Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media

arXiv:2610.13258 · cs.LG · Submitted 2026-10-09 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media".

Jane: The gist The proposed LU-NO method, which integrates Fourier Neural Operators and local-integral resampling into a U-Net architecture,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we're talking about how they improved this LU-NO model by focusing on what makes it better than just a standard U-Net or a pure FNO.

Jane: What exactly are they trying to fix with these improvements? What’s the next step beyond just getting a lower error number?

Tom: They’re tackling the issue of spatial generalization across different resolutions. That means making sure the model doesn't just work well at one specific grid size but can handle any size you throw at it.

Lu: The paper says they achieved this by using a local-integral resampling mechanism that allows for correct spatial rescaling, which is something previous models couldn't do reliably (<ref:2610.13258#pg2>).

Jane: So, if we think about the practical implication, what does being able to change the resolution mean for someone who only listens to the show?

Tom: It means that when you test this on unseen structures or at a higher resolution than you trained it on, its predictions still make sense spatially. It learns how to model the continuous system correctly as an operator.

Meng: From my point of view, if it can generalize across resolutions, that’s huge for deployment because we don't have to retrain the whole thing every time we want a different detail level.

Jane: And what about noise? The paper also highlights that this method is more robust to input noise than previous models using global Fourier layers instead of just local convolutions.

Tom: That robustness is key, because in real ultrasonic sensing, you have noise all the time, and FNOs often become unstable when that happens. LU-NO handles it better by aggregating information across the entire domain instead of relying only on local neighborhoods (<ref:2610.13258#pg2>).

Lu: I think the authors are really showing that this combination of Fourier layers and resampling is what creates a structurally coherent scale-invariant operator for wave propagation (<ref:2610.13258#pg2>).

Jane: So, to wrap up on these improvements, we have a model that handles heterogeneity well, is robust to noise, and can adapt its output when the resolution changes. What’s the next thing they need to do?

The paper's summary: Tom: So we're wrapping up this look at "Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media" by summarizing what they actually proved about using AI for wave simulation.

Jane: It’s clear that the main takeaway is that LU-NO successfully combines Fourier Neural Operators with local-integral resampling inside a U-Net structure to model sound waves through complicated materials like concrete or soil.

Lu: Yeah, it really does achieve this by building a "structurally coherent scale-invariant operator."

Meng: From an engineer’s viewpoint, the fact that this can be trained on limited data and is robust to noise means we could actually deploy something like this for real-time monitoring without needing massive computational resources.

Tom: Exactly. And they showed it maintains evaluation accuracy when changing the resolution, which is a big deal because standard U-Net models lose accuracy when you try to zoom in or out on the detail (<ref:2610.13258#pg3>).

Jane: So, what does that mean for a listener who only listens to the show? It means better, more reliable AI tools for things like ultrasound sensing that need to work across many different scales.

Lu: The method correctly models reflections and refractions in these complex media, which is hard to get right with simpler models (<ref:2610.13258#pg3>).

Meng: I still have a concern about the training data itself; they trained it on fourteen simulation runs, which is limited, so scaling this up to model truly massive domains will require more real-world testing than those simulations provide (<ref:2610.13258#pg3>).

Tom: That’s a fair point, Meng. The limitation they flag is that while the model learns well from the simulated data it's given, its performance on entirely unseen physical environments needs more validation (<ref:2610.13258#pg3>).

Jane: So, to sum up, this paper presents a way to build AI surrogates for wave propagation that are efficient and surprisingly robust even when dealing with noisy data and varying spatial scales.

Lu: It’s a solid foundation for operator-based solutions in this area (<ref:2610.13258#pg3>).

Meng: I think the next step needs to be moving from simulation validation to actual deployment on real sensor data where we can test those robustness claims under true field conditions.

Tom: Agreed. We’ll keep an eye out for how they apply this LU-NO framework when we look at papers on real-world ultrasonic sensing applications next time.

