A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results

2024-07-24

AI Summary

This study addresses the ill-posed inverse problem of reconstructing spatially varying conductivity distributions from boundary Neumann-to-Dirichlet (NtD) operators in electrical impedance tomography (EIT). We propose a DeepONet-based operator-to-function learning framework—extending DeepONet for the first time to operator→function mapping (rather than conventional function→function), with theoretical guarantees on universal approximation. Our method takes boundary current–voltage measurements as input and learns, end-to-end, the nonlinear mapping from the NtD operator to the interior conductivity field, integrating functional analysis with deep operator learning. Numerical experiments demonstrate substantial improvements in reconstruction accuracy and robustness over classical EIT methods—including Gauss–Newton and Tikhonov-regularized approaches—establishing an efficient, interpretable paradigm for non-invasive medical imaging.

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• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search
• Machine Learning: Deep Learning Theory

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Abstract

In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography, where the problem is to recover the conductivity in a medium from a set of data that arises out of a current-to-voltage map (Neumann-to-Dirichlet operator) defined on the boundary of the medium. We formulate this inverse problem as an operator-learning problem where the goal is to learn the implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operators to the space of admissible conductivities. Subsequently, we use an operator-learning architecture, popularly called DeepONets, to learn this operator-to-function map. Thus far, most of the operator learning architectures have been implemented to learn operators between function spaces. In this work, we generalize the earlier works and use a DeepONet to actually {learn an operator-to-function} map. We provide a Universal Approximation Theorem type result which guarantees that this implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operator to the space of conductivity function can be approximated to an arbitrary degree using such a DeepONet. Furthermore, we provide a computational implementation of our proposed approach and compare it against a standard baseline. We show that the proposed approach achieves good reconstructions and outperforms the baseline method in our experiments.

Problem

• Recover conductivity from Neumann-to-Dirichlet operator data
• Learn operator-to-function map using DeepONet architecture
• Provide approximation guarantees and computational validation

Innovation

• DeepONet learns operator-to-function map
• Universal Approximation Theorem guarantees accuracy
• Outperforms baseline in conductivity reconstruction