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arxiv: 2606.28158 · v1 · pith:SXP3PC4Knew · submitted 2026-06-26 · 💻 cs.LG · cs.NA· eess.IV· math.NA

Recovering Sharp Conductivity Features in the Finite-Data Calder\'on Problem with Physics-Informed Neural Networks

Ali AlHadi Kalout , Pablo Tejerina-P\'erez , Konstantin Karchev , Pedro Taranc\'on-\'Alvarez , Leonid Sarieddine , Raul Jimenez , Max Engelstein , Guy David This is my paper

Pith reviewed 2026-06-29 04:43 UTC · model grok-4.3

classification 💻 cs.LG cs.NAeess.IVmath.NA
keywords physics-informed neural networksCalderón problemconductivity reconstructionFourier feature encodingDirichlet-to-Neumann mapinverse problemselectrical impedance tomographywavelet excitations
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The pith

Physics-informed neural networks recover dominant conductivity structures from finite boundary measurements with 3-12% relative error.

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

The paper develops a neural network approach to the Calderón inverse problem of recovering an unknown conductivity inside a domain from electrical boundary measurements. Separate networks represent the conductivity and the electric potentials, with the elliptic PDE enforced through interior residuals and the available Dirichlet-to-Neumann data matched at the boundary. Randomized wavelet functions generate multiscale boundary excitations, and Fourier feature encodings are tested to capture sharp conductivity changes. On synthetic test cases the method reconstructs dominant features including inclusions and interfaces.

Core claim

The central claim is that representing conductivity and potentials with separate neural networks conditioned on randomized wavelet boundary excitations, then minimizing a physics-informed loss that combines elliptic PDE residuals with finite boundary DtN losses, reconstructs conductivity fields containing inclusions, sharp interfaces, smooth profiles, and heterogeneous media from limited data, with relative errors of 3% to 12%. Fourier feature encoding improves recovery of localized sharp features while raw-coordinate networks remain competitive for smoother fields.

What carries the argument

Separate neural networks for conductivity and the family of electric potentials, conditioned on boundary excitations, with physics-informed residuals enforcing the elliptic conductivity equation and Fourier feature encoding to represent sharp spatial variations.

If this is right

  • Fourier feature encoding improves reconstruction of localized sharp features such as inclusions and interfaces.
  • Raw-coordinate networks perform competitively when the conductivity field is smoother.
  • Multiscale wavelet boundary excitations enable recovery of dominant structures from limited data.
  • The framework applies to conductivity fields that include both sharp and smooth components.

Where Pith is reading between the lines

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

  • The relative performance of Fourier versus raw encodings indicates that input representation choice is critical when neural networks approximate discontinuous inverse-problem solutions.
  • The same separation of conductivity and potential networks could be tested on other elliptic inverse problems that supply only partial boundary maps.
  • If the synthetic-data assumption holds in practice, the method supplies a route to hybrid physics-ML solvers for electrical impedance tomography.

Load-bearing premise

The synthetic data generated by the finite-difference forward solver provides an accurate representation of the finite Dirichlet-to-Neumann map for the elliptic conductivity problem.

What would settle it

Reconstructing a known conductivity field from boundary data generated by a different numerical solver or from laboratory measurements and checking whether the relative error stays inside the reported 3-12% range.

Figures

Figures reproduced from arXiv: 2606.28158 by Ali AlHadi Kalout, Guy David, Konstantin Karchev, Leonid Sarieddine, Max Engelstein, Pablo Tejerina-P\'erez, Pedro Taranc\'on-\'Alvarez, Raul Jimenez.

Figure 1
Figure 1. Figure 1: Examples of wavelet-based BCs used in the data generation process. [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: PINN architecture used for conductivity reconstruction. The potential network [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructed conductivities for sharp, discontinuous profiles. We compare the [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the true data and PINN predictions for a representative [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstructed conductivities for smooth profiles. We compare the ground [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstructed conductivities for heterogeneous profiles with random Gaussian noise in the frequency domain, denoted “random clouds”. We compare the ground through (left column) with PINN reconstructions with FFE (middle column) and without FFE (right column). The random cloud cases in [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of scale-dependent, radially averaged, normalized power spectra [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Metrics for the recovery of the conductivity [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Metrics for the recovery of the conductivity [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calder\'on inverse problem from limited boundary data. In this work, we revisit neural Calder\'on inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calder\'on inversion.

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

1 major / 2 minor

Summary. The paper introduces a PINN framework for the Calderón inverse problem that represents conductivity and electric potentials with separate networks conditioned on boundary excitations. It employs randomized wavelet-based multiscale excitations, enforces the elliptic PDE via physics residuals, and matches finite DtN data through boundary losses. Synthetic data from a finite-difference solver is used to test recovery on inclusions, sharp interfaces, smooth profiles, and heterogeneous media, claiming 3–12% relative errors with FFE aiding localized sharp features (though not universally optimal).

