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THE LEARNED RANGE TEST METHOD FOR THE INVERSE INCLUSION PROBLEM.

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Title: THE LEARNED RANGE TEST METHOD FOR THE INVERSE INCLUSION PROBLEM.
Authors: SUN, SHIWEI, ALBERTI, GIOVANNI S.
Source: SIAM Journal on Applied Mathematics; 2025, Vol. 85 Issue 4, p1881-1905, 25p
Subject Terms: ARTIFICIAL neural networks, INVERSE problems, CAUCHY problem, COMPUTER simulation, IMAGE reconstruction algorithms, SUPERVISED learning
Abstract: We consider the inverse problem consisting of the reconstruction of an inclusion B contained in a bounded domain Ω ⊂ Rd from a single pair of Cauchy data (...), where Δu = 0 in ... and u = 0 on ∂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 pretrained 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. [ABSTRACT FROM AUTHOR]
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Abstract:We consider the inverse problem consisting of the reconstruction of an inclusion B contained in a bounded domain Ω ⊂ R<sup>d</sup> from a single pair of Cauchy data (...), where Δ<subscript>u</subscript> = 0 in ... and u = 0 on ∂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 pretrained 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. [ABSTRACT FROM AUTHOR]
ISSN:00361399
DOI:10.1137/24M170716X