Existing methods for seismic wave scattering analysis in complex canyon sites face challenges in simultaneously meeting the demands for accuracy, efficiency, and broadband adaptability. Analytical solutions are usually limited to regular geometries. Domain-based methods such as the finite difference method and the finite element method need artificial absorbing boundaries. The boundary element method can reduce the spatial dimensionality. It also automatically satisfies the radiation condition at infinity. However, it requires the assembly and inversion of large dense matrices. This becomes computationally expensive when many parametric cases are considered. To address this problem, we incorporate the boundary integral equation of the indirect boundary element method （IBEM） into a neural network framework as a physics constraint. This study investigates whether this approach can serve as a fast and reliable alternative to conventional boundary element solutions. This approach also eliminates the need for artificial absorbing boundaries and avoids the explicit inversion of dense matrices. The method is evaluated by analyzing surface displacement responses and virtual source density errors under different incident angles and dimensionless frequencies. The goal is to establish a rapid parametric analysis tool for seismic response assessment of complex canyon sites.

The proposed indirect boundary element method-constrained physics-guided neural network （IBEM-PGNN） uses the boundary virtual source density as the network output. The traction-free boundary integral equation of the IBEM serves as the physical constraint. For each boundary element, the network input includes boundary coordinates, geometric parameters, free-field responses, and dimensionless frequency. The output contains the real and imaginary parts of the virtual source density in the x- and y-directions. Thus, each boundary element corresponds to four scalar outputs. The network adopts a fully connected feedforward structure with five hidden layers. The numbers of neurons are 128, 128, 256, 256, and 64. The hyperbolic tangent function is used as the activation function. Multi-scale Fourier feature mapping is introduced in the input layer. It projects the original coordinates into a high-dimensional periodic feature space. This enhances the network’s ability to represent high-frequency spatial oscillations. It also reduces the spectral bias commonly observed in standard networks. The total loss function consists of a data loss term, a physics loss term, and a regularization term. The physics loss term is based on the traction-free boundary condition. The regularization term suppresses nonphysical amplification of the virtual source density. The data loss is computed as the mean squared error between the predicted virtual source density and the IBEM reference solution. The physics loss is computed by substituting the predicted virtual source density into the boundary integral equation. Then, the residual traction on the canyon surface is evaluated. An adaptive weighting strategy based on the Sigmoid function is used. It dynamically adjusts the data loss weight and the physics loss weight according to the training progress. This gradually shifts the training from data-dominated to physics-dominated learning. An exponential moving average is used to normalize the loss scales. This reduces the influence of magnitude differences among the loss terms. The transfer learning strategy follows a two-stage approach. The first stage is low-frequency pretraining. The second stage is target-frequency fine-tuning. Low-frequency pretraining provides a physically reasonable initialization. Target-frequency fine-tuning only needs to learn additional details caused by the frequency change.

A semi-circular canyon under plane P-wave incidence is used as an example. The incident angles are α＝0°, 30°, 45°, and 60°. The dimensionless frequency for low-frequency pretraining is η＝0.5. The target frequencies are η＝1.0, 1.5, and 2.0. After approximately 500−1000 training iterations in the low-frequency pretraining stage, the relative error of the virtual source density is reduced to below 2%. With transfer learning, the relative errors for all target-frequency cases are below 0.3%. Without transfer learning, the relative errors for medium-to-high frequency and large incident angle cases can reach 20%−40%. This comparison shows that transfer learning substantially improves prediction accuracy. The surface displacement amplitudes predicted by IBEM-PGNN agree well with the IBEM reference solutions. The predicted responses capture the main features of the surface motion. These features include local amplification near the canyon edges and spatial variation caused by wave interference. The training process converges stably in all tested cases. However, the number of iterations required for stable convergence increases with the dimensionless frequency. This is because the wave field exhibits stronger spatial oscillations at higher frequencies. Variations in incident angle also affect convergence. They change the asymmetry of the wave field and the distribution of local peaks and troughs.

The results show that the proposed IBEM-PGNN can effectively simulate P-wave scattering by canyon sites. It does not require artificial absorbing boundaries. It also avoids the explicit assembly and inversion of large dense matrices. The boundary integral equation of the IBEM reduces the spatial dimensionality of the problem. It automatically satisfies the radiation condition at infinity. These properties are directly incorporated into the neural network training process. The combined use of multi-scale Fourier features, adaptive loss weights, and transfer learning enables stable convergence and improved accuracy in broadband parametric analysis. The core conclusion is that a neural network constrained by boundary integral equations can provide a fast and reliable alternative to the IBEM for canyon site seismic response. This framework can be further extended to other wave types and more complex site conditions. Therefore, the proposed method offers a reliable basis for rapid seismic response analysis and seismic damage assessment of irregular topographic sites.