Recovering Sharp Conductivity Features in the Finite-Data Calderón Problem with Physics-Informed Neural Networks

2026-06-26

AI Summary

This work addresses the challenge of reconstructing conductivity distributions with sharp features—such as inclusions and interfaces—from limited boundary measurements in the Calderón inverse problem. The authors propose a novel physics-informed neural network (PINN) framework that employs multiscale stochastic wavelet boundary excitations and Fourier feature encoding to separately model the conductivity and electric potential fields. The approach jointly optimizes the residual of the governing elliptic partial differential equation and a finite set of Dirichlet-to-Neumann map data. Experiments on synthetic datasets demonstrate that the method effectively recovers dominant conductivity structures with relative errors ranging from 3% to 12%. Notably, the incorporation of Fourier feature encoding substantially enhances reconstruction accuracy for sharp features, highlighting the critical influence of coordinate representation and excitation design on the performance of neural inversion strategies.

Abstract

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.

Problem

• Calderón problem
• sharp conductivity features
• inverse problem
• limited boundary data
• electrical impedance tomography

Innovation

• Physics-Informed Neural Networks
• Calderón Problem
• Fourier Feature Encoding
• Multiscale Boundary Excitations
• Sharp Conductivity Reconstruction