pith. sign in

arxiv: 2605.19621 · v1 · pith:ON6IPDAHnew · submitted 2026-05-19 · 📡 eess.IV · cs.LG· cs.NA· math.NA

Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

Giovanni S. Alberti , Damiana Lazzaro , Serena Morigi , Matteo Santacesaria , Shibo Wang This is my paper

Pith reviewed 2026-05-20 01:58 UTC · model grok-4.3

classification 📡 eess.IV cs.LGcs.NAmath.NA
keywords electrical impedance tomographydiffusion modelsgraph neural networksinverse problemsposterior samplingregularizationconductivity reconstructionnonlinear inverse problems
0
0 comments X

The pith

Regularized diffusion posterior sampling on graphs produces stable and accurate conductivity reconstructions for electrical impedance tomography.

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

The paper extends diffusion posterior sampling to graph-structured data to address inverse problems governed by PDEs on unstructured meshes. It trains an unconditional score-based diffusion model directly on a 2D triangular mesh to learn a prior over physical conductivity distributions. A regularized variant called RDPS adds explicit terms such as total variation and generalized Tikhonov regularization to handle severe ill-posedness when conditioning on boundary measurements. Experiments on synthetic and real 2D EIT datasets show that this yields reconstructions that are more accurate, artifact-reduced, and physically plausible than current solvers while generalizing to new geometries and remaining robust to noise.

Core claim

By training a score-based diffusion model unconditionally on triangular meshes and then performing posterior sampling with added regularization, RDPS learns an effective prior over the space of physical solutions that, when conditioned on noisy boundary data, produces stable conductivity maps for nonlinear EIT inverse problems.

What carries the argument

Regularized Diffusion Posterior Sampling (RDPS) on graphs, which adapts score-based diffusion to mesh data and augments the implicit prior with explicit regularization such as total variation to mitigate ill-posedness.

If this is right

  • RDPS yields stable, physically plausible conductivity reconstructions on both synthetic and real 2D EIT datasets.
  • The method reduces artifacts and improves accuracy relative to GPnP-BM3D and DP-SGS.
  • Reconstructions generalize to out-of-distribution inclusion geometries.
  • Performance remains robust under varying levels of measurement noise.

Where Pith is reading between the lines

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

  • The same graph-diffusion-plus-regularization pattern could transfer to other mesh-based inverse problems in medical imaging or geophysical sensing.
  • Explicit regularization may prove necessary whenever purely generative priors encounter strong measurement noise or incomplete data.
  • Training diffusion models on finite-element meshes rather than grids could improve priors for any simulation that already uses unstructured discretizations.

Load-bearing premise

An unconditional score-based diffusion model trained on 2D triangular mesh data can learn a prior over physical conductivity that remains useful and accurate when conditioned on boundary voltage measurements.

What would settle it

On a held-out set of real or synthetic EIT measurements, if RDPS produces higher reconstruction error or more unphysical artifacts than GPnP-BM3D or DP-SGS, the performance claim would be refuted.

Figures

Figures reproduced from arXiv: 2605.19621 by Damiana Lazzaro, Giovanni S. Alberti, Matteo Santacesaria, Serena Morigi, Shibo Wang.

Figure 1
Figure 1. Figure 1: (a) Forward and reverse diffusion process, in the orange box we show the graph-based score [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 1: Comparison of different regularizers; (RMSE, RelErr) values are reported for [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example 2: Robustness to measurement noise. Reconstructions obtained with adaptive [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 3: Out-of-distribution reconstruction results. Columns (a)–(b) correspond to [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example 3: Reconstructions on the 2D EIT dataset [20]. Each column corresponds to [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 4: Reconstructions obtained by (row-wise from 2-6) RTO-MH, GPnP-BM3D, DP [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.

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

3 major / 2 minor

Summary. The manuscript proposes RDPS, a regularized extension of diffusion posterior sampling (DPS) adapted to graph-structured data for nonlinear inverse problems, with application to electrical impedance tomography (EIT). An unconditional score-based diffusion model is trained directly on 2D triangular meshes to serve as a prior over conductivity fields; this is then conditioned on boundary measurements via a regularized DPS procedure that adds explicit total-variation and generalized Tikhonov terms. Experiments on synthetic and real 2D EIT datasets are reported to show stable, physically plausible reconstructions that outperform baselines such as GPnP-BM3D and DP-SGS in accuracy and artifact reduction, with claimed robustness to noise and generalization to out-of-distribution inclusion geometries.

