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

TL;DR

This work tackles reconstructing an unknown coefficient $k$ in a 2D elliptic PDE from boundary data using the Dirichlet-to-Neumann map. It introduces a self-supervised framework that couples a neural network for $\tilde{k}$ with a FEM forward solve in the inner loop, forming a differentiable pipeline that minimizes Neumann-data mismatches on the observed boundary. A key contribution is the integration of physics-based FEM with learning, including adjoint-based gradients and loss designs that handle discontinuities and partial boundary data, plus theoretical discussion of DtN-based uniqueness. The approach demonstrates robust, high-fidelity reconstructions across constant, spatially varying, phantom, and discontinuous coefficients, under fully and partially observed data, highlighting practical applicability to boundary-only inverse problems in imaging and subsurface contexts.

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 present a new approach to solving the inverse problem for elliptic PDEs, using only boundary data. Our method leverages the Dirichlet-to-Neumann (DtN) map, which captures the relationship between boundary inputs and flux responses. This enables the reconstruction of the unknown physical properties within the domain from boundary measurements alone. Our framework employs a self-supervised machine learning algorithm that integrates a Finite Element Method (FEM) in the inner loop for the forward problem, ensuring high accuracy. Moreover, our approach illustrates its effectiveness in challenging scenarios with only partial boundary observations, which is often the case in real-world scenarios. In addition, the proposed algorithm effectively handles discontinuities by incorporating carefully designed loss functions. 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.

Figures (23)

• Figure 1: Illustration of the problem. The dashed line indicates the partially observed boundary measurement.
• Figure 2: Framework for reconstructing $k$ using a neural network and FEM.
• Figure 3: Example 1-1. Training dynamics and reconstruction accuracy. (a) Neumann data loss (blue, left axis) and relative error of $\tilde{k}$ (red, right axis) over training epochs (log scale). Each point represents the averaged value over a 10-epoch interval. (b) Absolute error distribution between the exact coefficient $k$ and the reconstructed coefficient $\tilde{k}$.
• Figure 4: Example 1-1. Illustration of the Dirichlet-to-Neumann (DtN) map. The solution $p(x,y)$ is shown and the Dirichlet boundary values $p(x,1)$ and $p(1,y)$ mapped to their corresponding Neumann data via the DtN map are presented.
• Figure 5: Example 1-2. (Left) Boundary regions and (Right) Dirichlet boundary condition $g$. The values for which $g=0$ indicate that no data is available.

Theorems & Definitions (1)

• Remark 1