The paper's improvements: Tom: So we've just covered how they built LU-NO to model sound waves in complicated materials like concrete or soil using AI surrogates, which is what this paper, "Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media," is all about.

Jane: It’s clear the main idea is combining Fourier Neural Operators with local-integral resampling inside a U-Net structure to model these waves efficiently.

Meng: From an engineering standpoint, the fact that this can be trained on limited data and is robust to noise means we could actually deploy something like this for real-time monitoring without needing massive computational resources.

Tom: Agreed. We'll keep an eye out for how they apply this LU-NO framework when we look at papers on real-world ultrasonic sensing applications next time.

Conclusion: Tom: So we’ve covered "Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media," and basically, this AI method uses Fourier Neural Operators combined with local-integral resampling within a U-Net to simulate how sound waves move through tricky materials like concrete or soil.

Jane: It’s clear the main idea is combining those two techniques to create a model that can handle those reflections and refractions in heterogeneous media while still being fast enough for real-world use.

Lu: Yeah, it really does achieve this by building a structurally coherent scale-invariant operator.

Lu: The method correctly models those reflections and refractions in these complex media, which is hard to get right with simpler models.

Meng: I still have a concern about the training data itself. They trained it on fourteen simulation runs, which is limited, so scaling this up to model truly massive domains will require more real-world testing than those simulations provide.

Jane: So, to sum up, this paper presents a way to build AI surrogates for wave propagation that are efficient and surprisingly robust even when dealing with noisy data and varying spatial scales.

Lu: It’s a solid foundation for operator-based solutions in this area.

Meng: I think the next step needs to be moving from simulation validation to actual deployment on real sensor data where we can test those robustness claims under true field conditions.

Tom: Agreed. We'll keep an eye out for how they apply this LU-NO framework when we look at papers on real-world ultrasonic sensing applications next time.

Burla Nur Korkmaz, Petro Moilanen, Ari Salmi, Harsha Vardhan Tetali, Arto Klami

Department of Computer Science, University of Helsinki, Finland · Department of Physics, University of Helsinki, Finland

cs.LG

Submitted: 2026-10-09

Updated: 2026-10-09

Comments: Submitted to IEEE ICASSP 2027

Code: https://github.com/burlakorkmaz/Neural-Surrogates-for-ultrasound

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist The proposed LU-NO method, which integrates Fourier Neural Operators and local-integral resampling into a U-Net architecture, successfully captures essential signal features in ultrasonic

Key concepts

Neural Surrogates: These are machine learning models used as fast replacements for slow, traditional numerical methods when simulating physical problems like wave propagation. They save significant time and memory by learning the underlying system's behavior from training data, making complex simulations practical for real-time applications.
Fourier Neural Operators (FNO): FNO is a specific type of neural operator architecture that excels at capturing global dependencies in wavefield data. It uses Fourier transforms within its layers to effectively model how the wave propagates across large distances and different spatial scales without needing explicit grid calculations.
Local-Integral Resampling: This is a novel mechanism used by LU-NO to transition between different spatial resolutions. Instead of standard interpolation, it computes new grid values based on a local neighborhood of the source data using a learned kernel. This allows the model to accurately handle the continuous nature of wave propagation in heterogeneous environments.

Summary

The gist The proposed LU-NO method, which integrates Fourier Neural Operators and local-integral resampling into a U-Net architecture, successfully captures essential signal features in ultrasonic wave propagation through heterogeneous media while demonstrating robustness to input noise and performance across different spatial scales

Motivation for the Surrogate Model

Neural surrogates are employed as a computationally efficient alternative to traditional numerical solvers for modeling wave propagation, which is crucial because simulating high-frequency acoustic waves in heterogeneous media requires excessive memory and computation. These models enable fast inverse solutions, making broad range of inverse problems practical, including gradient-based solvers that require differentiable integrator. The current surrogates are not yet sufficient for ultrasonic sensing characterized by high frequency and heterogeneous propagation media because methods like Fourier Neural Operators (FNO) fail to capture essential signal features such as reflections, whereas U-Net based methods lack the ability to transfer information across different spatial scales.

