Wave equations used in seismic modeling are typically classified into acoustic and elastic formulations. As a representative example, we consider the two-dimensional frequency-domain acoustic wave equation in a homogeneous medium.

where ω = 2 π f denotes the angular frequency with f being the frequency, c = c ( x , z ) is the wave speed, u ( x , z ) represents the wavefield, Δ denotes the Laplacian operator, s is the source term, and D denotes the computational domain.

The discretization of equation (1) is carried out with NAD scheme, where the partial derivatives along the x and z directions are computing respectively.

Due to computational limitations, the domain D must be truncated with artificial boundary. Such boundary, however, generate spurious reflections that compromise the accuracy of both forward and inverse computation. To mitigate these reflection, we employ the widely adopted Perfectly Matched Layer (PML) absorbing boundary condition [5], resulting in a modified formulation of the wave equation incorporating PML [22]:

Equation (3) is discretized using a fourth-order NAD finite-difference scheme, with detailed derivations provided in AppendixA1. Subsequently, the computational nodes are ordered sequentially in a row-wise manner, from left to right within each row, and the wavefield values together with their corresponding gradient terms at each node are stored contiguously. Under this ordering strategy, the discretized system can be written in the following linear system:

where C denotes the impedance matrix, which is complex-valued owing to the incorporation of PML, while u and s represent the discrete wavefield vector and the discrete source vector, respectively. The sparse structure of C is [22].

B i , j ( i = 1 , ⋯ , 9 ; j = 1 , ⋯ , N ) represents 3 × 3 nonzero sub-block matrix, N = ( n x − 2 ) ⋅ ( n z − 2 ) denotes the total number of grid nodes (i.e. problem scale) and n x , n z is the number grid nodes along horizontal and vertical direction respectively. The blank positions are all zero elements, and ‘ ⋯ ︷ n x − 3 ’ denotes there are n x − 3 zero sub-block matrices. It is obvious that the impedance matrix C has nine nonzero diagonals and the number of its non-zero elements is 47 ( n x − 4 ) ( n z − 4 ) + 62 ( n x + n z − 8 ) + 8 = O ( 47 N ) . For the specific expressions of the elements in B i , j , see AppendixA2.

To assess the numerical difficulty of the resulting linear system, we investigate the spectral properties of the impedance matrix under a representative discretization with n x = n z = 66 . As shown in Figure 1, the eigenvalues are clustered near the origin along the real axis and exhibit both positive and negative values, indicating that the impedance matrix is indefinite and nearly singular. Moreover, the introduction of the perfectly matched layer (PML) absorbing boundary condition renders the matrix asymmetric [22].

Figure 1. Typical eigenvalue distribution of the impedance matrix.

The coexistence of asymmetry, indefiniteness, and near-singularity poses substantial challenges for numerical solution. In particular, classical iterative methods such as the conjugate gradient (CG) algorithm require the coefficient matrix to be symmetric and positive definite [12][23], and their direct application may result in poor convergence or even breakdown. Among Krylov subspace methods, BICGSTAB retains the low memory requirements characteristic of CG while enhancing robustness through biconjugate search directions and a local stabilization mechanism [13]. As a result, it is well suited for nonsymmetric and indefinite linear systems. Nevertheless, when the system is severely ill-conditioned, the convergence of standard BICGSTAB iterations can still be slow or numerically unstable [16].

To further enhance the convergence behavior and numerical stability of the BICGSTAB algorithm, the introduction of preconditioning strategy is essential [12]. An appropriately designed preconditioner can alleviate the ill-conditioning of linear system while preserving solution accuracy, thereby reducing iterative numbers and improving overall efficiency. In particular, the original linear system (4) can be equivalently transformed into the following preconditioned form:

where P denotes preconditioner [11], equation (5) has the same solutions as the original equation (4). Algorithm 1 presents the preconditioned BICGSTAB implementation.

Algorithm 1. Preconditioned BICGSTAB method.

