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Mitigating Spectral Bias in Neural Operators for Underwater Transmission Loss Prediction

This paper proposes a Spectral-Spatial Residual Learning (S2RL) framework that mitigates the spectral bias of Fourier Neural Operators by decomposing underwater transmission loss prediction into a global spectral propagation and a local spatial refinement, thereby achieving rapid, accurate, and fine-grained predictions that outperform existing baselines.

Original authors: Yifan Sun, Shikai Fang, Chao Zhang, Lei Cheng, Jianlong Li, Peter Gerstoft

Published 2026-08-20

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CC BY 4.0

Original authors: Yifan Sun, Shikai Fang, Chao Zhang, Lei Cheng, Jianlong Li, Peter Gerstoft

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

The ocean is a vast, shifting maze where sound travels in complex, invisible waves. For scientists and engineers working underwater, predicting how far a sound will travel and how much of it will fade away—known as transmission loss—is a matter of critical importance. This knowledge guides everything from placing sensitive sensors to charting safe paths for submarines. Traditionally, figuring out these patterns has been a slow, heavy task. Researchers rely on powerful numerical solvers, which are essentially digital simulations that crunch through the physics of sound waves step by step. While these simulations are reliable, they are so computationally expensive that they cannot be run quickly enough for real-time decisions or large-scale planning.

To speed things up, researchers have turned to artificial intelligence, training computer models to learn the relationship between the ocean's environment and the resulting sound patterns. One promising type of model, called a Fourier Neural Operator, has shown great ability to understand the broad, global behavior of these waves. However, these models have a blind spot: they are excellent at seeing the big picture but tend to blur out the fine, intricate details. They smooth over the sharp, jagged interference patterns that are actually present in the water, much like a low-resolution photograph that captures the shape of a face but loses the texture of the skin. A new study from researchers at Zhejiang University and the Technical University of Denmark addresses this specific weakness, offering a way to keep the speed of these fast models while recovering the lost details.

The researchers, led by Yifan Sun and Lei Cheng, proposed a two-stage system they call Spectral-Spatial Residual Learning. Instead of asking a single artificial intelligence model to do everything perfectly at once, they broke the job into two distinct steps. In the first stage, a standard Fourier Neural Operator acts as a "Global Propagator." Its job is to quickly generate a rough, broad prediction of how sound moves through the water. This model is very good at capturing the overall trends and the large-scale structure of the sound field, but as expected, it produces a result that is too smooth, missing the sharp edges and fine ripples.

The second stage introduces a "Local Refiner," a different type of neural network designed to look at the mistakes made by the first model. This refiner does not try to predict the whole sound field again; instead, it focuses entirely on the difference, or the "residual," between the rough prediction and the true, detailed reality. By training specifically on these missing details, the refiner learns to reconstruct the high-frequency interference patterns that the first model smoothed over. The final answer is simply the sum of the rough prediction and the detailed corrections, resulting in a complete picture that is both globally consistent and locally sharp.

To test this approach, the team used a dataset of underwater acoustic data from the South China Sea, simulating sound propagation at a frequency of 200 hertz. They compared their new two-stage method against the standard models and the traditional, slow numerical solvers. The results were striking. The traditional solver took about 333 milliseconds to produce a single prediction, which is too slow for many real-time applications. The standard fast models were incredibly quick, taking less than 4 milliseconds, but their predictions were blurry, with an error rate of nearly 3 decibels. The new two-stage method added only a tiny amount of time, bringing the total to about 8 milliseconds, yet it slashed the error rate down to roughly 1.5 decibels.

The visual evidence confirmed what the numbers showed. When looking at the raw output of the standard model, the intricate interference fringes—the complex patterns where sound waves overlap and cancel each other out—were washed out and indistinct. The new method, however, restored these sharp, precise patterns, making the prediction look almost identical to the high-fidelity simulation. Furthermore, the researchers analyzed the energy distribution of the sound waves and found that the standard model was losing energy at higher frequencies, which correspond to those fine details. Their two-stage system successfully recovered this lost energy, proving that the "refiner" was effectively filling in the gaps left by the "propagator."

This work demonstrates that it is possible to have both speed and precision in underwater acoustic prediction. By acknowledging that different parts of the problem require different tools, the researchers created a system that leverages the speed of global models for the big picture and the precision of local models for the details. The result is a tool that can predict underwater sound fields with high accuracy in milliseconds, offering a practical solution for real-time ocean applications without sacrificing the fine-grained details that make the physics of the ocean so complex.

Technical Summary: Mitigating Spectral Bias in Neural Operators for Underwater Transmission Loss Prediction

1. Problem Statement

Accurate and rapid prediction of underwater acoustic transmission loss (TL) is essential for real-time ocean applications such as sensor placement optimization and vehicle path planning. Traditional numerical solvers, like the Range-Dependent Acoustic Model (RAM), provide reliable results but suffer from high computational costs, rendering them unsuitable for large-scale or real-time inference.

While deep learning surrogate models, particularly Fourier Neural Operators (FNO), have emerged as efficient alternatives due to their global receptive fields, they exhibit a critical limitation known as spectral bias. The standard FNO architecture employs a frequency truncation mechanism, retaining only the lowest $k$ frequency modes to compute global correlations. This operation acts as a low-pass filter, causing FNO models to produce over-smoothed predictions. Consequently, while FNOs capture global propagation trends effectively, they fail to resolve fine-grained interference patterns and sharp spatial gradients intrinsic to underwater acoustic fields.

