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

arXiv: https://arxiv.org/abs/2602.11197 · HTML: https://mlsanity.com/paper/2602.11197/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 62 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 19 of 19)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Swin Transformer V2: Scaling Up Capacity and Resolution](https://mlsanity.com/paper/2111.09883/) (2021)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)

## Built on by (most cited first, 1 of 1)

- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)

## Abstract (arXiv)

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.