PINNs with wavelet excitations recover sharp conductivity from limited data

The Calderón inverse problem seeks to reconstruct internal conductivity from boundary voltage and current measurements—a challenge in medical imaging and geophysics. Traditional solvers require dense data, but this paper tackles the realistic finite-data scenario using physics-informed neural networks (PINNs). The authors propose a novel framework that represents the unknown conductivity and electric potentials with separate neural networks, conditioned on applied boundary excitations. They replace standard sinusoidal inputs with multiscale randomized wavelet functions and incorporate Fourier-feature encoding (FFE) to capture high-frequency details. The elliptic PDE is enforced via physics-informed residuals, and finite Dirichlet-to-Neumann data are used as boundary losses.

Experiments on synthetic conductivity fields with inclusions, sharp interfaces, and heterogeneous media show that the method recovers dominant structures with relative errors between 3% and 12%. FFE significantly improves reconstruction of sharp features like inclusions and interfaces, but raw-coordinate networks are competitive for smoother profiles. This work demonstrates that careful design of boundary excitation (wavelet-based) and coordinate representation (FFE versus raw) are key factors in neural Calderón inversion. The approach could improve electrical impedance tomography (EIT) for medical diagnostics or industrial non-destructive testing, where only limited boundary measurements are available.