When P − 1 provides a good approximation to C − 1 , such that CP − 1 is close to the identity matrix, the condition number of the preconditioned system is significantly reduced, enabling convergence within a small number of iterations. Consequently, an effective preconditioner should satisfy P − 1 ≈ C − 1 . At the same time, the construction of P must account for computational efficiency: the cost of solving the auxiliary system P u = v should be kept minimal. These two requirements (accurate approximation of C − 1 and efficient application of P − 1 ) are inherently competing. In practical implementations, an appropriate balance between approximation quality and computational cost is therefore required to achieve overall efficiency. A variety of classical preconditioning techniques, such as ILU factorization and the complex-shift M₀ preconditioner, have been proposed and successfully applied in many settings. However, for the impedance matrix characterized by strong indefiniteness and near-singularity, these traditional preconditioners often exhibit slow convergence, motivating the development of more effective and adaptive preconditioning strategy.

To meet the requirements of approximating the inverse impedance operator, this study adopts a modified U-Net as the core network architecture [24]. The encoder-decoder structure not only ensures matching input and output dimensions but also provides a natural multi-resolution representation, which is particularly suitable for capturing both local high-frequency interactions and global low-frequency components inherent in the inverse operator [25]. Skip connections preserve fine-scale information across layers, playing a role analogous to residual correction mechanisms commonly employed in multigrid solvers [26]. The depth and scalability of this architecture further enhance its approximation capability, enabling it to accommodate problems with varying spatial scales and structural complexity. The specific modified U-Net used in this work (Figure 2) comprises approximately 28 layers, including 4 downsampling and 4 upsampling blocks.

Figure 2.Network architecture diagram used for numerical experiments.

In the encoder, the feature extractor consists of repeated modules composed of convolutional layers with 5 × 5 kernel and Tanh activation function. The use of larger convolutional kernel effectively expands the receptive field, enabling the capture of broader contextual information [27]. The Tanh activation function introduces nonlinearity while allowing negative values to be activated, thereby enhancing the ability of networks to learn complicated features [28]. The ordered repetition of this module allows the feature extractor to progressively learn higher-level features. Additionally, the dimensionality adjustment module achieves feature compression by repeated convolutional operations, halving the spatial size and doubling the number of channels to enrich feature representation after each downsampling step [29].

In the decoder, a combination of nearest-neighbor interpolation and convolutional operation is employed to progressively reconstruct fine-grained spatial feature [30]. Nearest-neighbor interpolation is used to expand the spatial dimensions of the feature maps, while subsequent convolutional layers refine channel dimensions and extract features. To compensate for information loss during downsampling, skip connections are introduced after each upsampling stage to concatenate the corresponding encoder feature maps. This mechanism enhances the recovery of spatial details and ensures that networks can reconstruct complete and fine spatial feature distributions [31].

The NAD discretization expresses higher-order derivatives as linear combinations of the wavefield and its gradients. Accordingly, in equation (4), the solution vector consists of the wavefield and gradient components, which are defined independently and stored adjacently. To embed physical structure into the learning process, we adopt a multi-channel feature separation strategy. Specifically, the wavefield and its gradient components are assigned to separate channels, and the real and imaginary parts of each component are further processed independently to accommodate real-valued neural networks. As a result, the network input is represented as a six-channel real-valued tensor, which preserves physical information while remaining compatible with real-valued network operations.

