We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $\Omega\subset\mathbb{R}^d$ from a single pair of Cauchy data $(u|_{\partial\Omega},\partial_\nu u|_{\partial\Omega})$, where $\Delta u=0$ in $\Omega\setminus\overline B$ and $u=0$ on $\partial B$. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.

翻译：我们考虑一个反问题：从单对柯西数据$(u|_{\partial\Omega},\partial_\nu u|_{\partial\Omega})$重构有界域$\Omega\subset\mathbb{R}^d$内的夹杂$B$，其中在$\Omega\setminus\overline B$内满足$\Delta u=0$且在$\partial B$上满足$u=0$。我们证明了基于区域测试（一种区域采样方法）的重构算法可以表示为具有特定架构的神经网络。我们提出在监督学习框架下学习该网络的权重，并将其与预训练的分类器相结合，以根据夹杂物与边界的距离对其进行区分。数值模拟表明，这种学习型区域测试方法能够准确且稳定地重构多边形夹杂。此外，其结果优于标准区域测试方法（无学习）以及端到端的全连接深度神经网络（一种纯数据驱动方法）。