Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields

TL;DR

The paper tackles efficient high-frequency acoustic wavefield simulation by embedding explicit Gabor basis functions into a Physics-Informed Neural Network for the Helmholtz equation. It redefines the network to learn a mapping from coordinates to a Gabor coordinate system, enabling oscillatory behavior to be captured by fixed Gabor terms while keeping the trainable parameter count constant. Coupled with an integrated Perfectly Matched Layer, the approach achieves faster convergence and higher accuracy across challenging velocity models such as Marmousi and Overthrust, and demonstrates robustness to weight initialization compared to prior Gabor-PINN variants. This work advances practical seismic modeling by offering an accurate, mesh-free, and computationally efficient PINN framework for scattered wavefields, setting the stage for broader seismic applications and future hybrid optimization strategies to further mitigate residual bias.

Abstract

Physics-Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh-free formulation. However, their low-frequency bias limits their accuracy and convergence speed for high-frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor-based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN-based seismic applications.

Figures (21)

• Figure 1: Real part of a 10 Hz Gabor function displayed as a function of $(d_x, d_z)$.
• Figure 2: Schematic representation of the neural network architecture. Dashed connections have no associated weight. The network learns a mapping from the input $(x,z)$ to $(d_x,d_z)$ in the Gabor functions that produce the oscillatory behavior of the wave field $u_s$.
• Figure 3: (a) Real part of the Gabor function plotted as a function of $(d_x, d_z)$. The superimposed paths represent the trajectories produced by the network for vertical and horizontal sets of input collocation points $(x, z)$. (b) Output values of this Gabor basis function, illustrating how the learned mapping from $(x, z)$ to $(d_x, d_z)$ enables the Gabor basis function to contribute to the wavefield solution.
• Figure 4: A simple velocity model and the finite difference (FD) modeled 10Hz wave field using a fine grid spacing. We use this simulation result to calculate the error of different PINN predictions.
• Figure 5: Comparison of the convergence behavior of the simple PINN and the proposed Gabor-PINN for simulating a 10 Hz wavefield (Test 1) in the simple velocity model shown in Figure \ref{['fig:10Hz_fd_wavefoeld']}. (a) Loss evolution for the simple PINN, (b) loss evolution for the proposed Gabor-PINN, (c) evolution of prediction errors on validation points (relative to the finite difference result) for both methods. The proposed Gabor-PINN demonstrates faster convergence and higher accuracy.