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

Mohammad Mahdi Abedi; David Pardo; Tariq Alkhalifah

physics.geo-phFeb 24, 2025

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

Authors: Mohammad Mahdi Abedi, David Pardo, Tariq Alkhalifah

Organizations: Basque Center for Applied Mathematics, Bilbao, Spain · University of the Basque Country, Department of Mathematics, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain · King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

Abstract

Physics-Informed Neural Networks (PINNs) have gained attention for solving partial differential equations, including the scattered Helmholtz equation, due to their flexibility and mesh-free formulation. However, their performance suffers from low-frequency bias, particularly in high-frequency wavefield simulations, limiting convergence speed and accuracy. To address this, we propose a novel and simplified PINN framework that incorporates explicit, trainable Gabor basis functions to efficiently capture the localized and oscillatory nature of wavefields. Unlike previous Gabor-based PINNs that rely on multiplicative filters or auxiliary networks to learn Gabor parameters, our approach redefines the network's task as learning a nonlinear mapping from input coordinates to a custom Gabor coordinate system, where a Gabor function captures the dominant oscillatory behavior of the wavefield. This formulation absorbs the effect of two Gabor parameters into the learned mapping, reducing computational complexity and eliminating the need for manual tuning of hyperparameters. We also present an efficient formulation for incorporating a Perfectly Matched Layer (PML) into the training by deriving real-valued loss components and introducing an analytical expression for the background wavefield. Numerical experiments on various velocity models show that our Gabor-PINN achieves faster convergence, higher accuracy, and greater robustness to architectural design and initialization compared to both traditional PINNs and prior Gabor-based methods. The improvement lies not in adding architectural complexity --as is common in enhanced PINNs -- but in absorbing this complexity into the learned coordinate transformation, making the method both simpler and more effective. Our implementation is publicly available to support reproducibility and future research.

Figures & tables

Figure 1: Real part of a 10 Hz Gabor function displayed as a function of $(d_{x},d_{z})$ , where $\sigma=0.1$ controls the Gaussian envelope width and $\theta=-\pi/4$ is the initial rotation angle of the wavefront.

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)

(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)

(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 in Test 1.

Figure 5: Convergence comparison between a simple PINN and the proposed Gabor-PINN for simulating a 10 Hz wavefield (Test 1) in the velocity model from Figure 4 . The only difference between the models is the proposed addition of Gabor functions. (a) Loss evolution for the simple PINN, (b) loss evolution for the Gabor-PINN, and (c) validation error for both methods. The Gabor-PINN shows faster convergence and improved accuracy.

Panel of Figure 6 — Figure 6: Real part of the predicted 10 Hz wavefield from the proposed simple PINN and the proposed Gabor-PINN at different training epochs (500, 10,000, and 100,000) for Test 1. Both methods share the same training settings, number of trainable parameters, and network architecture, except for the inclusion of Gabor functions in the Gabor-PINN. The Gabor-PINN exhibits faster convergence. The reference finite difference (FD) simulated wavefield is shown in Figure 4 .

Panel of Figure 7 — Figure 7: Comparison of a simple PINN and the proposed Gabor-PINN for simulating a 20 Hz scattered wavefield (Test 2) in the velocity model from Figure 4 . The validation error curves demonstrate significantly lower errors for the Gabor-PINN, highlighting its improved accuracy in high-frequency wavefield modeling.

Panel of Figure 8 — Figure 8: The Marmousi velocity model (Test 3) is shown in panel (a), and the reference finite difference (FD) wavefield in panel (b). Panels (c) and (d) display the Gabor-PINN wavefield predictions without and with PML, respectively. The corresponding loss and validation error curves are presented in the next figure.

Panel of Figure 9 — Figure 9: Comparison of training and validation losses for the simple PINN and proposed Gabor-PINN methods with and without PML (Test 3, related to Figure 8 ). The Gabor-PINN demonstrates better convergence and accuracy, while the simple PINN shows a significant delay in convergence. Validation errors for both methods are shown, illustrating the improved final accuracy when PML is applied.

Test	Layers	Neurons	$K$	Params (k)	MACs (k)	Collocation Points
Test 1 (10 Hz)	3	64	3	9.66	9.57 (no Gabor)	2,601
					9.76 (with Gabor)
Test 2 (20 Hz)	3	128	5	36.74	36.54 (no Gabor)	90,601
					36.93 (with Gabor)
Test 3 (Marmousi)	4	128	3	52.22	51.97 (no Gabor)	15,251 (no PML)
					52.35 (with Gabor)	30,351 (with PML)

Table 1: Summary of network hyperparameters, computational complexity represented by Multiply-Accumulate operations (MACs), positional encoding parameter

K

, and number of collocation points per epoch (full batch). Parameters and MACs are in thousands (k). Note that adding our Gabor functions does not change the number of trainable parameters.

Panel of Figure 10 — Figure 10: Test 4, (a) the Overthrust velocity model, (b) training loss and (c) validation error evolution for the simple PINN and proposed Gabor-PINN. The simple PINN exhibits unstable training, while the Gabor-PINN shows stable convergence. (d) and (e) show wavefield predictions on validation grid points for the simple PINN and Gabor-PINN, respectively. (f) and (g) present the prediction differences with the finite difference reference wavefield. The Gabor-PINN prediction is more accurate but still affected by low-frequency bias.

Panel of Figure 11 — Figure 11: Comparison of training losses, validation errors, and scattered wavefield predictions for a 4 Hz wavefield using three different initializations of weights (Test 5). Panels (a) and (b) show the loss curves for Alkhalifah and Huang (2024) and our Gabor-PINN, respectively, (c) presents the validation errors for both methods. (d)–(f) display the real part of the wavefield predictions and the finite difference (FD) reference. Our method shows higher accuracy and lower sensitivity to initialization.

Panel of Figure 12 — Figure 12: Comparative analysis of the proposed Gabor-PINN, Huang and Alkhalifah (2023), and Alkhalifah and Huang (2024) for a 16 Hz wavefield simulation (Test 6). The first row shows the loss evolution for both small and large models of each method. The second row displays the best predicted wavefields. The third row presents the corresponding prediction errors relative to the reference finite-difference solution. Quantitative results, including network architectures, computational complexity, and final prediction errors, are summarized in Table 2 .

Model	Neurons	Gabor	Params	MACs	NMSE	NMAE
	per layer	#	(k)	(k)
Our proposed Smaller model	128-128-128	64	36.22	36.42	0.0107	0.0990
Larger model	128-128-128-128	64	52.74	52.87	0.0102	0.1004
Huang and Alkhalifah (2023) Smaller model	128-128-128	512	53.89	51.84	0.0203	0.1377
Larger model	128-128-128-128	640	71.43	68.87	0.0179	0.1304
Alkhalifah and Huang (2024) Smaller model	128-128-64-64-64+Aux	64	35.82	39.74	0.0722	0.2580

Table 2: Comparison of Gabor-PINN models for a 16Hz wavefield simulation (Figure 12 ): architecture complexity and prediction accuracy after 50,000 epochs. MACs : Multiply-Accumulate operations (in thousands), representing computational complexity. Params : Number of trainable parameters, in thousands. NMSE : Normalized Mean Squared Error. NMAE : Normalized Mean Absolute Error. Gabor : Number of Gabor functions used in the network. Aux : Refers to the additional auxiliary network.

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Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

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