A Deep Learning-Enhanced Fourier Method for the Multi-Frequency Inverse Source Problem with Sparse Far-Field Data
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917180073836544 |
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| author | Chen, Hao Chang, Yan Guo, Yukun Wang, Yuliang |
| author_facet | Chen, Hao Chang, Yan Guo, Yukun Wang, Yuliang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Deep Learning-Enhanced Fourier Method for the Multi-Frequency Inverse Source Problem with Sparse Far-Field Data Chen, Hao Chang, Yan Guo, Yukun Wang, Yuliang Analysis of PDEs Numerical Analysis 35R30, 76M21, 78A46, 68T07 G.1.8 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. |
| title | A Deep Learning-Enhanced Fourier Method for the Multi-Frequency Inverse Source Problem with Sparse Far-Field Data |
| topic | Analysis of PDEs Numerical Analysis 35R30, 76M21, 78A46, 68T07 G.1.8 |
| url | https://arxiv.org/abs/2601.00427 |