A Robust Learning-Based Method for the Helmholtz Equation in Dissipative Media and Complex Domains
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914395175518208 |
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| author | Song, Lifu Li, Tingyue Cheng, Jin |
| author_facet | Song, Lifu Li, Tingyue Cheng, Jin |
| contents | To mitigate pollution effects in high-frequency Helmholtz problems, Learning-based Numerical Methods (LbNM) reconstruct solution operators using complete systems of exact solutions. However, the previously used fundamental-solution (FS) basis suffers from instability in dissipative media and requires sensitive geometric tuning. In this paper, we propose a robust alternative using a Bessel basis (BB). From a learning theory perspective, the BB forms a complete hypothesis space of standing waves, ensuring immunity to dissipation-induced signal loss. We establish a convergence result that depends on intrinsic regularity. Numerical experiments demonstrate that the proposed method achieves machine-precision accuracy in dissipative regimes where FS fails, significantly outperforms the Finite Element Method (FEM) in efficiency, and demonstrates the framework's geometric extensibility via a multi-center strategy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Robust Learning-Based Method for the Helmholtz Equation in Dissipative Media and Complex Domains Song, Lifu Li, Tingyue Cheng, Jin Numerical Analysis To mitigate pollution effects in high-frequency Helmholtz problems, Learning-based Numerical Methods (LbNM) reconstruct solution operators using complete systems of exact solutions. However, the previously used fundamental-solution (FS) basis suffers from instability in dissipative media and requires sensitive geometric tuning. In this paper, we propose a robust alternative using a Bessel basis (BB). From a learning theory perspective, the BB forms a complete hypothesis space of standing waves, ensuring immunity to dissipation-induced signal loss. We establish a convergence result that depends on intrinsic regularity. Numerical experiments demonstrate that the proposed method achieves machine-precision accuracy in dissipative regimes where FS fails, significantly outperforms the Finite Element Method (FEM) in efficiency, and demonstrates the framework's geometric extensibility via a multi-center strategy. |
| title | A Robust Learning-Based Method for the Helmholtz Equation in Dissipative Media and Complex Domains |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.14193 |