Extension and neural operator approximation of the electrical impedance tomography inverse map

TL;DR

This paper develops a neural operator approach for the noise-robust solution of the electrical impedance tomography inverse problem, extending the inverse map to a Hilbert space and demonstrating effective reconstructions with Fourier neural operators.

Contribution

It introduces a novel extension of the inverse map to a Hilbert space enabling neural operator approximation and applies Fourier neural operators to noisy EIT reconstructions.

Findings

Fourier neural operators effectively reconstruct conductivities in noisy EIT scenarios.

The extended inverse map maintains stability properties similar to the original.

Numerical experiments show successful reconstructions beyond theoretical assumptions.

Abstract

This paper considers the problem of noise-robust neural operator approximation for the solution map of Calder\'on's inverse conductivity problem. In this continuum model of electrical impedance tomography (EIT), the boundary measurements are realized as a noisy perturbation of the Neumann-to-Dirichlet map's integral kernel. The theoretical analysis proceeds by extending the domain of the inversion operator to a Hilbert space of kernel functions. The resulting extension shares the same stability properties as the original inverse map from kernels to conductivities, but is now amenable to neural operator approximation. Numerical experiments demonstrate that Fourier neural operators excel at reconstructing infinite-dimensional piecewise constant and lognormal conductivities in noisy setups both within and beyond the theory's assumptions. The methodology developed in this paper for EIT exemplifies a broader strategy for addressing nonlinear inverse problems with a noise-aware operator learning framework.