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Learning based numerical methods for Helmholtz equation with high frequency

Yu Chen, Jin Cheng, Tingyue Li, Yun Miao

TL;DR

The paper addresses solving high-frequency Helmholtz equations by learning a boundary-to-interior solution operator from training data, leveraging Green's representation and $\text{Tikhonov}$ regularization to stabilize the reconstruction using fundamental solutions. The resulting learning-based numerical method (LbNM) is mesh-free and enables fast updates for new boundary inputs, with rigorous error estimates anchored in the method of fundamental solutions and quantitative Runge approximation across three analytic-continuation scenarios. Three case studies provide Lipschitz-type, Hölder-type, and logarithmic-type convergence results, respectively, and numerical experiments on a flower-shaped domain validate high accuracy and computational efficiency even at large $k$. The approach offers interpretability and local computational flexibility, suggesting practical applicability to localized engineering problems and potential extensions to mixed, inhomogeneous, or variable-coefficient Helmholtz problems and Cauchy-type problems.

Abstract

High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.

Learning based numerical methods for Helmholtz equation with high frequency

TL;DR

The paper addresses solving high-frequency Helmholtz equations by learning a boundary-to-interior solution operator from training data, leveraging Green's representation and regularization to stabilize the reconstruction using fundamental solutions. The resulting learning-based numerical method (LbNM) is mesh-free and enables fast updates for new boundary inputs, with rigorous error estimates anchored in the method of fundamental solutions and quantitative Runge approximation across three analytic-continuation scenarios. Three case studies provide Lipschitz-type, Hölder-type, and logarithmic-type convergence results, respectively, and numerical experiments on a flower-shaped domain validate high accuracy and computational efficiency even at large . The approach offers interpretability and local computational flexibility, suggesting practical applicability to localized engineering problems and potential extensions to mixed, inhomogeneous, or variable-coefficient Helmholtz problems and Cauchy-type problems.

Abstract

High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.
Paper Structure (16 sections, 10 theorems, 88 equations, 10 figures, 2 tables)

This paper contains 16 sections, 10 theorems, 88 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The learning based algorithm
  3. Motivation
  4. Algorithm
  5. Preliminaries
  6. Theoretical results
  7. Case 1: The solution can be analytically continued to a sufficiently larger domain.
  8. Case 2: The solution can be analytically continued to a slightly larger domain.
  9. Case 3: The solution cannot be analytically continued to a larger domain
  10. Numerical examples
  11. Case 1
  12. Case 2
  13. Case 3
  14. Concluding remarks
  15. Detailed deduction of
  16. ...and 1 more sections

Key Result

Lemma 3.1

(Beretta2016 and Ruland2021) Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n, f \in H^{1 / 2}(\partial \Omega)$. Then, there exists a discrete set $E \subset \mathbb{R}$ such that, for every $k^2 \in\mathbb{R} \backslash E$, there exists a unique solution $u \in H^1(\Omega)$ of eq:Helm-e where $C$ depends only on $\Omega$. More precisely, $E$ denotes the set of the Dirichlet eigenvalue

Figures (10)

  • Figure 3.1: Geometry for the MFS in the unit disk.
  • Figure 4.1: Illustration of domains.
  • Figure 4.2: Illustration of Case 1.
  • Figure 4.3: Illustration of Case 2.
  • Figure 4.4: Illustration of Case 3.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 10 more