Significance. If the central claims are substantiated, the work would provide a concrete demonstration that score-based diffusion models can be trained on unstructured meshes and successfully conditioned for severely ill-posed PDE inverse problems. The explicit combination of an implicit learned prior with classical regularizers addresses a practical gap in applying generative models to mesh-based physical domains; reproducible code or mesh-specific diagnostics would further strengthen its utility for the EIT and broader inverse-problems communities.

major comments (3)
  1. [§3 and §4] §3 (Method) and §4 (Experiments): The central performance claims rest on the assertion that the unconditional graph diffusion model learns an accurate prior over physically admissible conductivity fields. However, the manuscript provides no diagnostics—such as marginal statistics, positivity checks, or comparison of sampled fields against training-distribution properties or known EIT solution characteristics—to verify that the learned score encodes physically consistent distributions rather than generic smoothness.
  2. [§4.3] §4.3 (Ablation and comparison): No ablation isolates the contribution of the graph diffusion prior from the added TV and Tikhonov regularization terms. Consequently, it remains unclear whether the reported gains over GPnP-BM3D and DP-SGS are attributable to the diffusion component or primarily to the explicit regularizers that are already known to stabilize EIT reconstructions.
  3. [§2.2] §2.2 (Regularized DPS formulation): The regularization terms are introduced to “complement the implicit diffusion prior,” yet the paper does not quantify the relative weighting or demonstrate that the diffusion prior remains load-bearing once the explicit terms are present; this weakens the claim that the graph diffusion model itself meaningfully constrains the nonlinear inverse map.
minor comments (2)
  1. [§2] Notation for the graph Laplacian and score network should be introduced once and used consistently; several equations in §2 reuse symbols without redefinition.
  2. [Figure 5] Figure captions for the real-data reconstructions should explicitly state the noise level and the number of independent runs used to compute reported error metrics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, indicating where we will revise the paper to strengthen the presentation and validation of our claims.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (Method) and §4 (Experiments): The central performance claims rest on the assertion that the unconditional graph diffusion model learns an accurate prior over physically admissible conductivity fields. However, the manuscript provides no diagnostics—such as marginal statistics, positivity checks, or comparison of sampled fields against training-distribution properties or known EIT solution characteristics—to verify that the learned score encodes physically consistent distributions rather than generic smoothness.

    Authors: We agree that explicit diagnostics would better substantiate that the learned score model encodes a physically consistent prior rather than generic smoothness. In the revised manuscript we will add (i) marginal statistics (mean, variance, and histogram comparisons) of conductivity values sampled from the unconditional model versus the training distribution, (ii) positivity checks confirming that sampled fields respect the physical non-negativity constraint on conductivity, and (iii) quantitative comparisons of geometric properties (e.g., inclusion size, contrast, and boundary smoothness) of generated fields against both training data and known analytic EIT solutions. These additions will directly address the concern and strengthen the claim that the graph diffusion prior is physically meaningful. revision: yes

  2. Referee: [§4.3] §4.3 (Ablation and comparison): No ablation isolates the contribution of the graph diffusion prior from the added TV and Tikhonov regularization terms. Consequently, it remains unclear whether the reported gains over GPnP-BM3D and DP-SGS are attributable to the diffusion component or primarily to the explicit regularizers that are already known to stabilize EIT reconstructions.

    Authors: We acknowledge the value of an explicit ablation to isolate the diffusion prior’s contribution. In the revision we will include a new ablation study that (a) removes the diffusion prior entirely while retaining the same TV and generalized Tikhonov terms, (b) varies the relative weight of the diffusion term while keeping the explicit regularizers fixed, and (c) reports reconstruction metrics for each variant on the same synthetic and real EIT test sets. This will quantify how much of the reported improvement over GPnP-BM3D and DP-SGS is due to the learned graph prior versus the classical regularizers. revision: yes

  3. Referee: [§2.2] §2.2 (Regularized DPS formulation): The regularization terms are introduced to “complement the implicit diffusion prior,” yet the paper does not quantify the relative weighting or demonstrate that the diffusion prior remains load-bearing once the explicit terms are present; this weakens the claim that the graph diffusion model itself meaningfully constrains the nonlinear inverse map.

    Authors: We will add a sensitivity analysis in the revised §2.2 and §4 that systematically varies the relative weighting λ between the diffusion posterior term and the explicit regularizers. We will report reconstruction error, artifact levels, and stability metrics across a range of λ values, including the limiting case where the diffusion term is removed. In addition, we will show that performance degrades noticeably when the diffusion prior is down-weighted while the explicit terms remain at their tuned strengths, thereby demonstrating that the graph diffusion model continues to play a load-bearing role in constraining the nonlinear inverse map even in the presence of the classical regularizers. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical performance claims rest on external benchmarks and prior DPS literature.

full rationale

The manuscript trains an unconditional score-based diffusion model on 2D triangular meshes and applies a regularized DPS variant (RDPS) to the nonlinear EIT inverse problem. Performance is evaluated via direct comparison to GPnP-BM3D and DP-SGS on held-out synthetic and real datasets, with no derivation step that renames a fitted quantity as a prediction, equates the learned prior to the target solution by construction, or relies on a self-citation chain whose validity is internal to this paper. The central claims therefore remain falsifiable against independent measurements and do not collapse to the training inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard diffusion modeling assumptions plus the domain-specific choice of mesh discretization; no new invented entities are introduced.

