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Neural Surrogate Modeling of Ultrasonic Wave Propagation in Heterogeneous Media

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

arXiv:2610.13258v1cs.LG

TL;DR

Ultrasonic sensing needs accurate modeling of high-frequency wave propagation in heterogeneous media, where existing surrogates have limitations. LU-NO combines Fourier layers and local-integral resampling in a U-Net, modeling reflections and refractions while maintaining accuracy across spatial resolutions.

  • Problem

    Neural surrogates remain insufficient for high-frequency ultrasonic propagation in heterogeneous media, where accurate modeling must capture interface effects.

    Source: §1 · p. 1, §2 · p. 1

  • Method

    LU-NO combines Fourier layers within a U-Net with learned local-integral resampling between spatial resolutions.

    Source: §3 · p. 2

  • Results

    LU-NO correctly models reflections and refractions and maintains evaluation accuracy when spatial resolution changes, unlike prior FNO-based methods that fail to generalize across resolutions.

    Source: §5 · p. 4, §4.3 · p. 4

  • Takeaways & Limitations

    Resolution-stable accuracy suggests LU-NO learns the underlying continuous system as an operator for heterogeneous media.

    Source: §5 · p. 4

    LU-NO's accuracy does not yet match that of a simpler convolutional U-Net, and the models were trained on very limited data.

    Source: §5 · p. 4

Abstract

from arXiv · show

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.

1. INTRODUCTION

Ultrasonic sensing requires modeling high-frequency waves in heterogeneous media, but numerical simulation is costly and neural surrogates still face accuracy challenges. Existing FNO and U-Net approaches each have limitations that motivate a hybrid.

1. INTRODUCTION

  • High-frequency waves in heterogeneous media interact with material interfaces and boundaries through abundant reflections and refractions.Realistic time-domain FEM or FD simulations can take hours, days, or weeks, and inverse algorithms typically require repeated simulations.Source: §1 · p. 1
  • Neural surrogates can amortize simulation costs, but their accuracy remains challenging for ultrasound's high frequencies and precise interface effects.Once trained, they can simulate new structures in seconds and support inverse problems requiring differentiable integrators.Source: §1 · p. 1
  • FNO-based methods have limited success with acoustic wavefields, while U-Nets handle local and global features better but learn only at one resolution.The paper builds on hybrid operator architectures and modifies resolution-invariant shrinking and expanding operations.Source: §1 · p. 1

2. PROBLEM FORMULATION

The task is to predict the development of ultrasonic pressure in a two-dimensional heterogeneous medium from its history and material properties. The model receives coordinates and source-location information and predicts the next block of pressure frames.

2. PROBLEM FORMULATION

  • The target is the ultrasonic pressure field p(x, t) over a two-dimensional domain with spatially varying wave velocity c(x).The domain coordinates are x = (x, y).Source: §2 · p. 1
  • Inputs include a historical pressure window, medium velocity, spatial and temporal coordinates, and source coordinates.The coordinates and source location are supplied as extra input channels.Source: §2 · pp. 1–2
  • The surrogate maps T input frames directly to the next non-overlapping T frames to capture longer-term temporal dynamics.Figure 1 depicts inputs on the left and outputs on the right.Source: §2 · p. 2, Figure 1 · p. 2

3. METHOD

LU-NO combines Fourier layers within a U-Net with learned local-integral resampling between spatial resolutions. The design uses global operator modeling alongside multi-resolution features and grid-aware transitions.

3. METHOD

  • LU-NO combines Fourier layers for global wavefield dependencies, a U-Net for multiple spatial resolutions, and local-integral kernels for transitions between resolutions.The method is designed to learn an operator for contiguous pressure fields.Source: §3 · p. 2
  • Each Fourier layer parameterizes a neural operator's non-local integral kernel in Fourier space, and LU-NO uses these layers within each U-Net resolution.The Fourier transform is applied over both spatial dimensions and time.Source: §3 · p. 2
  • Fourier layers at each U-shaped level use corresponding-level skip connections; coarser representations encourage modeling features such as reflections from secondary media.The architecture uses two Fourier layers at each level, unlike U-NO's one layer per level.Source: §3 · p. 2
  • Instead of grid-dependent interpolation or pooling, local-integral resampling computes each target feature from a learned, distance-based local neighborhood of source-grid features.The kernel uses triangular basis functions with learned coefficients.Source: §3 · p. 2
  • The local integration uses a radius defined as a fixed fraction of the source-domain length, with quadrature-based numerical integration in practice.A fixed radius fraction corresponds to the same fraction of the physical domain at different grid resolutions.Source: §3 · p. 2
  • Resampling is separable across spatial dimensions while preserving time, and the final layer dynamically maps latent features back to the input resolution.Intermediate spatial resolutions remain fixed across the encoder-decoder hierarchy.Source: §3 · p. 2

