A Robust Learning-Based Method for the Helmholtz Equation in Dissipative Media and Complex Domains
This addresses a domain-specific issue for computational physics and engineering by providing a more robust method for high-frequency wave problems in complex domains.
The paper tackled the problem of instability in learning-based numerical methods for the Helmholtz equation in dissipative media by proposing a Bessel basis, achieving machine-precision accuracy where previous methods failed and outperforming the Finite Element Method in efficiency.
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.