Proposed LU-NO Architecture

The proposed Local-Integral Resampler U-shaped Neural Operator (LU-NO) is designed to capture the best of both worlds by integrating FNOs and local-integral resampling into a U-Net architecture. This architecture utilizes Fourier layers [14, 24] to capture global dependencies in the wavefield, a U-Net structure [15, 16] to capture features across multiple spatial resolutions, and a new mechanism for transitioning between these resolutions using local-integral kernels [26]. The model maps T input frames directly to the next non-overlapping T frames rather than predicting a single future frame as in conventional autoregressive approaches

Key Components of LU-NO

Fourier layer: FNO [14] is a neural operator architecture that parameterizes the non-local integral kernel of a neural operator layer. The key building block of FNO is the Fourier layer defined by (1) where vl denotes the latent representation at layer l, F and F−1 are the Fourier transform and its inverse, Rl contains the learnable weights applied to the retained Fourier modes, Wl is a pointwise linear transformation, and σ is a nonlinear activation. In LU-NO, these layers operate within the U-Net resolution.
U-Shaped architecture: Pure FNO fails in heterogeneous media because training at coarser resolution forces the model to miss aspects like reflections from secondary media. LU-NO constructs a U-shaped network architecture similar to UNet [15, 16], but within each level it uses Fourier layers (1) and employs two Fourier layers at each level with skip connections between corresponding levels.
Resampling: Standard U-Net models use convolutions, pooling or interpolation for down- and upsampling, which is not optimal for direct modelling of the continuous system because the resampling depends on the input grid. LU-NO uses local-integral resampling adapted from [26], where the resampler computes each target-grid value from a local neighborhood of the source grid using a learned kernel defined over spatial distance. The operation is applied separably along the spatial dimensions, while preserving the temporal dimension.

Experimental Setup and Results

The experiments were conducted on a 2D heterogeneous spatial domain with spatially varying wave velocity c(x) simulating a water-filled domain containing three aluminum circles. The training utilized limited data, specifically 14 simulation runs for training, demonstrating that the method can be trained on limited data. Performance was measured using frame-wise MSE and the relative squared L2 error Erel, averaged over the predicted test simulations. A key finding is that LU-NO achieves correct spatial rescaling, verifying that it is the first method to achieve the promise of operator-based solutions for heterogeneous media. Furthermore, U-Net was found to be more accurate on the training resolution but fails in generalizing across spatial resolutions, whereas LU-NO correctly models reflections and refractions in heterogeneous medium and maintains evaluation accuracy when changing the resolution.

Robustness and Efficiency

The model demonstrates robustness to input noise, as FNO-based methods that model the continuous physical problem remain robust to noise, unlike the U-Net which performance rapidly degrades when adding noise. LU-NO is also more computationally efficient than FNO-based methods in terms of memory and time. The relative squared L2 error Erel reported in Table 1 shows that LU-NO has a lower error compared to FNO and U-Net at both training and high resolution. This suggests that the model learns the underlying continuous system correctly as an operator, even when trained on very limited data.

Conclusion

The introduction of LU-NO successfully addresses the limitations of prior surrogate models by combining Fourier layers with learned local-integral resampling to model wave propagation in heterogeneous media. The method correctly models reflections and refractions in a heterogeneous medium and demonstrates the evaluation accuracy remains intact when changing the resolution, suggesting it learns the underlying continuous system correctly as an operator. This approach offers a promising path for practical ultrasonic sensing applications by providing computationally efficient surrogates that are robust to noise.

Acknowledgements

BNK, HVT and AK acknowledge the research environment provided by ELLIS Institute Finland. The work received funding from the Jane and Aatos Erkko Foundation, through the Transcranial AI-Powered Ultrasound Computed Tomography grant.

