The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908850236424192 |
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| author | Zhao, Peiyao Wang, Rui Wu, Tingting Xu, Yuesheng |
| author_facet | Zhao, Peiyao Wang, Rui Wu, Tingting Xu, Yuesheng |
| contents | The Helmholtz equation is fundamental to wave modeling in acoustics, electromagnetics, and seismic imaging, yet high-frequency regimes remain challenging due to the ``pollution effect''. We propose FD-MGDL, an adaptive framework integrating finite difference schemes with Multi-Grade Deep Learning to efficiently resolve high-frequency solutions. While traditional PINNs struggle with spectral bias and automatic differentiation overhead, FD-MGDL employs a progressive training strategy, incrementally adding hidden layers to refine the solution and maintain stability. Crucially, when using ReLU activation, our algorithm recasts the highly non-convex training problem into a sequence of convex subproblems. Numerical experiments in 2D and 3D with wavenumbers up to $κ=200$ show that FD-MGDL significantly outperforms single-grade and conventional neural solvers in accuracy and speed. Applied to an inhomogeneous concave velocity model, the framework accurately resolves wave focusing and caustics, surpassing the 5-point finite difference method in capturing sharp phase transitions and amplitude spikes. These results establish FD-MGDL as a robust, scalable solver for high-frequency wave equations in complex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning Zhao, Peiyao Wang, Rui Wu, Tingting Xu, Yuesheng Numerical Analysis The Helmholtz equation is fundamental to wave modeling in acoustics, electromagnetics, and seismic imaging, yet high-frequency regimes remain challenging due to the ``pollution effect''. We propose FD-MGDL, an adaptive framework integrating finite difference schemes with Multi-Grade Deep Learning to efficiently resolve high-frequency solutions. While traditional PINNs struggle with spectral bias and automatic differentiation overhead, FD-MGDL employs a progressive training strategy, incrementally adding hidden layers to refine the solution and maintain stability. Crucially, when using ReLU activation, our algorithm recasts the highly non-convex training problem into a sequence of convex subproblems. Numerical experiments in 2D and 3D with wavenumbers up to $κ=200$ show that FD-MGDL significantly outperforms single-grade and conventional neural solvers in accuracy and speed. Applied to an inhomogeneous concave velocity model, the framework accurately resolves wave focusing and caustics, surpassing the 5-point finite difference method in capturing sharp phase transitions and amplitude spikes. These results establish FD-MGDL as a robust, scalable solver for high-frequency wave equations in complex domains. |
| title | The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.20719 |