The overall dataset construction procedure is illustrated in Figure 3. To capture the dynamic behavior of the iterative solver, the residual vectors p k is selected as the primary feature, as it reflect the evolving error structure during the iteration. To enhance sample diversity and mitigate overfitting to specific residual trajectories, additive noise vectors ξ k are incorporated as a form of regularization. Each noise vector is generated with entries independently sampled from a uniform distribution, ξ k ~ U ( − δ , δ ) , where δ is a predefined noise level parameter. The signal-to-noise ratio (SNR) is uniformly sampled within the range of 10 dB to 20 dB and defined as 20 log 10 ( max ( | p k | ) | ξ k | ) . The noise magnitude is scaled accordingly to satisfy the sampled SNR value, and the enhanced vector x k is constructed via element-wise addition: x k = p k + ξ k . Subsequently, the matrix–vector product y k = C x k is computed to obtain the corresponding output vector. During training, y k is used as the input sample, while x k serves as the reference target sample. This supervised pairing encourages the network to learn an implicit approximation of the inverse impedance operator, i.e., to approximate the mapping C − 1 .

Figure 3. The construction process of training data (Re(.) and Im(.) represent the real part and imaginary part, respectively).

During preprocessing, both X and Y undergo structured transformations, including stride-3 sampling, separation of real and imaginary parts, and channel reorganization based on row-major ordering. The stride-3 operation is consistent with the underlying interleaved storage layout ( u , ∂ u / ∂ x , ∂ u / ∂ z ) , where the three quantities are stored sequentially at each spatial node. Applying stride-3 along the storage dimension simply separates these physically coupled variables into distinct channels without discarding any information. The subsequent channel reorganization arranges them into contiguous feature maps suitable for convolutional processing.

At the beginning of networks training, the layer parameters are initialized following the default Kaiming scheme [32]. Then, the parameter is gradually optimized by the backpropagation algorithm to construct a nonlinear mapping function X = N e t ( Y , θ ) from the input data Y to the label data X . θ = { K w , b w } is the network parameter (including convolution weight coefficient k w and bias variable b w ). Given that the gradient of the L1 loss function is not excessively amplified by large errors from outliers, this property effectively prevents the model parameters from being dominated by a few extreme samples, thereby enhancing the robustness of networks [33]. To be specific, the objective function is optimized by

where m represents batch size, and the variables ( x i , y i ) can be selected from the training set. For the optimization of equation (7), the Adam optimizer (combining both momentum mechanisms and adaptive learning rate adjustment) is typically employed for parameter update. By integrating the first-order moment estimate of the gradients with a second-order moment correction term, its parameter update rule can be formulated [34]

with

where α denotes the line search step length, fuzz factor ε = 10 − 8 , β 1 = 0.9 , and β 2 = 0.999 . Although the Adam optimizer adapts the learning rate, applying an appropriate learning rate decay strategy can further enhance the performance of optimizer [35]. Thus, a scheduler (StepLR) is employed to decay the learning rate. During the training process, the networks continuously optimize its parameters by minimizing the L1 loss function, thereby progressively enhancing its representational capability.

After training, the neural network with optimized parameters θ ^ defines a prediction operator N e t ( ⋅ , θ ^ ) , which serves as a data-driven approximation to the inverse impedance operator. This operator is embedded into the BICGSTAB framework by replacing the conventional preconditioning steps (Steps 8 and 11 in Algorithm 1) with network-based predictions, resulting in the proposed DNNs-preconditioned BICGSTAB method. During each iteration, the network directly maps the residual-related vectors to approximate solutions of the corresponding preconditioned systems, thereby avoiding the explicit solution of auxiliary linear systems. The complete iterative procedure of the proposed solver is summarized in Algorithm 2.

Algorithm 2. DNNs-preconditioned BICGSTAB method.

It is worth noting that directly approximating the exact inverse of the impedance matrix using a deep neural network (i.e., enforcing N e t ( ⋅ , θ ^ ) ≈ C − 1 ) would, in principle, lead to near one-step convergence of the iterative solver. However, in high-dimensional settings, achieving such an approximation requires an exponentially growing number of training samples and optimization iterations, resulting in prohibitive computational cost. To balance approximation accuracy and efficiency, we adopt an optimized training strategy that enables the network to capture the dominant inverse characteristics of the impedance matrix without explicitly learning its full inverse. This design yields an effective data-driven preconditioner that significantly improves convergence while maintaining practical computational complexity.

