In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with deep neural networks as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the deep neural network approximations of the conductivity 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 multiple 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：本研究提出一种利用深度神经网络从全域单次解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构，采用标准最小二乘目标函数，以深度神经网络作为拟设函数同时逼近电导率和通量。我们针对连续损失和经验损失两种情形，系统分析了深度神经网络对电导率的逼近性能，给出了包含噪声水平、各类惩罚参数及神经网络架构参数（深度、宽度和参数界）显式表达的严格误差估计。我们还在二维及多维空间中开展了多项数值实验，验证了该方法对数据噪声具有优异稳定性且能够求解高维问题等显著特征。