Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lifu Song, Tingyue Li, Jin Cheng

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.

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

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.
Paper Structure (29 sections, 5 theorems, 57 equations, 9 figures, 4 tables, 1 algorithm)

This paper contains 29 sections, 5 theorems, 57 equations, 9 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Learning-Based Method with a Bessel basis
  3. The General Learning Algorithm
  4. The Bessel basis
  5. Theoretical Analysis
  6. Two Discretizations of the Single-Layer Potential
  7. Preliminaries
  8. Convergence Analysis
  9. Numerical Validation
  10. General Experimental Setup
  11. Governing Equation and Physics
  12. Exact Solutions and Error Metric
  13. Solver Regularization Strategies
  14. Test 1: Stability and Efficiency in Dissipative Media
  15. Objective
  16. ...and 14 more sections

Key Result

Lemma 3.1

Let $\Omega$ be a bounded domain. If $u \in H^1(\Omega)$ satisfies the Helmholtz equation $\Delta u + k^2 u = 0$ in $\Omega$ and $k^2$ is not a Dirichlet eigenvalue of the Laplacian on $\Omega$, then the the following estimation holds: where the constant $C_{stab}$ depends on the domain $\Omega$ and the distance of $k^2$ to the Dirichlet spectrum KuttlerSigillito1978Bounding.

Figures (9)

  • Figure 1: Exponential decay of the fundamental solution $H_0^{(1)}(kr)$ in a dissipative medium ($k=100+10\mathrm{i}$).
  • Figure 2: Illustration of the computational setup for Test 1. The figures show the interior solving points (blue dots), the collocation points on the boundary (red stars), and the exterior pole points used by the benchmark FS-LbNM (black circles).
  • Figure 3: Error vs. Damping Coefficient ($\sigma$). The BB-LbNM (Red) remains stable regardless of dissipation, while the FS-LbNM (Blue) degrades rapidly in high-loss media.
  • Figure 4: Convergence rate comparison for rounded-kite domain.
  • Figure 5: Computational Efficiency Comparison. The BB-LbNM (Red) achieves machine precision ($10^{-14}$) in seconds, whereas the FEM (Blue) struggles to reach $10^{-2}$ even after 120 seconds. The dashed line shows the extremely fast Online evaluation time for BB-LbNM.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Lemma 3.1: Interior Stability Estimate
  • Lemma 3.2: Completeness of the Bessel Basis
  • proof
  • Lemma 3.3: Quantitative Runge Approximation
  • Theorem 3.4: Bessel Series Approximation
  • proof
  • Theorem 3.5: BB-LbNM Error
  • proof
  • Remark 3.6: General Cases