In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.

翻译：本文提出了一种新颖算法，称为混合最小二乘深度神经网络（MLS-DNN），用于从解的内部测量数据中恢复各向异性电导率张量。该方法基于将最小二乘公式应用于椭圆问题的混合形式，并同时使用深度神经网络逼近内部通量和电导率张量。我们通过总体损失和经验损失对所得近似值提供了误差界。该分析依赖于典型源条件、深度神经网络的逼近理论以及统计学习理论。我们还通过多个数值实验说明了该方法的性能，并与标准伽辽金有限元方法和物理信息神经网络进行了比较研究。结果表明，该方法能够在二维和三维情况下准确恢复各向异性电导率，数据中噪声水平最高可达10%。