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arxiv: 2407.17182 · v4 · submitted 2024-07-24 · 💻 cs.LG

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

Anuj Abhishek , Thilo Strauss This is my paper

Pith reviewed 2026-05-23 22:44 UTC · model grok-4.3

classification 💻 cs.LG
keywords electrical impedance tomographyDeepONetoperator learningNeumann-to-Dirichlet operatoruniversal approximationinverse problems
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The pith

DeepONets can approximate maps from Neumann-to-Dirichlet operators to conductivities arbitrarily well.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper treats the inverse problem of recovering conductivity from boundary measurements in electrical impedance tomography as the task of learning a map from Neumann-to-Dirichlet operators to conductivity functions. It employs a Deep Operator Network to learn this map and proves that DeepONets can approximate any such operator-to-function map to arbitrary accuracy. Numerical tests confirm that the learned map produces accurate reconstructions that are robust and generalize well, outperforming the iteratively regularized Gauss-Newton method.

Core claim

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 method. Our results show that the proposed framework yields accurate and robust reconstructions, outperforms the baseline, and demonstrates strong generalization.

What carries the argument

DeepONet architecture adapted to the operator-to-function regime, where the network accepts a Neumann-to-Dirichlet operator as input and produces the corresponding conductivity distribution as output.

If this is right

  • DeepONets supply a direct, non-iterative procedure for recovering conductivities from boundary operators.
  • The universal approximation result ensures that DeepONets of sufficient size can reach any prescribed accuracy for operator-to-function maps.
  • Numerical performance exceeds that of the iteratively regularized Gauss-Newton method in both accuracy and generalization on the tested EIT instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same operator-to-function learning strategy could be applied to other inverse problems that map boundary operators to interior coefficients.
  • The approximation theorem may be specialized to yield explicit convergence rates once the function spaces native to EIT are fixed.
  • Training-data generation via forward solvers could be tuned to enlarge the region of reliable generalization beyond the simulated cases shown.

Load-bearing premise

The inverse map from Neumann-to-Dirichlet operators to admissible conductivities exists as a learnable function that finite samples can approximate.

What would settle it

A set of Neumann-to-Dirichlet operators for which the trained DeepONet produces conductivity outputs that deviate substantially from the true distributions, even when the training set is dense in the relevant operator space.

Figures

Figures reproduced from arXiv: 2407.17182 by Anuj Abhishek, Thilo Strauss.

Figure 1
Figure 1. Figure 1: The true map G † ext is approximated by a composition of three maps, encoder E, approximator A and reconstructor R. The resultant error in the approximation thus comprises of encoder, approximator, and reconstructor errors. and they have been used effectively in solving several inverse problems, [55]. With this background in mind, and to set the stage for formulating our learning problem as learning an ope… view at source ↗
Figure 2
Figure 2. Figure 2: DeepONet architecture used in this manuscript. It feeds the EIT measurements [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of reconstructions of our DeepONet with the classical IRGN [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper formulates the EIT inverse problem as learning the map from Neumann-to-Dirichlet operators to admissible conductivities via DeepONets, extending operator learning to the operator-to-function setting. It states a universal approximation theorem guaranteeing arbitrary approximation of such maps by DeepONets, implements the approach numerically, and reports that it yields accurate reconstructions, outperforms the iteratively regularized Gauss-Newton method, and generalizes well.

Significance. If the universal approximation result holds under a suitable topology on the space of N-t-D operators and the numerical comparisons are robust, the work would supply the first rigorous approximation-theoretic justification for DeepONet-based inversion in EIT, strengthening the case for operator networks in nonlinear inverse problems.

