Construct deep neural networks based on direct sampling methods for solving electrical impedance tomography

This work investigates the electrical impedance tomography problem when only limited boundary measurements are available, which is known to be challenging due to the extreme ill-posedness. Based on the direct sampling method (DSM) introduced in [Y. T. Chow, K. Ito, and J. Zou, Inverse Problems, 30 (2016), 095003], we propose deep direct sampling methods (DDSMs) to locate inhomogeneous inclusions in which two types of deep neural networks (DNNs) are constructed to approximate the index function (functional): fully connected neural networks and convolutional neural networks. The proposed DDSMs are easy to be implemented, capable of incorporating multiple Cauchy data pairs to achieve high-quality reconstruction and highly robust with respect to large noise. Additionally, the implementation of DDSMs adopts offline-online decomposition, which helps to reduce a lot of computational costs and makes DDSMs as efficient as the conventional DSM proposed by Chow, Ito, and Zou. The numerical experiments are presented to demonstrate the efficacy and show the potential benefits of combining DNN with DSM.