# The learned range test method for the inverse inclusion problem

arXiv: https://arxiv.org/abs/2411.00463 · HTML: https://mlsanity.com/paper/2411.00463/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 71 references that are in the map.
- First posted on arXiv 1 Nov 2024.

## Builds on (most cited first, 9 of 9)

- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography](https://mlsanity.com/paper/2006.01620/) (2020)
- [Convex regularization in statistical inverse learning problems](https://mlsanity.com/paper/2102.09526/) (2021)
- [A Direct Sampling-Based Deep Learning Approach for Inverse Medium Scattering Problems](https://mlsanity.com/paper/2305.00250/) (2023)
- [Electrical Impedance Tomography with Deep Calder'on Method](https://mlsanity.com/paper/2304.09074/) (2023)
- [Learning a Gaussian Mixture for Sparsity Regularization in Inverse Problems](https://mlsanity.com/paper/2401.16612/) (2024)

## Abstract (arXiv)

We consider the inverse problem consisting of the reconstruction of an inclusion B contained in a bounded domain Omegasubsetℝd from a single pair of Cauchy data (u|partialOmega,partial_nu u|partialOmega), where Delta u=0 in Omegasetminusoverline 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.