We firstly perform a numerical simulation using a homogeneous medium defined over a 5.2 km × 5.2 km square domain, with a constant wave velocity of 2 km/s. A source with the frequency of 20 Hz is placed at coordinates (2.6 km, 0.45 km). The comparison results are presented in Figure 4(a) and Table 2, where Figure 4(a) shows the convergence curves of three solvers and the column 2 of Table 2 lists the number of iteration and computing time. The results indicate that both ILU-BICGSTAB, M₀-BICGSTAB, and DNNs-BICGSTAB accelerate convergence to some extent compared to the baseline None-BICGSTAB. It is worth noting that DNNs-BICGSTAB shows better convergence performance than other preconditioners, demonstrating its superior numerical efficiency. These findings suggest that deep learning techniques great potential for enhancing preconditioning strategies in numerical simulations. In addition, wavefield snapshot obtained using DNNs-BICGSTAB is presented (Figure 5(a)) to validate the accuracy and reliability of the proposed method. Since the same stopping criteria are used in the iterative solver, the use of a preconditioner does not affect the numerical consistency of the solution.

To further investigate the applicability and robustness of the proposed method under discretization schemes of different orders, equation (1) is additionally solved using a sixth-order NAD formulation. For consistency and fairness, all numerical parameters are kept identical to those adopted in the previously discussed fourth-order NAD scheme. After assembling the resulting linear system, the four solvers are applied, and the corresponding BICGSTAB convergence histories are presented in Figure 4(d). As illustrated in the figure, the proposed DNNs-BICGSTAB method continues to demonstrate clear performance advantages even when a higher-order discretization is employed. In particular, it converges significantly faster than ILU-BICGSTAB, M₀-BICGSTAB, and the unpreconditioned BICGSTAB. These results indicate that the effectiveness of the proposed approach is not restricted to a specific discretization order, but remains robust under higher-order NAD schemes, underscoring its suitability for more demanding and complex numerical simulations.

To evaluate how the training overhead influences the overall computational efficiency of the proposed approach in multi-source scenarios, a comparative study is performed. To be specific, simulation is conducted at a fixed frequency of 20 Hz, where seismic sources are uniformly distributed along the horizontal direction. The number of sources is gradually increased from 50 to 500, corresponding to source spacings ranging from 104 m to 10.4 m, thereby forming a sequence of test cases with increasing computational complexity. Figure 6 summarizes the total runtime of different solvers as a function of the number of sources. Several important observations can be drawn from the results. Although the DNN-based preconditioning solver introduces additional training costs for each individual linear system, its overall computational efficiency improves progressively as the number of sources increases. In particular, when the number of sources exceeds 150, the total runtime of DNNs-BICGSTAB becomes lower than that of the unpreconditioned BICGSTAB (None-BICGSTAB). As the source count further increases to 300, it also outperforms ILU-BICGSTAB, and at approximately 330 sources, it surpasses the M₀-BICGSTAB method. More importantly, among all compared solvers, DNNs-BICGSTAB exhibits the slowest growth rate in computational time, clearly demonstrating its superior scalability and efficiency in large-scale multi-source simulations.

Table 2. The number of iterative steps (Iter) and computing time (Time: second) with respect to the listed preconditioned BICGSTAB methods for various media at a single source and f = 20   Hz .

(a) (b)

(c) (d)

Figure 4. Convergence curves of different preconditioned BICGSTAB methods applied to various velocity media: (a) Homogeneous medium, (b) Two-layer medium, (c) Marmousi medium, and (d) Homogeneous medium discretized using the sixth-order NAD scheme.

(a) (b)

(c)

Figure 5. Mono-frequency wavefield snapshot at frequency f = 20   Hz in various media: (a) Homogeneous medium, (b) Two-layer medium, (c) Marmousi medium.

