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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.20719 |
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Table of 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.