Significance. If the central empirical claims hold after addressing discretization consistency, the work would provide concrete evidence that coordinate representations and excitation design matter for neural Calderón inversion on limited boundary data. The tests across multiple conductivity classes and the observation that raw-coordinate networks can compete for smooth fields are useful empirical contributions.

major comments (1)
  1. [§3 (data generation) and §2.2 (loss)] The evaluation uses boundary data generated by a finite-difference forward solver while the physics-informed loss enforces the continuous elliptic conductivity PDE. This mismatch is especially relevant near discontinuities, where FD truncation and numerical diffusion are absent from the residual; the reported improvement from FFE on sharp inclusions and interfaces may therefore partly reflect fitting of discrete artifacts rather than genuine recovery of the continuous inverse problem. (See data-generation paragraph in §3 and the residual formulation in §2.2.)
minor comments (2)
  1. [Abstract and §4] The abstract states relative errors of 3%–12% without error bars, baseline comparisons, or details on data exclusion / hyperparameter sensitivity; these should be added to the results section for reproducibility.
  2. [§2.1] Notation for the family of potentials and the conditioning on excitations is introduced without an explicit equation reference; a single clarifying equation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment highlighting the discretization consistency issue. We address it directly below and outline targeted revisions.

read point-by-point responses

  1. Referee: [§3 (data generation) and §2.2 (loss)] The evaluation uses boundary data generated by a finite-difference forward solver while the physics-informed loss enforces the continuous elliptic conductivity PDE. This mismatch is especially relevant near discontinuities, where FD truncation and numerical diffusion are absent from the residual; the reported improvement from FFE on sharp inclusions and interfaces may therefore partly reflect fitting of discrete artifacts rather than genuine recovery of the continuous inverse problem. (See data-generation paragraph in §3 and the residual formulation in §2.2.)

    Authors: We acknowledge the discretization mismatch between the finite-difference forward solver (used to generate synthetic DtN data) and the continuous elliptic PDE residual in the PINN loss. This is a valid concern, especially near discontinuities where FD numerical diffusion may introduce artifacts not present in the continuous model. The finite-difference solver is a conventional choice for synthetic data in Calderón problem studies, and sufficiently refined grids provide a reasonable approximation; however, it does not eliminate the inconsistency. Regarding FFE, while it may partially interact with discrete effects, the empirical pattern—that FFE aids sharp inclusions/interfaces while raw coordinates compete for smooth fields—suggests it primarily enhances representation capacity rather than solely fitting artifacts. In the revision we will (i) expand the data-generation paragraph in §3 to explicitly discuss the FD approximation and its limitations, (ii) add a brief limitations paragraph noting the mismatch and suggesting future consistent discretizations (e.g., finite-element or spectral forward models), and (iii) include a short grid-resolution sensitivity remark if space allows. These changes clarify the scope without altering the core empirical claims. revision: partial

Circularity Check

0 steps flagged

No significant circularity; empirical PINN inversion on independent synthetic data

full rationale

The paper implements a standard PINN setup for the Calderón problem: separate networks for conductivity and potentials, physics residuals enforcing the elliptic PDE, and boundary losses matching finite DtN data generated by an external finite-difference solver. Reported 3-12% relative errors are direct numerical outcomes on test fields (inclusions, interfaces, smooth profiles). No derivation step reduces by construction to its inputs, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from prior author work are invoked. The method is self-contained against the provided synthetic benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Abstract-only access prevents identification of specific numerical free parameters such as network widths or exact wavelet randomization seeds; the ledger captures the core modeling assumptions visible in the abstract.

free parameters (2)
  • Fourier feature mapping parameters
    Parameters chosen to help represent sharp conductivity variations; specific frequencies or scales not provided in abstract
  • Wavelet scale and randomization parameters
    Multiscale randomized wavelet functions for boundary excitations; tuning details absent from abstract
axioms (2)
  • domain assumption The electric potential satisfies the conductivity equation ∇ · (σ ∇ u) = 0 inside the domain
    Enforced through physics-informed residuals in the proposed framework
  • domain assumption Separate neural networks conditioned on boundary excitations can represent the conductivity and the associated family of potentials
    Core architectural choice stated in the reconstruction framework description

pith-pipeline@v0.9.1-grok · 5788 in / 1655 out tokens · 75814 ms · 2026-06-29T04:43:17.436669+00:00 · methodology

discussion (0)

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Reference graph

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