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

1. Problem Statement

The paper addresses the challenge of surrogate modeling for wave propagation and scattering in heterogeneous media, specifically in high-contrast regimes (e.g., subsurface models containing salt bodies).

• Context: Accurate forward modeling is critical for seismic imaging, inversion, and applications like hydrocarbon exploration and CO2 sequestration. Traditional solvers (FDM, FEM) are computationally prohibitive for inverse problems due to the need for repeated forward solves.
• The Gap: Existing data-driven surrogates, such as Fourier Neural Operators (FNOs), excel at modeling smooth wave propagation but struggle with high-contrast media. In these regimes, sharp velocity contrasts induce strong scattering, reflections, and diffractions, creating complex, geometry-dependent spatial interactions that violate the inductive biases of standard neural operators.
• The Challenge: Learning the solution operator for high-frequency Helmholtz equations with strong contrasts requires capturing both smooth background propagation and singular, localized scattering effects simultaneously, which leads to poor generalization and high parameter requirements for single-architecture models.

2. Methodology

The authors propose a hybrid architecture that decomposes the scattering operator into two distinct physical components, assigning each to the neural architecture best suited for its mathematical character.

A. Physics-Based Decomposition

The core innovation is splitting the wavespeed $v(x)$ and the resulting pressure field $p(x, \omega)$ into smooth and residual components:

1. Wavespeed Decomposition: $v(x)$ is split into a smooth background $v_{bg}(x)$ (obtained via mollifying convolution) and a sharp residual contrast $\delta v(x) = v(x) - v_{bg}(x)$.
2. Field Decomposition: The total pressure field is split into a background field $p_{bg}$ and a scattering correction $\delta p$.
3. Operator Splitting: The forward map $F$ is decomposed into:
   • $F_{bg}$ (Smooth Background): Maps source and smooth wavespeed to the background field.
   • $F_{sc}$ (Scattering Corrector): Maps the background field and the velocity contrast to the pressure residual. This is formulated as a Lippmann–Schwinger integral equation where the contrast acts as a source term.

B. Hybrid Architecture

The model combines two distinct neural networks in a sequential pipeline:

1. Fourier Neural Operator (FNO):
   • Role: Learns the smooth background map $F_{bg}$.
   • Mechanism: Uses spectral mixing (Fourier layers) to efficiently capture globally coupled wave propagation in smooth media.
   • Output: Generates globally coupled feature tokens representing $p_{bg}$.
2. Vision Transformer (specifically scOT):
   • Role: Learns the high-contrast scattering corrector $F_{sc}$.
   • Input: Takes the background field $p_{bg}$ (from the FNO) and the velocity contrast $\delta v$.
   • Mechanism: Uses patch-based tokenization and shifted-window self-attention (Swin-style). This allows the model to capture strong, geometry-dependent spatial interactions and long-range dependencies induced by scattering obstacles without the quadratic complexity of standard attention.
   • Output: Predicts the pressure residual $\delta p$.

Final Output: The total wavefield is recovered via superposition: $p = p_{bg} + \delta p$.

3. Key Contributions

1. Scattering-Aware Operator Decomposition: The paper introduces a novel formulation that isolates localized wavespeed discontinuities into a separate operator-learning problem. This decomposition aligns the learning task with the underlying physics, creating a better-conditioned problem for neural networks.
2. Hybrid Neural Operator–Transformer Architecture: The authors propose a specific architecture where an FNO handles global smooth propagation and a Vision Transformer handles localized scattering corrections. This leverages the strengths of both: FNOs for smooth spectral coupling and Transformers for content-adaptive, geometry-dependent interactions.
3. Theoretical Justification: The work provides a theoretical framework interpreting patch-based transformers as consistent discretizations of continuum operators acting on measure-valued contexts, justifying their use for scattering problems where regularity is low.
4. Empirical Validation: Extensive experiments on high-frequency (40 Hz) Helmholtz benchmarks demonstrate that this hybrid approach outperforms standalone FNOs and Transformers in both accuracy and parameter efficiency.

4. Results

The model was evaluated on high-contrast Helmholtz problems involving salt bodies:

• Accuracy: The hybrid model achieved substantially improved phase and amplitude accuracy compared to standalone FNOs or Transformers. It significantly reduced prediction errors, particularly in regions surrounding obstacles where strong scattering and diffraction occur.
• Task Specialization:
  • On the Smooth Task ($v_{bg} \to p_{bg}$), FNOs outperformed Transformers.
  • On the Residual Task ($p_{bg}, \delta v \to \delta p$), Transformers (scOT) outperformed FNOs, confirming that attention mechanisms are superior for modeling scattering corrections.
• Parameter Efficiency: The hybrid architecture exhibited favorable accuracy–parameter scaling. It outperformed both baseline architectures trained end-to-end on the full "sharp" task, even though the hybrid components were trained separately on their respective sub-tasks.
• Generalization: The decomposition approach allowed the model to resolve complex wave phenomena (interference fringes, fine-scale features) that single architectures failed to capture accurately.

5. Significance

• Advancing Seismic Imaging: This work provides a computationally efficient surrogate for forward modeling in complex geological settings, potentially accelerating Full Waveform Inversion (FWI) and subsurface characterization workflows.
• Bridging Architectural Gaps: It demonstrates that high-contrast wave scattering constitutes a distinct regime where decomposition-based hybrid models are necessary. It challenges the notion that a single architecture can efficiently model all physical regimes, suggesting that "global spectral coupling followed by localized refinement" is a superior design principle for such problems.
• Future Applications: The framework's ability to accurately predict wavefields and their derivatives (via adjoint-state methods) opens the door for direct application in gradient-based inverse problems, offering a pathway to more robust and faster seismic inversion in high-contrast environments.