A learning-based hybrid reconstruction scheme for an inverse boundary value problem

Abstract We investigate the problem of recovering unknown inclusions in a heat conductor from boundary measurements, formulated mathematically as an inverse boundary value problem for the heat equation. A sampling-type approach, the linear sampling method (LSM), has been successfully applied to tackle this problem. This method is easy to implement and generates reasonable reconstructions using a well-designed indicator function. However, several practical limitations of the LSM remain evident. First, the decision boundary determined by the threshold of the indicator function is ambiguous and the threshold often varies with different inclusions. Second, high-resolution reconstructions require the highly accurate Neumann-to-Dirichlet map on the entire boundary, especially in cases involving multiple inclusions. Finally, although the time-dependent setting naturally provides multi-temporal indicator functions, the conventional LSM lacks a systematic strategy to integrate them effectively. To address these issues, we propose a learning-based hybrid strategy to realize high-quality reconstructions. Specifically, the LSM is first used to obtain rough reconstructions, which serve as noisy versions of true inclusions. Motivated by the remarkable success of deep learning in image processing, we then utilize the U-net neural network, which extracts the dominant features of inclusions, to refine the reconstructions. The network incorporates the total variation regularization and supports multi-temporal indicator functions as multi-channel inputs to exploit temporal information. Various numerical experiments demonstrate that the proposed method successfully overcomes the fundamental limitations of the LSM. We highlight that this framework, which combines physics-based feature extraction with learning-based post-processing, is extendable to various inverse problems for PDEs models. It is particularly applicable for time-dependent equations, where multi-temporal measurements naturally provide complementary information for data-driven refinement.