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NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

Fukai Chen, Ziyang Liu, Guochang Lin, Junqing Chen, Zuoqiang Shi

TL;DR

NSNO introduces a Neumann-series neural operator to learn the Helmholtz solution operator in inhomogeneous media, reformulating the problem so that a single operator from source terms to solutions can be learned and then extended via a truncated Neumann series. The architecture combines Fourier-based operators with a U-shaped multiscale network (UNO) to capture complex, oscillatory, and multi-scale wave phenomena, yielding substantial accuracy gains (often >60% relative $L^2$-error reduction) and about half the training cost compared to prior methods. The approach demonstrates strong forward performance across diverse datasets and enables a fast, physics-informed surrogate for inverse scattering problems, including MNIST-based scatterers, with significant speedups while maintaining accuracy. Limitations include a convergence constraint on the inhomogeneity magnitude and wavenumber, and future work aims to extend NSNO to other PDEs and higher-contrast regimes using alternative iterative schemes and architectures.

Abstract

In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential equation (PDE) with applications in various scientific and engineering fields. However, efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber. Recently, deep learning has shown great potential in solving PDEs especially in learning solution operators. Inspired by Neumann series in Helmholtz equation, we design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature. Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60\% lower relative $L^2$-error, especially in the large wavenumber case, and has 50\% lower computational cost and less data requirement. Moreover, NSNO can be used as the surrogate model in inverse scattering problems. Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

TL;DR

NSNO introduces a Neumann-series neural operator to learn the Helmholtz solution operator in inhomogeneous media, reformulating the problem so that a single operator from source terms to solutions can be learned and then extended via a truncated Neumann series. The architecture combines Fourier-based operators with a U-shaped multiscale network (UNO) to capture complex, oscillatory, and multi-scale wave phenomena, yielding substantial accuracy gains (often >60% relative -error reduction) and about half the training cost compared to prior methods. The approach demonstrates strong forward performance across diverse datasets and enables a fast, physics-informed surrogate for inverse scattering problems, including MNIST-based scatterers, with significant speedups while maintaining accuracy. Limitations include a convergence constraint on the inhomogeneity magnitude and wavenumber, and future work aims to extend NSNO to other PDEs and higher-contrast regimes using alternative iterative schemes and architectures.

Abstract

In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential equation (PDE) with applications in various scientific and engineering fields. However, efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber. Recently, deep learning has shown great potential in solving PDEs especially in learning solution operators. Inspired by Neumann series in Helmholtz equation, we design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature. Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60\% lower relative -error, especially in the large wavenumber case, and has 50\% lower computational cost and less data requirement. Moreover, NSNO can be used as the surrogate model in inverse scattering problems. Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.
Paper Structure (33 sections, 2 theorems, 28 equations, 14 figures, 4 tables)

This paper contains 33 sections, 2 theorems, 28 equations, 14 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Neumann Series
  4. Preliminaries
  5. Neumann Series
  6. Network Architecture and Training Process
  7. Network Architecture
  8. Network Architecture of $G_{\bm{\theta}}$
  9. Fourier Neural Operator
  10. U-shaped Neural Operator
  11. Multiple-Input Multiple-Output (MIMO) Encoder
  12. Fourier Layers
  13. Multiple-Input Single-Output (MISO) Decoder
  14. Training Process
  15. Experiments
  16. ...and 18 more sections

Key Result

theorem 1

The variation problem variation has a unique solution in $H^1(\it\Omega)$. Moreover, if the wavenumber $k>1$, there exists a constant $C$, which only depends on the domain $\it\Omega$, such that for any $f\in L^2(\it\Omega)$, the solution to problem variation satisfies

Figures (14)

  • Figure 1: Overall Network Architecture of NSNO
  • Figure 2: Network architecture of FNO
  • Figure 3: Network architecture of UNO
  • Figure 4: Network architecture of the encoding block in UNO
  • Figure 5: Examples of $q$. (a)-(c): T-shaped distribution. (d)-(f): Random circle distribution with the number of circles from 1 to 3. (g)-(i): Smoothed random circle distribution with the number of circles from 1 to 3.
  • ...and 9 more figures

Theorems & Definitions (2)

  • theorem 1
  • theorem 2: Convergence of Neumann series