Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media

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

cs.LGOct 9, 2026

Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media

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

Organizations: Department of Computer Science, University of Helsinki, Finland

Abstract

Neural surrogates provide computationally efficient auto-regressive modeling of wave propagation, enabling e.g. fast inverse solutions. Despite recent progress, the current surrogates are not yet sufficient for the needs of ultrasonic sensing, characterized by high frequency and heterogeneous propagation media. We show how methods building on Fourier Neural Operators (FNO) fail to capture essential signal features, such as reflections, whereas U-Net -based methods learn the wave fields well but lack the ability to transfer information across different spatial scales. We then propose a new surrogate that captures the best of the two worlds, by integrating FNOs and local-integral resampling into a U-Net architecture. We show it can be trained on limited data, and that the predictions are robust to input noise and work across spatial scales.

Figures & tables

Figure 1: LU-NO: Time autoregressive neural surrogate for wavefields, using Fourier operator layers (blue) and local-integral resampling (red) to learn structurally coherent scale-invariant operator.

Model	Relative squared L2 error		Mem. (GB)	Time (s)	# Params.
Model	High resolution	Training resolution	Mem. (GB)	Time (s)	# Params.
U-Net	$1.160\pm 0.025$	$\mathbf{0.160\pm 0.076}$	0.31	21	1.5M
FNO	$0.405\pm 0.073$	$0.374\pm 0.085$	17.32	59	944M
U-NO	$0.453\pm 0.047$	$0.295\pm 0.078$	16.66	61	1B
LU-NO	$\mathbf{0.305\pm 0.083}$	$0.295\pm 0.088$	11.91	34	739M

Table 1: Error on two spatial resolutions, with standard deviations over 16 test instances. Memory cost is for one training instance and the time is for full epoch, and # Params indicates the number of trainable parameters.

Panel of Figure 2 — Figure 2: Frame-wise relative squared $L_{2}$ error over 16 test instances, computed as prediction SSE divided by the ground-truth signal energy at each frame. Solid lines show the mean, with shading indicating $\pm$ one standard deviation across test instances. (a) Training-resolution evaluation. (b) High-resolution evaluation. (c) Snapshots of $(t=36\,\mu\mathrm{s})$ acoustic wave propagation. Our method best matches the ground truth in qualitative terms, as the only method that correctly models the main reflection also in high resolution (See App. B for more results).

Panel of Figure 3 — Figure 3: Sensitivity to additive white Gaussian noise of $\sigma_{n}=\alpha\sigma_{p}$ , where $\sigma_{p}$ is the RMSE of the mean-subtracted clean wavefield, for the initial pressure-field inputs ( Left: Training resolution, Right: High resolution). The results are averaged over the 16 test simulations, with confidence interval (one std) over 10 independent noise seeds. U-Net with substantially worse overall accuracy is excluded from the high resolution plot.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Illustration of the physical domain. ( First three plots: In-domain structure, Last plot: Out-of-domain structure.) Arrows indicate pulse locations; green ones are used during training, and red ones are used for testing. Blue circles show aluminum areas.

Figure 5: Snapshots of acoustic wave propagation at different time points.

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Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

Abstract

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