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：在本文中，我们开发了一种新的方法，使用深度神经网络从一个内部测量中重建椭圆问题中的传导率分布。该方法基于混合重构的控制方程，并利用标准的最小二乘目标函数同时逼近传导率和通量，采用深度神经网络作为解答函数。我们对连续和经验损失的神经网络近似进行了彻底的分析，包括严格的误差估计，这些误差估计明确地与噪声水平、各种惩罚参数和神经网络架构参数（深度、宽度和参数边界）有关。我们还提供了广泛的二维和多维数值实验，以说明该方法的不同特点，例如，对数据噪声的极佳稳定性和解决高维问题的能力。