The learned range test method for the inverse inclusion problem
This work addresses a specific inverse problem in applied mathematics, offering incremental improvements for domain sampling methods in inclusion reconstruction.
The paper tackles the inverse inclusion problem by reconstructing an inclusion from Cauchy data, proposing a learned range test method that combines a neural network with a pre-trained classifier to improve accuracy and stability. Numerical simulations show superior results compared to standard and purely data-driven methods, with accurate reconstructions of polygonal inclusions.
We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $Ω\subset\mathbb{R}^d$ from a single pair of Cauchy data $(u|_{\partialΩ},\partial_νu|_{\partialΩ})$, where $Δu=0$ in $Ω\setminus\overline B$ 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.