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

arXiv: https://arxiv.org/abs/2407.17182 · HTML: https://mlsanity.com/paper/2407.17182/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 78 references that are in the map.
- First posted on arXiv 24 Jul 2024.

## Builds on (most cited first, 21 of 21)

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- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Texture Fields: Learning Texture Representations in Function Space](https://mlsanity.com/paper/1905.07259/) (2019)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Deep D-bar: Real time Electrical Impedance Tomography Imaging with Deep Neural Networks](https://mlsanity.com/paper/1711.03180/) (2017)
- [Learning Implicit Surface Light Fields](https://mlsanity.com/paper/2003.12406/) (2020)
- [Solving Electrical Impedance Tomography with Deep Learning](https://mlsanity.com/paper/1906.03944/) (2019)
- [Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators](https://mlsanity.com/paper/2308.05141/) (2023)
- [A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling](https://mlsanity.com/paper/2301.11402/) (2023)
- [Beltrami-Net: Domain Independent Deep D-bar Learning for Absolute Imaging with Electrical Impedance Tomography (a-EIT)](https://mlsanity.com/paper/1811.12830/) (2018)
- [Classification of stroke using Neural Networks in Electrical Impedance Tomography](https://mlsanity.com/paper/2004.06546/) (2020)

## Abstract (arXiv)

In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography (EIT), where the goal is to recover the conductivity in a medium from boundary current-to-voltage measurements, i.e., the Neumann-to-Dirichlet (N--t--D) operator. We formulate this inverse problem as an operator-learning task, where the aim is to approximate the implicitly defined map from N--t--D operators to admissible conductivities. To this end, we employ a Deep Operator Network (DeepONet) architecture, thereby extending operator learning beyond the classical function-to-function setting to the more challenging operator-to-function regime. We establish a universal approximation theorem that guarantees that such operator-to-function maps can be approximated arbitrarily well by DeepONets. Furthermore, we provide a computational implementation of our approach and compare it against the iteratively regularized Gauss--Newton (IRGN) method. Our results show that the proposed framework yields accurate and robust reconstructions, outperforms the baseline, and demonstrates strong generalization. To our knowledge, this is the first work that combines rigorous approximation-theoretic guarantees with DeepONet-based inversion for EIT, thereby opening a principled and interpretable pathway for use of DeepONets in such inverse problems.