-FNO methods can be effectively applied to complex models. However, by comparing Fig. 19c and e, it is evident that the differences obtained using the U-FNO method are significantly smaller than those using the FNO method. This indicates that our proposed method has higher prediction accuracy. Similar observations can also be seen in Fig. 20. It is worth noting that the difference in the

Snapshots of particle velocity in 314th shot at 0.17 s: True particle velocity

Snapshots of particle velocity in 314th shot at 0.17 s: true particle velocity
SSIM and SNR of the prediction by FNO and U-FNO with the partial Marmousi model.
Superscript . | FNO . | U-FNO . | |
---|---|---|---|
SSIM | 0.8490 | 0.8994 | |
0.8463 | 0.9012 | ||
SNR | 23.51 | 29.67 | |
24.99 | 30.42 |
Superscript . | FNO . | U-FNO . | |
---|---|---|---|
SSIM | 0.8490 | 0.8994 | |
0.8463 | 0.9012 | ||
SNR | 23.51 | 29.67 | |
24.99 | 30.42 |
SSIM and SNR of the prediction by FNO and U-FNO with the partial Marmousi model.
Superscript . | FNO . | U-FNO . | |
---|---|---|---|
SSIM | 0.8490 | 0.8994 | |
0.8463 | 0.9012 | ||
SNR | 23.51 | 29.67 | |
24.99 | 30.42 |
Superscript . | FNO . | U-FNO . | |
---|---|---|---|
SSIM | 0.8490 | 0.8994 | |
0.8463 | 0.9012 | ||
SNR | 23.51 | 29.67 | |
24.99 | 30.42 |
3.4. Computational performance
For a more comprehensive comparison of computational performance, we have detailed the computational costs in Table 5, with all training, testing, and prediction processes conducted on a GPU (GeForce RTX 3060). The computational performance of the traditional method is not included in Table 4 due to its implementation on the GPU using C code, rendering comparisons with CUDA-based Python implementations less meaningful without adequate context. Notably, the computational efficiency of the FNO method in generating wavefields has shown significant acceleration compared to traditional methods once the network is adeptly trained. This efficiency improvement is quantified as an enhancement of two to three orders of magnitude in computational speed. In our key comparison, we evaluated the computational performance of dataset prediction on the GPU for both FNO and U-FNO methods. We found that the computational cost for U-FNO is 0.61 times higher than that of the FNO method. This increase can be ascribed to the integration of an additional U-net layer in conjunction with the conventional Fourier layers. Nonetheless, U-FNO demonstrates a notable improvement in computational performance when contrasted with the traditional SGPS method.
Method . | Number of trainable parameters . | Time (s) . |
---|---|---|
FNO | 926 417 | 0.00 353 |
U-FNO | 1 156 337 | 0.00 571 |
Method . | Number of trainable parameters . | Time (s) . |
---|---|---|
FNO | 926 417 | 0.00 353 |
U-FNO | 1 156 337 | 0.00 571 |
Method . | Number of trainable parameters . | Time (s) . |
---|---|---|
FNO | 926 417 | 0.00 353 |
U-FNO | 1 156 337 | 0.00 571 |
Method . | Number of trainable parameters . | Time (s) . |
---|---|---|
FNO | 926 417 | 0.00 353 |
U-FNO | 1 156 337 | 0.00 571 |
4. Conclusions
Our research aims to develop a more accurate surrogate model to approximate the solution of viscoelastic wave equations while maintaining an acceptable computational cost. Compared with the traditional FNO-based method, our method introduces U-Fourier layers subsequent to the standard Fourier layers. This improvement effectively approximates the nonlinear mapping operator that governs solution of the equation through data-driven training, thereby markedly improving the accuracy solution of equations. Notably, our approach retains the benefits of the conventional FNO-based method, including obviating the need for the fractional Laplace operator and enhancing computational efficiency. Numerical examples show that our method exhibits better performance in approximating solutions to fractional viscoelastic equations compared to traditional FNO-based methods. Additionally, it can prove significant robustness and generalization ability by predicting solutions for multicomponent fractional viscoelastic equations, even beyond the confines of the dataset. The proposed method demonstrates significant advantages in approximating wave-equation solutions, showing potential for applications that require multiple iterations of simulation. In the future, by optimizing the network and integrating the velocity parameter model, this method could be further enhanced and more effectively applied to the field of parameter inversion.
Acknowledgements
Many colleagues have helped with suggestions for improving this papers.
Conflict of interest statement
The authors declared that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Funding
This work is supported by the National Natural Science Foundation of China (grant nos. 42274147 and 41874144).
Data availability
Data associated with this research are available and can be obtained by contacting the corresponding author.
Appendix A
SSIM function is a quantitative metric used to evaluate the similarity between the true snapshot (x) and the predicted snapshot (y). This evaluation is based on factors such as the local mean values (
and
Higher SSIM and SNR values indicate greater similarity between the true and predicted wavefield snapshots.