In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography (EIT), where the goal is to recover the conductivity in a medium from boundary current-to-voltage measurements, i.e., the Neumann-to-Dirichlet (N--t--D) operator. We formulate this inverse problem as an operator-learning task, where the aim is to approximate the implicitly defined map from N--t--D operators to admissible conductivities. To this end, we employ a Deep Operator Network (DeepONet) architecture, thereby extending operator learning beyond the classical function-to-function setting to the more challenging operator-to-function regime. We establish a universal approximation theorem that guarantees that such operator-to-function maps can be approximated arbitrarily well by DeepONets. Furthermore, we provide a computational implementation of our approach and compare it against the iteratively regularized Gauss--Newton (IRGN) method. Our results show that the proposed framework yields accurate and robust reconstructions, outperforms the baseline, and demonstrates strong generalization. To our knowledge, this is the first work that combines rigorous approximation-theoretic guarantees with DeepONet-based inversion for EIT, thereby opening a principled and interpretable pathway for use of DeepONets in such inverse problems.

翻译：本文研究了电阻抗成像（EIT）这一无创医学影像模态，其目标是通过边界电流-电压测量（即诺伊曼-狄利克雷（N-t-D）算子）恢复介质内的电导率分布。我们将该反问题表述为算子学习任务，旨在逼近从N-t-D算子到可容许电导率的隐式定义映射。为此，我们采用深度算子网络（DeepONet）架构，从而将算子学习从经典的函数到函数设定拓展至更具挑战性的算子到函数范畴。我们建立了普适逼近定理，保证了此类算子到函数映射可由DeepONet实现任意精度逼近。此外，我们给出了所提方法的计算实现，并与迭代正则化高斯-牛顿（IRGN）方法进行了对比。结果表明，所提框架能生成准确且鲁棒的重建结果，优于基线方法，并展现出强泛化能力。据我们所知，这是首个将严格近似理论保证与基于DeepONet的EIT反演相结合的工作，从而为DeepONet在此类反问题中的应用开辟了一条具备原理性与可解释性的新路径。