2. Methodology: Spectral-Spatial Residual Learning (S2RL)

To address the trade-off between global consistency and local detail, the authors propose a two-stage Spectral-Spatial Residual Learning (S2RL) framework. This approach decomposes the TL prediction task into a coarse-to-fine process based on an operator-theoretic interpretation of FNO as a low-pass propagator.

A. Theoretical Motivation

From an operator perspective, the FNO with truncated modes approximates the solution operator as a band-limited projection ( $U_{coarse} \approx P_{low}U$ ). The prediction error is not random noise but a structured residual residing in the complementary high-frequency space ( $U_{res} \approx P_{high}U$ ). The S2RL framework explicitly models the TL field as the sum of these components:
$U = P_{low}U + P_{high}U$

B. Framework Architecture

The framework consists of two cascaded neural networks:

Stage I: Global Propagator ( $G_1$ )
- Role: Generates a coarse, globally consistent prediction of the acoustic field.
- Architecture: Utilizes an FNO-based backbone. The authors evaluate two variants:
  - Vanilla FNO: Uses raw ocean sound speed fields as input.
  - Hankel-FNO: Augments inputs with physical priors, including the asymptotic behavior of the zeroth-order Hankel function (to capture range-dependent spreading loss) and a bathymetry embedding (to capture seabed reflection effects).
- Output: A low-frequency approximation ( $U_{coarse}$ ) that captures dominant propagation trends but lacks high-frequency details.
Stage II: Local Refiner ( $G_2$ )
- Role: Recovers high-frequency residuals to correct the over-smoothed output of Stage I.
- Architecture: A U-Net operating in the spatial domain. Unlike FNO, U-Net uses local convolutions and skip connections, making it adept at capturing edges, sharp gradients, and local textures.
- Input/Output: Takes $U_{coarse}$ as input and predicts the residual map ( $U_{res}$ ).
- Final Prediction: The final output is the sum of the coarse prediction and the refined residual: $U_{pred} = U_{coarse} + U_{res}$ .

C. Training Strategy

The networks are trained sequentially using a stage-wise strategy to ensure stability:

Phase 1 (Global Trend Learning): The Global Propagator ( $G_1$ ) is trained to minimize the masked L2 loss between its output and the ground truth. A binary mask excludes the sediment layer, focusing optimization on the water column.
Phase 2 (Residual Learning): Once $G_1$ converges, its parameters are frozen. The Local Refiner ( $G_2$ ) is then trained to minimize the reconstruction error of the final prediction ( $U_{coarse} + G_2(U_{coarse})$ ) against the ground truth, specifically learning the discrepancy between the coarse approximation and the true high-frequency details.

3. Key Contributions

Spectral-Spatial Decomposition: The paper introduces a principled framework that decouples global wave propagation modeling (spectral domain) from local detail reconstruction (spatial domain), directly addressing the spectral bias inherent in FNOs.
Hybrid Architecture: It proposes a specific cascaded design combining the efficiency of FNOs for global trends with the multi-scale feature extraction capabilities of U-Nets for high-frequency residuals.
Physical Priors Integration: The study demonstrates the integration of domain-specific knowledge (Hankel function asymptotics and bathymetry) into the FNO backbone to enhance the initial coarse prediction.
Efficient High-Frequency Recovery: Unlike methods that require denser discretization or iterative refinement (e.g., PINNs), S2RL achieves fine-scale recovery through a single forward pass of the refined network.

4. Experimental Results

Experiments were conducted on a dataset derived from the South China Sea, simulating transmission losses at 200 Hz using RAM. The dataset included 3,456 samples covering a 10 km range and 1.5 km depth.

Accuracy: The S2RL framework significantly outperformed baseline FNO models.
- Vanilla FNO: RMSE reduced from 2.93 dB to 1.56 dB.
- Hankel-FNO: RMSE reduced from 1.95 dB to 1.54 dB.
- Both backbones converged to a comparable high-precision level, demonstrating the robustness of the Local Refiner.
Efficiency: All neural models achieved millisecond-level inference speeds (approx. 3–10 ms), which is two orders of magnitude faster than the RAM solver (333 ms). The addition of the refinement stage introduced only a marginal overhead (~5 ms).
Spectral Analysis: Power Spectral Density (PSD) analysis confirmed that while coarse predictions aligned with ground truth at low frequencies, they decayed rapidly at higher frequencies due to spectral bias. The refined predictions successfully recovered this high-frequency energy, closely tracking the ground truth profile across the full spectrum.
Visual Quality: Visual comparisons showed that while vanilla FNO produced blurred interference fringes, S2RL rectified these discrepancies, recovering sharp and precise patterns.

5. Significance and Claims

The paper claims that the S2RL framework offers a principled alternative to simply increasing spectral resolution or network depth, which often yields diminishing returns. By explicitly decomposing the problem into low-frequency global propagation and high-frequency interference residuals, the method achieves a balance between computational efficiency and reconstruction fidelity.

The authors position S2RL as an efficient and accurate surrogate for complex acoustic wave propagation, capable of delivering predictions that are both globally consistent and locally precise. This makes the approach particularly suitable for real-time ocean acoustic applications where both speed and the accurate resolution of fine-grained interference patterns are critical. The framework maintains the speed advantages of neural operators while overcoming their primary limitation regarding spectral bias.