Next, for evaluating the performance of DNNs-BICGSTAB under varying medium configurations, a square two-layer velocity model of 5.2 km per side is constructed for numerical testing. The velocity in the upper layer is set to 2.0 km/s, and that in the lower layer to 2.5 km/s, simulating a typical geological interface with a clear velocity difference. The source is placed at the outer layer of the PML absorbing boundary with a frequency of 20 Hz. Figure 4(b) presents the residual convergence curves of four solvers, and the corresponding iterative numbers and computational times are listed in the third column of Table 2. These results show that None-BICGSTAB exhibits very low convergence efficiency, highlighting the necessity of introducing preconditioning techniques. While ILU and M₀ preconditioners improve convergence to some extent, the reduction in

Figure 6.Computing time (involving training time) versus the number of sources for various preconditioned methods applied to the wavefield simulation of Homogeneous medium.

the number of iterations remains limited. In contrast, DNNs-BICGSTAB method demonstrates stable and efficient convergence throughout the entire iterative process, outperforming the other three solvers. These findings confirm that DNNs-based preconditioning strategy maintains robust numerical performance even in the presence of strong velocity contrasts at media interfaces. Figure 5(b) displays the wavefield snapshot under this medium, further validating the accuracy and reliability of the proposed method in two-lay settings.

To further evaluate the general applicability of the proposed method over a range of frequencies, several representative frequency values commonly adopted in seismic exploration are selected for comparative analysis. For each frequency, the network is retrained independently to capture the corresponding characteristics of the impedance matrix. Table 3 summarizes the iterative counts and computational costs of different solvers at these frequencies. The results indicate that DNNs-BICGSTAB method consistently achieves robust performance at all tested frequencies, reducing the number of iterations compared with ILU-BICGSTAB, M₀-BICGSTAB, and the unpreconditioned BICGSTAB. These observations demonstrate that the proposed neural network-based preconditioning approach is effective and stable when applied across different frequency conditions.

Table 3. The iterative number (Iter) and running time (Time: second) of preconditioned BICGSTAB methods for solving single-source problem at various frequencies in Two-layer medium.

Finally, to assess the applicability of the proposed approach in geologically complex environments, the classical Marmousi velocity model (shown in Figure 7) is adopted as a challenging test case. The seismic source is positioned at (3.84 km, 0.36 km) and operates at a frequency of 20 Hz. The convergence histories of the different solvers are illustrated in Figure 4(c), while the corresponding iteration counts and computational times are reported in the fourth column of Table 2. Representative wavefield snapshots for this model are further presented in Figure 5(c). The results indicate that, even in the presence of strong lateral velocity variations and complex geological structures, the DNN-based preconditioning strategy maintains stable and efficient convergence, clearly demonstrating its robustness and effectiveness for large-scale seismic wavefield simulations in realistic geological settings.

Figure 7. Marmousi velocity structure of experimental medium.

In this paper, the spatial step sizes in the horizontal direction x and the vertical direction z are set to be equal and uniformly denoted as h . The classical fourth-order NAD finite difference scheme is employed, and the difference template for the high-order partial derivatives of equation (3) in x is

to derive the coefficients of the above equation (11), it is necessary to use the Taylor formula to expand, and the specific process is

solving the linear equations of (12) and (13) yields the coefficients of NAD method:

According to Lang and Yang [22] and Liu et al. [8], the discrete formulas for other higher-order partial differential terms in equation (3) are as follows:

when the above NAD grid difference template is used to discretize the frequency domain acoustic wave equation, the equation (3) can be discretized. The grid nodes are sorted row by row, and the corresponding node numbers correspond to: k = n x ⋅ ( j − 1 ) + i . Then, the wavefield and its gradient term at the same node are placed as ( ⋯ , u k , ∂ u k ∂ x , ∂ u k ∂ z , ⋯ ) T , and the sparse block structure of the impedance matrix can be formed [7][22][36].

The specific expression of the non-zero sub-block matrix in equation (4) is