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Multiscale Fourier Neural Operator for Inverse Wave Scattering in Highly Oscillatory Media

This paper proposes a multiscale Fourier neural operator (MscaleFNO) combined with a plug-and-play elucidated diffusion model to effectively solve inverse Helmholtz problems, enabling accurate reconstruction of highly oscillatory medium profiles from scattered wavefields in partial aperture scenarios.

Original authors: Zilin You, Zhenli Xu, Wei Cai

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zilin You, Zhenli Xu, Wei Cai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out what a hidden object looks like by throwing a bunch of pebbles at it and listening to the sound of the splash. This is essentially what scientists do in Inverse Wave Scattering. They send waves (like sound or radar) through a material, measure how the waves bounce back, and try to reconstruct the shape and texture of the hidden object inside.

The problem is, when the waves are very fast and wiggly (high frequency), the math becomes incredibly difficult. It's like trying to solve a puzzle where the pieces are moving too fast to see, and the picture is blurry.

Here is a simple breakdown of what this paper proposes to fix that problem:

1. The Old Way: The "Blurry Lens"

Traditionally, scientists use a method called Full Waveform Inversion (FWI). Think of this as a very strict, slow detective who checks every single clue by running a massive simulation on a supercomputer.

The Problem: When the waves wiggle very fast (high frequency), the computer has to do the math trillions of times. It's like trying to count every grain of sand on a beach while running a marathon. It takes too long and often gets stuck in a "local minimum"—a wrong answer that looks good enough to stop the detective, but isn't the real picture.
The AI Attempt: Recently, people tried using AI (Neural Networks) to speed this up. They trained a "surrogate" model to guess the answer instantly. However, standard AI models have a bad habit called "Spectral Bias."
- Analogy: Imagine a painter who is great at painting smooth, rolling hills but terrible at painting jagged, jagged lightning bolts. Standard AI models are like that painter; they smooth out the fast, wiggly details because they are "lazy" with high-frequency data. They miss the fine cracks and textures of the hidden object.

2. The New Solution: The "Multi-Scale Painter" (MscaleFNO)

The authors introduce a new AI model called MscaleFNO (Multiscale Fourier Neural Operator).

How it works: Instead of using one painter who tries to do everything at once, they use a team of painters, each specialized in a different "zoom level."
- One painter handles the big, smooth shapes (low frequency).
- Another handles the medium wiggles.
- A third handles the tiny, super-fast lightning bolts (high frequency).
The Result: By combining these specialized views, the model can finally see and reproduce those fast, wiggly details that the old AI models missed. It acts as a fast, accurate "surrogate" that predicts how waves will behave without needing the slow, heavy computer simulations every time.

3. The "Smart Editor" (Diffusion Prior)

Even with a great painter, if the clues (the wave measurements) are noisy or incomplete (like only hearing the splash from one side of the beach), the picture might still look weird or distorted.

The Fix: The authors add a "Smart Editor" based on a Diffusion Model.
Analogy: Imagine you are sketching a face, but your hand is shaking, and you only have a few blurry photos to guide you. You might draw a nose that looks like a potato. The "Smart Editor" is like a seasoned art teacher who looks at your sketch every few minutes and says, "Hey, noses usually look like this, not a potato. Let's nudge your drawing back toward a realistic shape."
This editor doesn't just smooth things out; it injects "common sense" about what the hidden material should look like, based on patterns it learned from thousands of previous examples. It fixes the weird artifacts without blurring the fine details.

4. What They Found

The paper tested this new system on 2D simulations (like a flat slice of the earth or a material).

Speed and Accuracy: The new "Multi-Scale Painter" was much better at predicting the fast, wiggly waves than the old AI models, especially when the waves were very fast.
Reconstruction: When trying to rebuild the hidden object, the new method produced much sharper, more accurate images, even when the data was noisy or the sensors were only looking at the object from a limited angle (like only looking at the object from the front, not the sides).
Comparison: They compared their method to standard smoothing techniques (Tikhonov regularization). The standard method made the image too smooth (blurring out the details), while their "Smart Editor" kept the sharp edges and textures intact while removing the noise.

Summary

In short, this paper presents a new way to "see" inside hidden materials using waves. They replaced the slow, old computer math with a fast, specialized AI team that can handle fast-wiggling waves, and they added a "Smart Editor" to clean up the picture when the data is messy. This allows for faster and clearer reconstructions of complex, high-frequency structures.

Technical Summary: Multiscale Fourier Neural Operator for Inverse Wave Scattering in Highly Oscillatory Media

Problem Statement
The paper addresses the challenge of solving inverse medium problems for the Helmholtz equation, specifically Full Waveform Inversion (FWI), in highly oscillatory (high-frequency) regimes. Traditional FWI relies on gradient-based optimization requiring repeated solutions of the forward wave equation. In high-frequency regimes, this becomes computationally prohibitive due to the need for fine mesh discretizations to resolve short wavelengths. Furthermore, the inverse problem is ill-posed, and standard numerical solvers suffer from the "pollution effect."