axioms (1)
  • domain assumption Physical domains for EIT can be accurately represented by 2D triangular meshes on which a score-based diffusion model can be trained directly.
    Stated in the description of developing the unconditional score-based diffusion model on a 2D triangular mesh.

pith-pipeline@v0.9.0 · 5750 in / 1205 out tokens · 42738 ms · 2026-05-20T01:58:39.554146+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    Alessandrini

    G. Alessandrini. Stable determination of conductivity by boundary measurements.Appl. Anal., 27(1-3):153–172, 1988

    work page 1988

  2. [2]

    Arridge, P

    S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch¨ onlieb. Solving inverse problems using data-driven models.Acta Numerica, 28:1–174, 2019

    work page 2019

  3. [3]

    J. M. Bardsley, A. Seppanen, A. Solonen, H. Haario, and J. Kaipio. Randomize-then-optimize for sampling and uncertainty quantification in electrical impedance tomography.SIAM/ASA Journal on Uncertainty Quantification, 3(1):1136–1158, 2015

    work page 2015

  4. [4]

    L. Borcea. Electrical impedance tomography.Inverse Problems, 18(6):R99–R136, 2002. 25

    work page 2002

  5. [5]

    Borsic, B

    A. Borsic, B. M. Graham, A. Adler, and W. R. Lionheart. In vivo impedance imaging with total variation regularization.IEEE transactions on medical imaging, 29(1):44–54, 2009

    work page 2009

  6. [6]

    C. A. Bouman and G. T. Buzzard. Generative plug and play: Posterior sampling for inverse problems. In2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1–7. IEEE, 2023

    work page 2023

  7. [7]

    T. A. Bubba, editor.Data-driven Models in Inverse Problems. De Gruyter, Berlin, Boston, 2025

    work page 2025

  8. [8]

    Calder´ on

    A. Calder´ on. On an inverse boundary value problem.Seminar on Numerical Analysis and its Applications to Continuum Physics, pages 65–73, 1980

    work page 1980

  9. [9]

    Cheney, D

    M. Cheney, D. Isaacson, J. Newell, S. Simske, and J. Goble. Noser: An algorithm for solving the inverse conductivity problem.Int J Imaging Syst Technol., 2(2):66–75, 1990

    work page 1990

  10. [10]

    Cheney, D

    M. Cheney, D. Isaacson, and J. C. Newell. Electrical impedance tomography.SIAM Rev., 41(1):85–101, 1999

    work page 1999

  11. [11]

    Chung, J

    H. Chung, J. Kim, M. T. Mccann, M. L. Klasky, and J. C. Ye. Diffusion posterior sampling for general noisy inverse problems. InThe Eleventh International Conference on Learning Represen- tations, 2023

    work page 2023

  12. [12]

    Colibazzi, D

    F. Colibazzi, D. Lazzaro, S. Morigi, and A. Samor´ e. Learning nonlinear electrical impedance tomography.J. Sci. Comput., 90(1), Jan. 2022

    work page 2022

  13. [13]

    Colibazzi, D

    F. Colibazzi, D. Lazzaro, S. Morigi, and A. Samor´ e. Deep-plug-and-play proximal gauss-newton method with applications to nonlinear, ill-posed inverse problems.Inverse Problems and Imaging, 17(6):1226–1248, 2023

    work page 2023

  14. [14]

    Denker, F

    A. Denker, F. Margotti, J. Ning, K. Knudsen, D. N. Tanyu, B. Jin, A. Hauptmann, and P. Maass. Deep learning based reconstruction methods for electrical impedance tomography.arXiv preprint arXiv:2508.06281, 2025

    work page arXiv 2025

  15. [15]

    M. A. G. Duff, N. D. F. Campbell, and M. J. Ehrhardt. Regularising inverse problems with generative machine learning models.J. Math. Imaging Vision, 66(1):37–56, 2024

    work page 2024

  16. [16]

    Eliasof, E

    M. Eliasof, E. Haber, and E. Treister. Pde-gcn: Novel architectures for graph neural networks motivated by partial differential equations. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Information Processing Systems, volume 34, pages 3836–3849. Curran Associates, Inc., 2021

    work page 2021

  17. [17]

    Eliasof, M

    M. Eliasof, M. S. R. Siddiqui, C.-B. Sch¨ onlieb, and E. Haber. Learning regularization for graph inverse problems. InProceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 16471–16479, 2025

    work page 2025

  18. [18]

    Feldman, M

    J. Feldman, M. Salo, and G. Uhlmann.The Calder´ on problem—an introduction, volume 253 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2025] ©2025

    work page 2025

  19. [19]