4. EXPERIMENTS

The experiments test whether LU-NO and three surrogate baselines generalize from limited low-resolution training data to unseen geometries and higher spatial resolution. LU-NO captures the main reflection at high resolution, while U-Net accuracy degrades under resolution changes and input noise.

4.1. Data and evaluation setup

  • Training uses 14 simulations from one geometry with different emission locations; testing includes unseen geometries and the original 301 × 301 resolution.The training simulations are downsampled to 75 × 121 × 121 and augmented by horizontal and vertical flips; evaluation also includes the 121 × 121 training resolution.Source: §4.1 · p. 3, §4 · p. 3

4.2. Baselines and architectural details

  • LU-NO combines Fourier layers in a U-shaped encoder-decoder with learned local-integral resampling; the baselines are FNO, convolutional U-Net, and U-NO.The reported LU-NO resampling maps spatial sizes 121 → 60 → 30 → 15 → 30 → 60 → 121 while preserving temporal resolution.Source: §4.2 · p. 3

4.3. Results

  • LU-NO is the only method that correctly models the main reflection at high resolution, and it achieves correct spatial rescaling.The comparison evaluates frame-wise relative squared L2 error and pressure-field snapshots; the qualitative high-resolution snapshot shows the reflection result.Source: Figure 2 · p. 4, §4.3 · p. 4
  • U-Net is more accurate than LU-NO at training resolution, but fails to model the pressure field at a different resolution.Its training-resolution advantage is most visible in reconstruction of longer wavelengths inside the aluminum circles.Source: §4.3 · p. 4
  • For noisy initial pressure fields, U-Net performance rapidly degrades when noise is added, while all FNO-based methods modeling the continuous physical problem remain robust; this may reflect global Fourier layers aggregating domain-wide information.Source: §4.3 · p. 4

5. CONCLUSION

The paper introduces LU-NO, which combines Fourier layers and learned local-integral resampling in a U-shaped neural operator for ultrasonic propagation across spatial resolutions. It reports correct modeling of reflections and refractions across resolutions, while noting that accuracy remains below a convolutional U-Net trained on limited data.

  • LU-NO combines Fourier layers with learned local-integral resampling in a U-shaped neural operator for ultrasonic wave propagation across spatial resolutions.Source: §5 · p. 4
  • LU-NO correctly models reflections and refractions in heterogeneous media, and its evaluation accuracy remains intact when resolution changes.Source: §5 · p. 4
  • LU-NO accuracy does not yet reach that of a convolutional U-Net, and the authors note that limited training data may be insufficient for learning larger models perfectly.Source: §5 · p. 4

A. DOMAIN STRUCTURE

The experimental domain contains water and three aluminum circles, with training geometries formed from the original domain and its horizontal and vertical flips. A separate geometry is used for out-of-domain testing.

B. ADDITIONAL RESULTS

Figure 5 presents LU-NO evaluations as acoustic wave-propagation snapshots at different time points.

B. ADDITIONAL RESULTS

C. LOCAL-INTEGRAL RESAMPLING DETAILS

The local-integral resampler maps features between spatial grids using learned local kernels, with separable processing across spatial dimensions and boundary-aware support.

C. LOCAL-INTEGRAL RESAMPLING DETAILS

D. METRICS

This section identifies the relative squared L2 error reported in Table 1 and describes MSE use for prediction evaluation and noise sensitivity analysis.

D. METRICS

  • The relative squared L2 error is reported in Table 1 for a predicted simulation.Source: D. METRICS · p. 7
  • MSE is computed for a predicted simulation used in Figure 3.Source: D. METRICS · p. 7
  • MSE is normalized for sensitivity analyses, with α denoting noise level and MSE(0) the corresponding MSE without added noise.Source: D. METRICS · p. 7
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