major comments (2)
  1. [§3] §3 (Universal Approximation Theorem): the statement that DeepONets can approximate operator-to-function maps arbitrarily well invokes the standard DeepONet UAT without specifying the topology (e.g., operator norm on suitable Sobolev spaces) under which the set of admissible N-t-D operators is compact or the target map is continuous; without this, the reduction from the classical function-to-function case does not transfer directly.
  2. [§4] §4 (Numerical Experiments): the reported outperformance over IRGN lacks error bars, details on the distribution of training N-t-D operators, or ablation on the branch-network encoding of the full operator input; these omissions make it impossible to assess whether the claimed robustness and generalization are load-bearing or sensitive to post-hoc choices.
minor comments (2)
  1. [Introduction] Notation for the N-t-D operator and the conductivity space should be introduced with explicit function-space assumptions (e.g., H^{1/2} boundary data) already in the introduction rather than deferred to §2.
  2. [Figures] Figure captions for the reconstruction examples should state the conductivity contrast range and the number of boundary electrodes used, to allow direct comparison with the IRGN baseline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (Universal Approximation Theorem): the statement that DeepONets can approximate operator-to-function maps arbitrarily well invokes the standard DeepONet UAT without specifying the topology (e.g., operator norm on suitable Sobolev spaces) under which the set of admissible N-t-D operators is compact or the target map is continuous; without this, the reduction from the classical function-to-function case does not transfer directly.

    Authors: We thank the referee for this observation. Our universal approximation result applies the standard DeepONet theorem to the operator-to-function setting by viewing the N-t-D operator as the input in an appropriate function space. In the EIT context, the admissible N-t-D operators form a compact set in the operator norm on bounded linear operators between Sobolev spaces H^{1/2}(∂Ω) and H^{-1/2}(∂Ω), with continuity of the map to conductivities following from the well-posedness results under the given admissibility conditions on the conductivity. We will revise §3 to explicitly name this topology and add a short justification paragraph, making the reduction from the classical case fully rigorous. revision: yes

  2. Referee: [§4] §4 (Numerical Experiments): the reported outperformance over IRGN lacks error bars, details on the distribution of training N-t-D operators, or ablation on the branch-network encoding of the full operator input; these omissions make it impossible to assess whether the claimed robustness and generalization are load-bearing or sensitive to post-hoc choices.

    Authors: We agree that these details will improve transparency. In the revised manuscript we will add error bars obtained from repeated training runs with different random seeds. We will also specify the exact sampling procedure and parameter ranges used to generate the training collection of N-t-D operators. Finally, we will include a short ablation comparing the branch-network input encoding (discretized matrix versus functional representation) to confirm that the reported performance is not an artifact of a single encoding choice. These additions will allow readers to evaluate the robustness claims directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; UAT and numerical results are independent of inputs

full rationale

The paper formulates the EIT inverse problem as learning an operator-to-function map, states a universal approximation theorem for DeepONets in this regime, and reports numerical comparisons against the external IRGN baseline. No quoted equations or claims reduce the UAT statement, the map approximation, or the reported performance gains to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The derivation chain remains self-contained against external benchmarks and does not invoke load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a well-defined, learnable operator-to-function map in EIT and on the applicability of DeepONet universal approximation theory to that map; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption There exists a well-defined map from Neumann-to-Dirichlet operators to admissible conductivities that can be learned from data.
    Invoked when the inverse problem is reformulated as an operator-learning task.

pith-pipeline@v0.9.0 · 5757 in / 1208 out tokens · 24378 ms · 2026-05-23T22:44:15.081869+00:00 · methodology

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Works this paper leans on

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    Introduction. In recent years tremendous progress has been made in de- veloping accurate, non-invasive medical imaging techniques for disease detection and diagnosis and to make it suitable for practical deployment. One such medical imaging technology is Electrical Impedance Tomography (EIT). EIT is a functional imaging technique that seeks to reconstruct...

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    The Learning Problem for EIT. Our goal in this article is to propose a DeepONet based neural-network architecture for learning to invert Neumann-to- Dirichlet operator relevant to the problem of EIT. To formally set up the learn- ing problem, let us consider the shifted abstract N-t-D operator ˜Λγ = Λ γ − Λ1, where Λ 1 is the background N-t-D operator. An...

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