A Machine Learning and Finite Element Framework for Inverse Elliptic PDEs via Dirichlet-to-Neumann Mapping

Abstract: Inverse problems for partial differential equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In this work, we address the inverse problem for elliptic PDEs by leveraging the Dirichlet-to-Neumann (DtN) map, which captures the relationship between boundary inputs and flux responses. Thus, this approach enables to solve the inverse problem that seeks the material properties inside the domain by utilizing the boundary data. Our framework employs an unsupervised machine learning algorithm that integrates a finite element method (FEM) in the inner loop for the forward problem, ensuring high accuracy. Moreover our approach is flexible to utilize partial observations of the boundary data, which is often the case in real-world scenarios. By incorporating carefully designed loss functions that accommodate discontinuities, the method refines coefficient reconstructions iteratively. This combined FEM and machine learning approach offers a robust, accurate solution strategy for a broad range of inverse problems, enabling improved estimation of critical parameters in applications from medical diagnostics to subsurface exploration.