A Deep Learning-Enhanced Fourier Method for the Multi-Frequency Inverse Source Problem with Sparse Far-Field Data

TL;DR

This work tackles the multi-frequency inverse source problem for the Helmholtz equation with sparse far-field data by coupling a physics-based Fourier truncation with a deep U-Net that maps band-limited reconstructions $S_N$ to high-fidelity sources $S$, effectively suppressing truncation artifacts. A neural network $\mathcal{G}_\Theta$ learns the inverse of the truncation operator, operating as an image-to-image translator that enhances resolution while preserving physical interpretability. The authors introduce a high-to-low noise transfer learning strategy, pretraining on very noisy data to capture global structural priors and then fine-tuning on cleaner data, which accelerates convergence and improves generalization. Numerical experiments show accurate reconstructions up to $100\%$ noise, outperforming traditional spectral methods under sparse data and generalizing to unseen geometries, including MNIST-like digits and letters, with substantial reductions in NMSE and high SSIM. The approach offers a computationally efficient, robust alternative for inverse source problems and paves the way for extensions to three dimensions and broader wave models.

Abstract

This paper introduces a hybrid computational framework for the multi-frequency inverse source problem governed by the Helmholtz equation. By integrating a classical Fourier method with a deep convolutional neural network, we address the challenges inherent in sparse and noisy far-field data. The Fourier method provides a physics-informed, low-frequency approximation of the source, which serves as the input to a U-Net. The network is trained to map this coarse approximation to a high-fidelity source reconstruction, effectively suppressing truncation artifacts and recovering fine-scale geometric details. To enhance computational efficiency and robustness, we propose a high-to-low noise transfer learning strategy: a model pre-trained on high-noise regimes captures global topological features, offering a robust initialization for fine-tuning on lower-noise data. Numerical experiments demonstrate that the framework achieves accurate reconstructions with noise levels up to 100%, significantly outperforms traditional spectral methods under sparse measurement constraints, and generalizes well to unseen source geometries.

Figures (9)

• Figure 1: Illustration of the inverse source scattering problem.
• Figure 2: Schematic of the U-Net architecture used in this study.
• Figure 3: Reconstruction results for disk sources under $5\%$, $50\%$, and $100\%$ noise levels. Columns from left to right: ground-truth source, classical Fourier reconstruction ($N=3$), U-Net-enhanced reconstruction ($N=3$), and classical Fourier reconstruction ($N=10$).
• Figure 4: Generalization performance of the network trained on $100\%$ noise when applied to lower-noise data. From left to right: ground-truth source, reconstruction for $5\%$ noise level, and reconstruction for $50\%$ noise level.
• Figure 5: Fine-tuned reconstruction results using the high-to-low noise transfer learning strategy. Rows from top to bottom: ground-truth sources, $5\%$ noise reconstructions, and $50\%$ noise reconstructions.