Hybrid operator learning of wave scattering maps in high-contrast media

TL;DR

The paper tackles forward modeling of high-frequency Helmholtz wave propagation in strongly scattering, high-contrast media. It introduces a hybrid operator framework that splits the forward map into a smooth background component learned by a Fourier Neural Operator and a high-contrast scattering corrector learned by a vision transformer, yielding $p = p_{bg} + \delta p$. Empirical results on challenging salt-contrast benchmarks show the Hybrid model achieves substantially better phase and amplitude accuracy than either baseline, with favorable scalability in parameter count. This approach offers a more efficient surrogate for seismic forward modeling and holds promise for improved inversion via adjoint methods.

Abstract

Surrogate modeling of wave propagation and scattering (i.e. the wave speed and source to wave field map) in heterogeneous media has significant potential in applications such as seismic imaging and inversion. High-contrast settings, such as subsurface models with salt bodies, exhibit strong scattering and phase sensitivity that challenge existing neural operators. We propose a hybrid architecture that decomposes the scattering operator into two separate contributions: a smooth background propagation and a high-contrast scattering correction. The smooth component is learned with a Fourier Neural Operator (FNO), which produces globally coupled feature tokens encoding background wave propagation; these tokens are then passed to a vision transformer, where attention is used to model the high-contrast scattering correction dominated by strong, spatial interactions. Evaluated on high-frequency Helmholtz problems with strong contrasts, the hybrid model achieves substantially improved phase and amplitude accuracy compared to standalone FNOs or transformers, with favorable accuracy-parameter scaling.

Figures (15)

• Figure 1: Schematic of a heterogeneous wave domain containing a high-contrast obstacle (i.e. salt body). On $\Gamma_{\mathrm{ABC}}$ we impose an absorbing (Sommerfeld) boundary condition so outgoing waves exit without spurious reflections. $\Gamma_{\mathrm{Free}}$ is a free-surface boundary, with $p = 0$.
• Figure 2: Decomposition of the Helmholtz forward map into a smooth background propagation $\mathcal{F}_{\mathrm{bg}}$ and a scattering correction $\mathcal{F}_{\mathrm{sc}}$. Our proposed hybrid model uses an FNO to learn the $F_{\mathrm{bg}}$ map from source $s$ and smoothed wavespeed $v_\text{bg}$ to the background pressure field $p_\text{bg}$, and a Vision Transformer to learn the $\mathcal{F}_{\mathrm{sc}}$ map from $p_\text{bg}$ and wavespeed residual $\delta v$ to the pressure residual $\delta p$. The final output is $p_\text{bg} + \delta p$. For the experiments in this work, we use a constant point source near the center of the free boundary; surrogate FNOs with varied sources for smooth wavespeed were investigated in lara_benitez2024ood.
• Figure 3: Helmholtz wavefield predictions using different architectures. (Top row) Real part of the predicted pressure field $p(\mathbf{x}, \omega)$ for the expected (numerical) solution and three architectures: FNO, scOT, and (ours) Hybrid. (Bottom row) Absolute prediction errors relative to the expected solution. The velocity model (bottom left) contains two obstacles (salt bodies) in a homogeneous background. The Hybrid architecture demonstrates superior accuracy with significantly reduced error compared to FNO and scOT approaches, particularly in capturing fine-scale wavefield features around the obstacles. Additional result visualizations are presented in appendix \ref{['appendix:results']}.
• Figure 4: Performance of FNO, scOT, and Hybrid architectures with different parameter sizes across three different learning tasks: (L to R) Smooth ($v_{\mathrm{bg}} \mapsto p_{\mathrm{bg}}$), Residual data ($p_{\mathrm{bg}}(\cdot, \omega), \delta v \mapsto \delta p$), and full end-to-end Helmholtz solution on sharp wavespeeds ($v \mapsto p$). The Hybrid architecture (this work) enables end-to-end learning of the forward operator in the high-contrast scattering case.
• Figure 5: Input wavespeeds sampled randomly from the training data. The red obstacle regions (e.g. salt) have a wavespeed of 4.5 km/s, and the blue background regions have a wavespeed of 1.5 km/s.