The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning

[2602.20719] The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2602.20719 (math) [Submitted on 24 Feb 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning Authors: Peiyao Zhao , Rui Wang , Tingting Wu , Yuesheng Xu View PDF HTML (experimental) Abstract: The Helmholtz equation is fundamental for modeling wave propagation in acoustics, electromagnetics, and geophysics; however, high-frequency regimes remain notoriously difficult due to the severe numerical ``pollution effect.'' We propose FD-MGDL, an adaptive framework that synergizes finite difference discretizations with Multi-Grade Deep Learning (MGDL) to efficiently resolve high-frequency wavefields. Unlike standard physics-informed neural networks (PINNs), which frequently suffer from spectral bias and heavy automatic differentiation overhead, FD-MGDL employs a progressive, grade-wise training strategy that incrementally incorporates shallow sub-networks to refine residual errors. By leveraging ReLU activations in refinement grades, the framework reformulates the highly non-convex global optimization problem into a sequence of tractable convex subproblems, dramatically improving training stability and convergence reliability. Extensive numerical experiments in two and three dimensions with wavenumbers up to $\kappa=200$ demonstrate that FD-MGDL significantly outperforms single-grade networks and standard benchmark neural solvers in both accuracy and computational efficiency. When applied to an inhomogeneous concave velocity model, the proposed method accurately captures wave focusing and caustic formations, markedly outperforming classical five-point finite difference schemes in resolving sharp phase transitions and peak amplitudes. These results establish FD-MGDL as a robust, scalable, and mathematically grounded paradigm for high-frequency wave simulation in complex media. Subjects: Numerical Analysis (math.NA) Cite as: arXiv:2602.20719 [math.NA] (or arXiv:2602.20719v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2602.20719 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rui Wang [ view email ] [v1] Tue, 24 Feb 2026 09:31:43 UTC (5,978 KB) [v2] Fri, 28 Aug 2026 08:25:18 UTC (6,495 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-02 Change to browse by: cs cs.NA math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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