用神经算子稳定重建含噪电导率，突破传统反问题瓶颈
Extension and neural operator approximation of the electrical impedance tomography inverse map
- 将反问题算子延拓至希尔伯特空间，实现噪声鲁棒的神经算子逼近
- 傅里叶神经算子在含噪条件下准确重构分段常数与对数正态电导率
- 为非线性反问题提供可推广的噪声感知学习框架，适合逆问题研究者
本文研究卡尔德隆反电导率问题解映射的噪声鲁棒神经算子近似。在连续体模型的电学阻抗断层成像（EIT）中，边界测量被建模为诺伊曼到狄利克雷映射积分核的含噪扰动。理论分析通过将反演算子定义域扩展至核函数的希尔伯特空间，保持了原反映射从核到电导率的稳定性，同时使算子可被神经算子逼近。数值实验表明，傅里叶神经算子在含噪场景下，无论是否满足理论假设，均能有效重构无限维的分段常数及对数正态电导率。本文提出的EIT方法展示了针对非线性反问题的噪声感知算子学习通用策略。
原文摘要 · Abstract (English)
This paper considers the problem of noise-robust neural operator approximation for the solution map of Calderón's inverse conductivity problem. In this continuum model of electrical impedance tomography (EIT), the boundary measurements are realized as a noisy perturbation of the Neumann-to-Dirichlet map's integral kernel. The theoretical analysis proceeds by extending the domain of the inversion operator to a Hilbert space of kernel functions. The resulting extension shares the same stability properties as the original inverse map from kernels to conductivities, but is now amenable to neural operator approximation. Numerical experiments demonstrate that Fourier neural operators excel at reconstructing infinite-dimensional piecewise constant and lognormal conductivities in noisy setups both within and beyond the theory's assumptions. The methodology developed in this paper for EIT exemplifies a broader strategy for addressing nonlinear inverse problems with a noise-aware operator learning framework.
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