In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one internal measurement. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective to approximate the conductivity and flux simultaneously, with deep neural networks as ansatz functions. We provide a thorough analysis of the neural network approximations for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters and neural network architectural parameters (depth, width and parameter bound). We also provide extensive numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

翻译：本文提出了一种利用深度神经网络从单一内部测量中重建椭圆问题电导率分布的新方法。该方法基于控制方程的混合重新表述，并利用标准最小二乘目标函数同时逼近电导率和通量，其中深度神经网络作为拟设函数。我们对连续损失和经验损失的神经网络逼近进行了深入分析，包括以噪声水平、各种惩罚参数及神经网络架构参数（深度、宽度和参数界限）显式表达的严格误差估计。此外，我们还在二维及多维空间中开展了大量数值实验，以说明该方法的显著特征，例如对数据噪声具有优异的稳定性以及解决高维问题的能力。