    Guillard

    H. Guillard. Node-nested multi-grid method with Delaunay coarsening. Research Report RR- 1898, INRIA, 1993

    work page 1993

  20. [20]

    Open 2D Electrical Impedance Tomography data archive

    A. Hauptmann, V. Kolehmainen, N. M. Mach, T. Savolainen, A. Sepp¨ anen, and S. Siltanen. Open 2d electrical impedance tomography data archive.arXiv preprint arXiv:1704.01178, 2017. 26

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Ehrhardt, and Sebastian Neumayer

    J. Hertrich, H. S. Wong, A. Denker, S. Ducotterd, Z. Fang, M. Haltmeier, ˇZ. Kereta, E. Kobler, O. Leong, M. S. Salehi, et al. Learning regularization functionals for inverse problems: A com- parative study.arXiv preprint arXiv:2510.01755, 2025

    work page arXiv 2025

  22. [22]

    J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models.Advances in neural information processing systems, 33:6840–6851, 2020

    work page 2020

  23. [23]

    D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. InProceedings of 3rd International Conference on Learning Representations, 2015

    work page 2015

  24. [24]

    Lauga, J

    W. Lauga, J. Rowbottom, A. Denker, ˇZ. Kereta, M. Eliasof, and C.-B. Sch¨ onlieb. Graph Neural Regularizers for PDE Inverse Problems.arXiv preprint arXiv:2510.21012, 2025

    work page arXiv 2025

  25. [25]

    Lazzaro, S

    D. Lazzaro, S. Morigi, and P. Zuzolo. A Physics-Informed Graph Neural Network for Computing Laplace–Beltrami Eigenfunctions on Manifolds.IEEE Access, 13:199647–199664, 2025

    work page 2025

  26. [26]

    Y. Ling, X. Ke, H. Wang, and Q. Zhou. Split gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography. 2025

    work page 2025

  27. [27]

    M. Lino, S. Fotiadis, A. A. Bharath, and C. D. Cantwell. Multi-scale rotation-equivariant graph neural networks for unsteady eulerian fluid dynamics.Physics of Fluids, 34(8):087110, 08 2022

    work page 2022

  28. [28]

    Mandache

    N. Mandache. Exponential instability in an inverse problem for the Schr¨ odinger equation.Inverse Problems, 17(5):1435–1444, 2001

    work page 2001

  29. [29]

    Monga, Y

    V. Monga, Y. Li, and Y. C. Eldar. Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing.IEEE Signal Processing Magazine, 38(2):18–44, 2021

    work page 2021

  30. [30]

    Pfaff, M

    T. Pfaff, M. Fortunato, A. Sanchez-Gonzalez, and P. Battaglia. Learning mesh-based simulation with graph networks. InInternational Conference on Learning Representations, 2021

    work page 2021

  31. [31]

    Somersalo, M

    E. Somersalo, M. Cheney, and D. Isaacson. Existence and uniqueness for electrode models for electric current computed tomography.SIAM J. Appl. Math., 52(4):1023–1040, Aug. 1992

    work page 1992

  32. [32]

    J. Song, C. Meng, and S. Ermon. Denoising diffusion implicit models. InInternational Conference on Learning Representations, 2021

    work page 2021

  33. [33]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations. InInternational Conference on Learning Representations, 2020

    work page 2020

  34. [34]

    T. N. Tallman and D. J. Smyl. Structural health and condition monitoring via electrical impedance tomography in self-sensing materials: A review.Smart Materials and Structures, 29(12), 2020

    work page 2020

  35. [35]

    D. N. Tanyu, J. Ning, A. Hauptmann, B. Jin, and P. Maass.Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches, pages 437–470. De Gruyter, Berlin, Boston, 2025

    work page 2025

  36. [36]

    M. L. Valencia, T. Pfaff, and N. Thuerey. Learning distributions of complex fluid simulations with diffusion graph networks. InThe Thirteenth International Conference on Learning Repre- sentations, ICLR 2025, Singapore, April 24-28, 2025. OpenReview.net, 2025

    work page 2025

  37. [37]

    Vauhkonen, M

    P. Vauhkonen, M. Vauhkonen, T. Savolainen, and J. Kaipio. Three-dimensional electrical impedance tomography based on the complete electrode model.IEEE Transactions on Biomedical Engineering, 46(9):1150–1160, 1999. 27

    work page 1999

  38. [38]

    S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg. Plug-and-play priors for model based reconstruction. In2013 IEEE Global Conference on Signal and Information Processing, pages 945–948. IEEE, 2013

    work page 2013

  39. [39]

    Q. Zhao, D. B. Lindell, and G. Wetzstein. Learning to solve PDE-constrained inverse problems with graph networks. InICML 2022 2nd AI for Science Workshop, 2022. 28

    work page 2022