Neumann Series-based Neural Operator for Solving Inverse Medium Problem

TL;DR

This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs, offering a scalable solution to traditionally complex inverse problems.

Abstract

The inverse medium problem, inherently ill-posed and nonlinear, presents significant computational challenges. This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance, even with varying scattering properties and noisy data. The robustness and adaptability of our framework provide crucial insights and methodologies, extending its applicability to a broad spectrum of scattering problems. These advancements mark a significant step forward in the field, offering a scalable solution to traditionally complex inverse problems.

Figures (18)

• Figure 1: Schematic diagram of the inverse medium problem.
• Figure 2: Network structure for the implicit approach. (a) $\odot$ refers to the element-by-element multiplication and $\oplus$ denotes the element-wise addition. (b) detail structure of the normalizing operation. $u^i = u_{\boldsymbol{\theta}}^{(0)}, u_{\boldsymbol{\theta}}^{(j)} = u\| qu_{\boldsymbol{\theta}}^{(j-1)}\|_{\infty}, j = 1,2,\cdots,L$.
• Figure 3: Training process for the explicit approach. $u_1,u_2$ are complex-valued network outputs, respectively, $u^{(j)} = u_1 \| \text{real}\|_{\infty} + iu_2 \| \text{imag}\|_{\infty}, j = 1,2,\cdots,L, u^{(0)} = u^i.$
• Figure 4: Comparison of predicted scattered fields at wavenumber k=60 for FNO, NSFNO, and FNONS.
• Figure 5: Network performance on out-of-distribution incident angles at various wavenumbers.

Theorems & Definitions (3)

• Remark 5.1
• Theorem A.1
• proof