Document Type

Article

Publication Date

2026

DOI

10.1007/s10915-026-03463-w

Publication Title

Journal of Scientific Computing

Volume

109

Issue

2

Pages

50 (53 pp.)

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 κ = 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.

Rights

© The Authors 2026

 This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 

Data Availability

Article states: "Enquiries about data availability should be directed to the authors."

Original Publication Citation

Zhao, P., Wang, R., Wu, T., & Xu, Y. (2026). The adaptive solution of high-frequency Helmholtz equations via multi-grade deep learning. Journal of Scientific Computing, 109(2), Article 50. https://doi.org/10.1007/s10915-026-03463-w

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