A Robust Learning-Based Method for the Helmholtz Equation in Dissipative Media and Complex Domains

TL;DR

This paper introduces a robust learning-based numerical method using Bessel basis functions for solving high-frequency Helmholtz equations in dissipative media, overcoming instability issues of previous methods and achieving high accuracy and efficiency.

Contribution

The paper proposes a Bessel basis approach for learning solution operators, providing stability in dissipative media and demonstrating superior accuracy and efficiency over existing methods.

Findings

Achieves machine-precision accuracy in dissipative regimes.

Outperforms Finite Element Method in efficiency.

Demonstrates geometric extensibility with a multi-center strategy.

Abstract

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.