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

TL;DR

This work addresses Calderón's inverse problem in electrical impedance tomography under noisy boundary data by formulating an operator-learning approach that treats the inverse map as a function of functions. The authors extend the inverse map to a Hilbert-space setting of kernel functions derived from Neumann-to-Dirichlet data and prove that a Fourier neural operator can uniformly approximate this extended inverse on compact, noise-perturbed inputs. They establish stability and compactness properties, develop a boundary-manifold patching scheme for neural operators, and prove an existence result for a noise-robust FNO that reconstructs conductivities from NtD kernels up to the noise level. Numerical experiments on shape detection, three-phase inclusions, and lognormal conductivities demonstrate accurate, fast reconstructions under substantial noise, with robustness beyond the theoretical assumptions. Overall, the paper provides a principled framework for data-driven solution of nonlinear inverse problems via noise-aware neural operators and suggests a general pathway for extending these ideas to other PDE-based inverses.

Abstract

This paper considers the problem of noise-robust neural operator approximation for the solution map of Calderón'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.

Figures (15)

• Figure 1: FNO reconstruction of three discontinuous conductivities as training dataset size $N$ increases. D-bar method knudsen2009regularized reconstructions are shown in comparison. The noise level is $1\%$.
• Figure 2: An overview of \ref{['thm:fno_approx_main']}'s proof structure.
• Figure 3: Sample reconstructions from the shape detection dataset.
• Figure 4: Shape detection noise robustness (top row) and sample complexity (bottom row) experiments. For training dataset size $N$, \ref{['subfig:noise_sweep_two_phase_noisy']} shows how the average relative $L^1(\mathbb{D})$ test error $\mathrm{Err}_{\delta, N}$ increases sublinearly as the training and test data noise level $\delta$ increases. \ref{['subfig:noise_log_two_phase_noisy']} plots the excess error versus $\log(1/\delta)$ to validate the logarithmic bound in \ref{['thm:fno_approx_main']}. \ref{['subfig:data_sweep_two_phase_noisy']} shows how $\mathrm{Err}_{\delta, N}$ decreases algebraically with $N$. \ref{['subfig:data_power_two_phase_noisy']} plots the excess error versus $N$ to obtain precise estimates of the convergence rate (\ref{['tab:rates_all']}). Purple lines are linear least squares fits on a logarithmic scale. Shaded bands denote two standard deviations from the mean (excess) error over five independent training runs.
• Figure 5: Sample reconstructions from the three phase inclusion dataset.

Theorems & Definitions (48)

• Theorem 3.1: Benyamin--Lindenstrauss extension
• Theorem 3.2: $L^2$ stability of the Calderón problem
• proof
• Remark 1: stronger metrics on the set of conductivities
• Lemma 3.3: NtD to DtN
• proof
• Lemma 3.4: uniform bounds on NtD maps
• Lemma 3.5: NtD inverse inequality: Hilbert--Schmidt norm to operator norm
• proof
• Lemma 3.6: NtD maps and Hilbert--Schmidt integral operators