In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.

翻译：本研究提出了一种新颖的神经网络方法，用于解决在方形晶格上通过离散狄利克雷-诺依曼映射反演网络边电导率分布的离散逆电导率问题。该方法的创新性在于：目标电导率并非直接作为神经网络的输出，而是被编码于训练后神经网络第二层的权重中。因此，训练后神经网络的权重具有明确的物理意义，这与大多数现有神经网络方法中权重通常不可解释的特点形成鲜明对比。本工作为设计具有可解释训练后权重的神经网络迈出了重要一步。数值实验表明，该方法在含噪完整数据与部分数据条件下均优于传统的Curtis-Morrow算法。