While deep learning offers data-driven alternatives, existing operator learning methods like the standard Fourier Neural Operator (FNO) suffer from "spectral bias," meaning they struggle to represent high-frequency oscillatory wavefields accurately. Conversely, methods that rely on repeated PDE solves (even with learned priors) remain computationally expensive. The core challenge is to develop a framework that is simultaneously computationally efficient, physically consistent, and capable of resolving fine-scale, high-frequency structures without the prohibitive cost of traditional solvers.

Methodology
The authors propose a novel inversion framework that integrates a Multiscale Fourier Neural Operator (MscaleFNO) as a differentiable surrogate for the forward wave propagation operator, coupled with a Plug-and-Play (PnP) Elucidated Diffusion Model (EDM) for regularization.

MscaleFNO as a Surrogate Forward Model:
- Instead of solving the Helmholtz equation numerically during inversion, the authors employ a pre-trained MscaleFNO to map medium perturbations ( $\alpha$ ) and source terms ( $f$ ) to wavefields ( $u$ ).
- Unlike standard FNOs, which truncate high-frequency modes and exhibit spectral bias, MscaleFNO utilizes multiple Fourier operator branches at different scales. Each branch $FNO_i$ operates with a specific scaling factor $c_i$ , allowing the network to learn representations across a spectrum of frequencies simultaneously.
- This architecture enables accurate prediction of highly oscillatory wavefields in high-wavenumber regimes ( $k \in [10, 60]$ ), serving as an efficient, differentiable surrogate that eliminates the need for repeated PDE solves.
Plug-and-Play Diffusion Prior:
- To address the ill-posedness of the inverse problem, the framework incorporates a pre-trained EDM as a learned prior.
- The inversion process alternates between data-consistency steps (updating the medium estimate $\alpha$ using Adam optimization based on the gradient of the data misfit computed via the MscaleFNO surrogate) and periodic prior-correction steps.
- Every $T$ iterations, the current estimate is refined by the EDM sampler, which denoises the reconstruction toward the learned manifold of physically plausible media. This "plug-and-play" approach injects structural priors without requiring explicit regularization functionals or additional forward PDE solves.
Optimization Framework:
- The inversion is formulated as minimizing a data fidelity term (mismatch between predicted and observed wavefields) guided by the diffusion prior.
- The process is initialized with a Gaussian random field and iteratively updated. The noise schedule in the EDM sampler is dynamically adjusted during inversion, shifting from strong prior correction in early iterations to fine-scale refinement in later stages.

Key Contributions

Superior Surrogate for High-Wavenumber Problems: The paper demonstrates that MscaleFNO overcomes the spectral bias of standard FNOs. It achieves faster inference, improved numerical stability, and higher solution accuracy for high-wavenumber Helmholtz equations by leveraging scale-adaptive spectral representations.
Efficient Differentiable Inversion: By embedding MscaleFNO as a differentiable forward model, the framework replaces expensive repeated PDE solves with efficient neural network evaluations, enabling gradient-based optimization for high-frequency FWI.
Robust Regularization via Diffusion: The integration of a PnP EDM prior significantly improves reconstruction accuracy and robustness to noise and limited-aperture data. It effectively suppresses artifacts while preserving fine-scale oscillatory details that traditional smoothness-based regularizers (like Tikhonov) tend to oversmooth.

Numerical Results

Forward Accuracy: In forward operator learning tasks, MscaleFNO maintained stable convergence and low relative $L_2$ errors across wavenumbers $k=10$ to $k=60$ . In contrast, standard FNO performance degraded significantly as $k$ increased, failing to capture high-frequency components.
Inverse Reconstruction: Under 10% noise, MscaleFNO-based inversion achieved substantially lower reconstruction errors than FNO-based inversion across all tested frequencies. The error decreased from 0.956 at $k=10$ to 0.129 at $k=60$ , confirming the benefit of high-frequency measurements when paired with the correct surrogate.
Limited Aperture and Noise: The method demonstrated robustness under varying noise levels (up to 20%) and limited-aperture configurations (source/receiver angles from $45^\circ$ to $360^\circ$ ). Results showed that while limited apertures increase error, the method successfully recovers main oscillatory structures, with performance improving as angular coverage increases.
Ablation Studies:
- Spectral Analysis: Fourier spectra of MscaleFNO branches revealed distinct frequency patterns, with larger scales capturing richer, structured high-frequency modes, validating the multiscale design.
- FNO Limitations: Increasing the number of Fourier modes and network width in standard FNO did not resolve its failure at high frequencies, suggesting the limitation is intrinsic to its single-scale representation and optimization instability, not just model capacity.
- Regularization Comparison: Compared to unregularized inversion (which produced spurious oscillations) and Tikhonov regularization (which oversmoothed features), the PnP-EDM approach preserved high-frequency details while removing artifacts.

Significance and Claims
The paper claims to provide a solution to the computational bottleneck of high-frequency FWI by replacing traditional solvers with a multiscale neural operator surrogate. It asserts that this approach enables accurate, efficient, and well-regularized reconstruction of highly oscillatory media properties, a task where standard neural operators fail due to spectral bias and traditional methods fail due to computational cost. The integration of diffusion models offers a statistically principled way to handle the ill-posed nature of the inverse problem without sacrificing the resolution of fine-scale features. The work positions MscaleFNO as a critical enabler for high-resolution wavefield imaging in complex, oscillatory environments.