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

2025-11-25

AI Summary

This work addresses the robust solution of the Calderón inverse conductivity problem in electrical impedance tomography (EIT) under noisy measurements. Methodologically, we propose a noise-aware operator learning framework based on neural operators. Our core innovation lies in extending the original nonlinear inverse operator to a reproducing kernel Hilbert space of integral kernels—preserving mapping stability while enabling efficient neural operator approximation. Specifically, we employ the Fourier neural operator to parameterize the continuous integral kernel and integrate theoretical guarantees from operator approximation under noise perturbations, achieving end-to-end reconstruction of infinite-dimensional piecewise-constant and log-normal conductivity fields. Experiments demonstrate high reconstruction accuracy under strong noise and superior generalization over classical regularization methods. The approach provides a data-driven paradigm for nonlinear inverse problems that is both theoretically grounded—via stability and approximation guarantees—and practically effective.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Approximating electrical impedance tomography inverse map with neural operators
• Extending inversion domain to handle noisy boundary measurement perturbations
• Reconstructing infinite-dimensional conductivities robustly in noisy experimental setups

Innovation

• Extending inverse operator domain to Hilbert space
• Using Fourier neural operators for approximation
• Developing noise-aware operator learning framework