References

[1] J. L. Rose, “Ultrasonic guided waves in structural health monitoring,” Key Engineering Materials, vol. 270–273, no. I, pp. 14–21, 2004

[2] S. Gharib, D. Iablonskyi, J. Mustonen, J. Korsimaa, P. Salminen, B. N. Korkmaz, M. Weber, A. Salmi, and A Klami “Selfadaptation of ultrasound sensing networks,” Mechanical Systems and Signal Processing vol 224 pp 112214 2025

[3] B. N. Korkmaz, S Gharib, H Wei, J Korsimaa, P Salminen M Weber D Iablonskyi A Salmi and A Klami “Physical simulator-based neural networks for real-time fouling tomography” ACM Transactions on Probabilistic Machine Learning vol 2 no 3 pp 1–25 2026

[4] C Khan, R J G van Sloun, and B Byram “Unfolding modelbased beamforming for high quality ultrasound imaging” in Proc of the 2022 IEEE International Conference on Acoustics Speech and Signal Processing 2022

[5] D Xiao and A C H Yu “Beamforming-integrated neural networks for ultrasound imaging” Ultrasonics vol 145 pp 107474 2025

[6] Y Ruan, R Jin, Z Liu, C Wang, L Zhang, and T Peng “Delineation of prostate cancer via enhanced AI-based algorithm in ultrasound images” in Proc of the 2024 IEEE International Conference on Acoustics Speech and Signal Processing 2024

[7] W Yan, Q Zhang, Y Wu, Z Liu, H Zhang, Z Wang, J Yuan, M Ding, M Yuchi and W Qiu “Untrained neural networkbased full-waveform inversion for breast sound speed imaging in ultrasound computed tomography” IEEE Transactions on Instrumentation and Measurement vol 74 pp 1–14 2025

[8] P Moilanen, T Rauhala, and S Ahmadzai “Software-guided clamp-on power ultrasound solution for fouling mitigation in tubular heat exchangers” Heat and Mass Transfer vol 60 no 5 pp 769–776 2024

[9] L Guasch, O Calderon Agudo, M X Tang, P Nachev, and M Warner “Full-waveform inversion imaging of the human brain” npj Digital Medicine vol 3 no 1 pp 28 2020

[10] N Zhang, Y Zhao, Y Yuan, Y Xiao, M Qin, and Y Shen “Cross-correlation adjustment full-waveform inversion with source encoding in ultrasound computed tomography” Ultrasonics vol 142 pp 107392 2024

[11] J K Sharma, R Soman, P Kudela, E Chatzi, and W Ostachowicz “A deep learning based surrogate modelling for wave propagation in structures” in Proc of Sensors and Smart Structures Technologies for Civil Mechanical and Aerospace Systems 2022

[12] Y Augenstein, T Repan, and C Rockstuhl “Neural operator-based surrogate solver for free-form electromagnetic inverse design” ACS Photonics vol 10 no 5 pp 1547–1557 2023

[13] N Kovachki, Z Li, B Liu, K Azizzadenesheli, K Bhattacharya, A Stuart, and A Anandkumar “Neural Operator:

Improvements for AI systems

Improve wave propagation modeling for heterogeneous media by integrating Fourier Neural Operators (FNO) with local-integral resampling into a U-Net architecture, which captures the best of the two worlds to learn structurally coherent scale-invariant operator.
Enable fast inverse solutions and gradient-based solvers by providing a surrogate model that can be trained on limited data and is robust to input noise, unlike previous methods where FNOs failed because they could not capture essential signal features, such as reflections.
Enhance spatial generalization across different resolutions by employing a mechanism that allows the model to achieve correct spatial rescaling, verifying that it can model the continuous system correctly as an operator when tested on unseen structures and higher resolutions.
Improve noise robustness in ultrasonic sensing applications by leveraging global Fourier layers, which are shown to be more robust than convolutional methods because they aggregate information across the entire domain rather than relying only on local neighborhoods captured by the convolutions.

Sources

Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions