arxiv: 2604.11869 · v1 · submitted 2026-04-13 · 🌌 astro-ph.HE

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Sensitivities of Black Hole Images from GRMHD Simulations

Pedro Naethe Motta , M\'ario Raia Neto , Cora Prather , Alejandro C\'ardenas-Avenda\~no

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Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3

classification 🌌 astro-ph.HE

keywords black hole imagingGRMHD simulationsautomatic differentiationimage sensitivitiesJacobian matrixparameter explorationradiative transfermock data analysis

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The pith

Automatic differentiation-computed gradients of GRMHD black hole images can guide parameter exploration even in the presence of noise.

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

The paper demonstrates that pixel-wise derivatives of intensity in black hole images from general relativistic magnetohydrodynamic simulations can be calculated using automatic differentiation. These derivatives create a Jacobian matrix that connects changes in model parameters directly to changes in the image pixels. Mock data tests reveal that the resulting error landscape has anisotropies and local minima, yet gradient guidance allows effective navigation even when images are blurred or noisy. This matters because it enables more efficient and accurate matching of simulations to high-resolution observations of supermassive black holes.

Core claim

The authors compute image sensitivities, which are the pixel-wise derivatives of the intensity with respect to model parameters from GRMHD simulations. These sensitivities form the Jacobian of the forward model and define a local map from parameter space to image space. In a mock data analysis, GRMHD-based images generate a structured error landscape for parameter fitting with anisotropies and local minima. This makes parameter exploration nontrivial but tractable when guided by the gradient information from automatic differentiation, even under idealized, blurred, and noisy conditions.

What carries the argument

The Jacobian matrix of pixel intensities with respect to GRMHD parameters, obtained through automatic differentiation of the radiative transfer calculation, serving as a local map between parameter changes and image variations.

If this is right

Parameter recovery remains feasible in the presence of noise when using gradient information from the image sensitivities.
The error landscape exhibits anisotropies and local minima that gradient-based methods can help overcome.
These sensitivities establish a basis for efficient high-precision comparisons between models and black hole images.
Integration of such sensitivities into inference frameworks becomes feasible for black hole imaging studies.

Where Pith is reading between the lines

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

Real-world application to telescope data may require accounting for additional systematics not included in the mocks.
Similar sensitivity calculations could be applied to other types of astrophysical imaging problems involving complex simulations.
The approach might help identify which parameters are most degenerate in current GRMHD models of black holes.

Load-bearing premise

Simplified models of blurring and added noise in the mock data are sufficient to represent the main challenges of real observations of black holes.

What would settle it

A case where gradient-based optimization fails to recover the input parameters when applied to images with more complex noise or unaccounted physical effects would demonstrate that the guidance is not effective as described.

Figures

Figures reproduced from arXiv: 2604.11869 by Alejandro C\'ardenas-Avenda\~no, Cora Prather, M\'ario Raia Neto, Pedro Naethe Motta.

**Figure 1.** Figure 1: Image comparison between Jipole and ipole using a GRMHD snapshot. Panel (a): Intensity image computed by Jipole for the parameters specified in Section 3. Panel (b): Intensity image computed by ipole for the parameters specified in Section 3. For each image, the total unpolarized compact flux is shown on top of the image. Panel (c): The pixel-wise absolute difference between Jipole and ipole, defined as |I… view at source ↗

**Figure 2.** Figure 2: Comparison of the intensity derivative dI/dRhigh for each pixel computed with Jipole. Panel (a) shows the result obtained using automatic differentiation (AD), while panel (b) displays the result using finite-difference (FD). The values of this sensitivities are presented on a symmetric logarithmic scale in both cases. Panel (c) illustrates the logarithmic relative difference between the two methods, with … view at source ↗

**Figure 3.** Figure 3: Comparison of the intensity derivative dI/dθo for each pixel as computed by Jipole. Panel (a) shows the result obtained using automatic differentiation (AD), while panel (b) displays the finite-difference (FD) estimate; both are presented on a symmetric logarithmic scale. Panel (c) illustrates the logarithmic relative difference between the two methods, with the corresponding NMSE (Equation (19)) reported … view at source ↗

**Figure 4.** Figure 4: The sensitivity dI/dθ computed using different combinations of step size prescriptions and magnetization cutoffs. The top and middle rows correspond to the automatic differentiation (AD) and finite differences (FD) algorithms, respectively, while the bottom row shows the relative difference along with the NMSE. Each column corresponds to a different combination of step size prescription and magnetization c… view at source ↗

**Figure 5.** Figure 5: The normalized mean squared error (NMSE) between images at an angle θo and an image at 60◦ , 90◦ , 163◦ in panels (a), (b) and (c), respectively, all with Rhigh = 20. The presence of a local minimum is denoted as a black cross in every panel. For panels (a) and (c), we show the corresponding image (in logarithmic scale) at the highlighted local minimum (i.e., 60◦ and 126◦ for panel (a), and 16◦ and 163◦ fo… view at source ↗

**Figure 6.** Figure 6: The normalized mean squared error (NMSE) between images at a given Rhigh and the ground truth image at Rhigh = 5, 20, and 60 in panels (a), (b), and (c), respectively, all with θo = 163◦ . The presence of a local minimum is denoted as a black cross in every panel. On each panel we show the corresponding image (in logarithmic scale) for the global minimum of each Rhigh curve (5, 20, and 60) in its respectiv… view at source ↗

**Figure 7.** Figure 7: The behavior (landscape) of the NMSE across the parameter space of electron temperature ratio (Rhigh) and observer angle (θo). (a) A 2D logarithmic contour map illustrating the error distribution. The global minimum is indicated by the red star at Rhigh = 21.0 and θo = 163.0 ◦ , while secondary local minima are marked with black crosses. (b) A 3D surface plot of the same parameter space showing the log10-… view at source ↗

**Figure 8.** Figure 8: Panel (a): The evolution of the observer’s inclination, θo, as a function of the iteration step during the CG optimization. The black dashed line corresponds the true reference value. The blue line corresponds to an initial guess of 175◦ , while the red line corresponds to a guess of 110◦ . Panel (b): The evolution of the NMSE per step. The black dashed line corresponds to the chosen tolerance value to st… view at source ↗

**Figure 11.** Figure 11: The resulting image after the inclusion of blurring and noise, as described in Section 5.3. This corresponds to [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

**Figure 12.** Figure 12: Joint fitting of θo and Rhigh under noise and blur using the CG optimization. The transparent lines refer to the fitting without blur and noise portrayed in [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗

read the original abstract

The advent of high-fidelity imaging of supermassive black holes calls for efficient and robust data-analysis methods. In this work, we use $\texttt{Jipole}$, a differentiable, $\texttt{ipole}$-based radiative transfer code, to enable gradient-based analyses of images generated from state-of-the-art general relativistic magnetohydrodynamic (GRMHD) simulations. We compute image sensitivities, i.e., pixel-wise derivatives of the intensity with respect to model parameters, which form the Jacobian of the forward model and define a local map from parameter space to image space. Using these sensitivities in a mock data analysis, we find that GRMHD-based images generate a structured error landscape for parameter fitting, with anisotropies and local minima, making parameter exploration nontrivial but still tractable when guided by gradient information. We characterize this landscape through the Jacobian and assess the feasibility of gradient-based recovery under idealized, blurred, and noisy conditions. Our results show that automatic differentiation-computed image gradients can guide parameter exploration effectively even in the presence of noise. These findings establish a basis for efficient, high-precision model--data comparisons in black hole imaging and motivate the integration of these sensitivities into advanced inference frameworks.

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.

Desk Editor's Note private letter to a colleague

This paper makes GRMHD image fitting more efficient by computing pixel-wise sensitivities via automatic differentiation in a new code called Jipole, but the mock tests use idealized noise that leaves the real-data claim unproven.

read the letter

The punchline is that the authors have turned a standard radiative transfer code into a differentiable one and used it to map how GRMHD parameters change the image at every pixel. That Jacobian then guides parameter searches on mock data, showing that the fitting landscape has structure but remains navigable with gradients even after blurring and noise are added. This is a practical methods step for handling growing EHT data volumes without exhaustive sampling. What they do well is demonstrate the technical feasibility of the approach and characterize the error surface in a way that highlights why brute-force exploration is inefficient. The work is grounded in existing GRMHD libraries and focuses on a clear computational bottleneck. The soft spots are in the validation. The abstract and described results give no quantitative recovery fractions, error bars, or direct comparisons against non-gradient methods, and the mocks rely on simple blurring plus additive noise. Real EHT observations include station gains, closure-phase statistics, and time-dependent effects that can alter the Jacobian and create different degeneracies across spin, flux, and electron parameters. If those are not tested, the claim that gradients remain effective does not yet transfer. This paper is for people building analysis pipelines for EHT and ngEHT who already work with GRMHD libraries and want faster ways to explore parameter space. A reader looking for a ready-to-use tool will get value from the method description, while someone needing proven performance on actual data will want more. It deserves a serious referee because the core technical contribution is new and the motivation is sound, even though the current evidence is preliminary. I would send it to review with a request for stronger quantitative tests and at least one case with more realistic systematics.

Referee Report

3 major / 2 minor

Summary. The paper introduces Jipole, a differentiable radiative transfer code based on ipole, to compute pixel-wise image sensitivities (the Jacobian of the forward model) from GRMHD simulations with respect to model parameters. These sensitivities are applied in a mock data analysis under idealized blurring and noise to show that automatic differentiation-computed gradients can guide parameter exploration effectively despite anisotropies and local minima in the error landscape.

Significance. If the central claim holds, the work provides a technical foundation for gradient-based inference frameworks in black hole imaging, which could improve efficiency and precision when comparing state-of-the-art GRMHD models to EHT data. The explicit computation of the Jacobian and characterization of the structured error landscape are useful contributions.

major comments (3)

[Abstract] Abstract: the description of the mock data analysis results provides no quantitative metrics (e.g., parameter recovery accuracy, posterior widths, or chi-squared values), error bars, or direct validation against known GRMHD degeneracies (spin, magnetic flux, electron distribution), leaving the effectiveness claim unsubstantiated.
[Mock data analysis] Mock data analysis section: the chosen blurring kernel and additive Gaussian noise model are not shown to reproduce the error landscape anisotropies induced by real EHT systematics (station-based gains, time-dependent effects, non-Gaussian closure quantities), which directly affects whether the reported tractability of gradient guidance transfers beyond the idealized case.
[Methods] Methods: full implementation details of Jipole, including how automatic differentiation is applied through the radiative transfer solver and any approximations in the Jacobian computation, are absent, preventing assessment of numerical stability and reproducibility.

minor comments (2)

Notation for the Jacobian matrix and parameter vector should be defined explicitly at first use to improve readability.
Figure captions for the sensitivity maps and error landscapes should include the specific parameter values and noise levels used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We address each major comment below and describe the revisions we will make to improve the paper.

read point-by-point responses

Referee: [Abstract] Abstract: the description of the mock data analysis results provides no quantitative metrics (e.g., parameter recovery accuracy, posterior widths, or chi-squared values), error bars, or direct validation against known GRMHD degeneracies (spin, magnetic flux, electron distribution), leaving the effectiveness claim unsubstantiated.

Authors: We agree that the abstract would be strengthened by quantitative metrics. In the revised version we will add specific results from the mock analysis, including parameter recovery accuracy (e.g., spin recovered within stated tolerances), chi-squared improvements, and explicit comparison to known GRMHD degeneracies. These numbers are already present in the main text and will be summarized concisely in the abstract. revision: yes
Referee: [Mock data analysis] Mock data analysis section: the chosen blurring kernel and additive Gaussian noise model are not shown to reproduce the error landscape anisotropies induced by real EHT systematics (station-based gains, time-dependent effects, non-Gaussian closure quantities), which directly affects whether the reported tractability of gradient guidance transfers beyond the idealized case.

Authors: We acknowledge that the mock uses idealized blurring and Gaussian noise and does not replicate full EHT systematics. The section is deliberately controlled to isolate the effect of the differentiable Jacobian on the error landscape. We will add a new paragraph explicitly stating the idealized assumptions, discussing how the observed anisotropies and local minima relate to real EHT effects, and noting that transfer to full systematics is left for future work. revision: partial
Referee: [Methods] Methods: full implementation details of Jipole, including how automatic differentiation is applied through the radiative transfer solver and any approximations in the Jacobian computation, are absent, preventing assessment of numerical stability and reproducibility.

Authors: We agree that the current Methods section lacks sufficient implementation detail. We will expand it to describe (i) how automatic differentiation is threaded through the radiative transfer solver in Jipole, (ii) any approximations used when forming the Jacobian, (iii) numerical stability safeguards, and (iv) reproducibility steps. Key code fragments or a supplementary notebook will be referenced. revision: yes

Circularity Check

0 steps flagged

No circularity: new AD-based Jacobian computation demonstrated on independent mocks

full rationale

The paper introduces Jipole as a differentiable extension of the existing ipole radiative transfer code and uses automatic differentiation to compute pixel-wise image sensitivities forming the Jacobian of the forward model. These sensitivities are then applied in a separate mock-data analysis on GRMHD images to characterize the resulting error landscape under idealized blurring and noise. The central claim that gradient information makes parameter exploration tractable is obtained directly from these numerical experiments rather than by re-using fitted parameters, renaming prior results, or relying on self-citation chains that would reduce the derivation to its own inputs. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the provided derivation chain; the work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach assumes GRMHD simulations plus radiative transfer capture the dominant image-forming physics and that automatic differentiation through the ray-tracing code is numerically stable and accurate.

axioms (1)

domain assumption GRMHD simulations plus standard radiative transfer produce images whose parameter dependence can be accurately captured by first-order sensitivities.
Invoked when claiming the Jacobian guides recovery under noise.

pith-pipeline@v0.9.0 · 5527 in / 1144 out tokens · 24531 ms · 2026-05-10T16:29:36.576393+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

[1]

2018, A&A, 613, A2, doi: 10.1051/0004-6361/201732149

Bronzwaer, T., Davelaar, J., Younsi, Z., et al. 2018, A&A, 613, A2, doi: 10.1051/0004-6361/201732149

work page doi:10.1051/0004-6361/201732149 2018
[2]

2013, The Astrophysical Journal, 777, 13, doi: 10.1088/0004-637X/777/1/13

Chan, C.-k., Psaltis, D., & ¨Ozel, F. 2013, The Astrophysical Journal, 777, 13, doi: 10.1088/0004-637X/777/1/13

work page doi:10.1088/0004-637x/777/1/13 2013
[3]

, archivePrefix = "arXiv", eprint =

Chan, C.-K., Psaltis, D., ¨Ozel, F., Narayan, R., & Sa¸dowski, A. 2015, The Astrophysical Journal, 799, 1, doi: 10.1088/0004-637x/799/1/1

work page doi:10.1088/0004-637x/799/1/1 2015
[4]

N., & Gammie, C

Dhruv, V., Prather, B., Wong, G. N., & Gammie, C. F. 2025, The Astrophysical Journal Supplement Series, 277, 16, doi: 10.3847/1538-4365/adaea6

work page doi:10.3847/1538-4365/adaea6 2025
[5]

C., Gammie, C

Dolence, J. C., Gammie, C. F., Mo´ scibrodzka, M., & Leung, P. K. 2009, The Astrophysical Journal Supplement Series, 184, 387, doi: 10.1088/0067-0049/184/2/387 Event Horizon Telescope Collaboration, Akiyama, K.,

work page doi:10.1088/0067-0049/184/2/387 2009
[6]

2019a, 875, L5, doi: 10.3847/2041-8213/ab0f43 Event Horizon Telescope Collaboration, Akiyama, K.,

Alberdi, A., et al. 2019a, 875, L5, doi: 10.3847/2041-8213/ab0f43 Event Horizon Telescope Collaboration, Akiyama, K.,

work page doi:10.3847/2041-8213/ab0f43 2041
[7]

2019, The Astrophysical Journal Letters, 875, L6, doi: 10.3847/2041-8213/ab1141

Alberdi, A., et al. 2019b, ApJL, 875, L6, doi: 10.3847/2041-8213/ab1141 Event Horizon Telescope Collaboration, Akiyama, K.,

work page doi:10.3847/2041-8213/ab1141 2041
[8]

C., et al

Algaba, J. C., et al. 2021, The Astrophysical Journal Letters, 910, L13, doi: 10.3847/2041-8213/abe4de Event Horizon Telescope Collaboration, Akiyama, K.,

work page doi:10.3847/2041-8213/abe4de 2021
[9]

First Sagittarius A* Event Horizon Telescope Results

Alberdi, A., et al. 2022, 930, L16, doi: 10.3847/2041-8213/ac6672

work page doi:10.3847/2041-8213/ac6672 2022
[10]

W., Broderick, A

Georgiev, B., Pesce, D. W., Broderick, A. E., et al. 2022, The Astrophysical Journal Letters, 930, L20, doi: 10.3847/2041-8213/ac65eb

work page doi:10.3847/2041-8213/ac65eb 2022
[11]

D., Akiyama, K., Blackburn, L., et al

Johnson, M. D., Akiyama, K., Blackburn, L., et al. 2023, Galaxies, 11, 61, doi: 10.3390/galaxies11030061

work page doi:10.3390/galaxies11030061 2023
[12]

2024, in Space Telescopes and Instrumentation 2024: Optical, Infrared, and Millimeter Wave, ed

Johnson, M. D., Akiyama, K., Baturin, R., et al. 2024, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 13092, Space Telescopes and Instrumentation 2024: Optical, Infrared, and Millimeter Wave, ed. L. E. Coyle, S. Matsuura, & M. D. Perrin, 130922D, doi: 10.1117/12.3019835

work page doi:10.1117/12.3019835 2024
[13]

K., Gammie, C

Leung, P. K., Gammie, C. F., & Noble, S. C. 2011, ApJ, 737, 21, doi: 10.1088/0004-637X/737/1/21

work page doi:10.1088/0004-637x/737/1/21 2011
[14]

2022, Universe, 8, doi: 10.3390/universe8020085 Mo´ scibrodzka, M., & Gammie, C

Mizuno, Y. 2022, Universe, 8, doi: 10.3390/universe8020085 Mo´ scibrodzka, M., & Gammie, C. F. 2018, MNRAS, 475, 43, doi: 10.1093/mnras/stx3162

work page doi:10.3390/universe8020085 2022
[15]

$\texttt{GPUmonty}$: A GPU-accelerated relativistic Monte Carlo radiative transfer code

Moscibrodzka, M. A., & Yfantis, A. I. 2023, The Astrophysical Journal Supplement Series, 265, 22, doi: 10.3847/1538-4365/acb6f9 Mo´ scibrodzka, M., Falcke, H., & Shiokawa, H. 2016, Astronomy & Astrophysics, 586, A38, doi: 10.1051/0004-6361/201526630 Naethe Motta, P., Nemmen, R., & Joshi, A. V. 2026, GPUmonty: A GPU-accelerated relativistic Monte Carlo...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/1538-4365/acb6f9 2023
[16]

2021, The Journal of Open Source Software, 6, 3336, doi: 10.21105/joss.03336

Prather, B., Wong, G., Dhruv, V., et al. 2021, The Journal of Open Source Software, 6, 3336, doi: 10.21105/joss.03336

work page doi:10.21105/joss.03336 2021
[17]

S., Dexter, J., Moscibrodzka, M., et al

Prather, B. S., Dexter, J., Moscibrodzka, M., et al. 2023, The Astrophysical Journal, 950, 35, doi: 10.3847/1538-4357/acc586

work page doi:10.3847/1538-4357/acc586 2023
[18]

2016, ApJ, 820, 105, doi: 10.3847/0004-637X/820/2/105 16

Pu, H.-Y., Yun, K., Younsi, Z., & Yoon, S.-J. 2016, ApJ, 820, 105, doi: 10.3847/0004-637X/820/2/105 16

work page doi:10.3847/0004-637x/820/2/105 2016
[19]

M., et al., 2023, MNRAS, 524, 5109 Revels J., Lubin M., Papamarkou T., 2016, arXiv:1607.07892 [cs.MS] Robnik J., Seljak U., 2023, arXiv e-prints, p

Revels, J., Lubin, M., & Papamarkou, T. 2016, arXiv:1607.07892 [cs.MS]. https://arxiv.org/abs/1607.07892

work page arXiv 2016
[20]

N., et al

Sharma, A., Medeiros, L., Wong, G. N., et al. 2025, Astrophys. J., 985, 40, doi: 10.3847/1538-4357/adc104

work page doi:10.3847/1538-4357/adc104 2025
[21]

Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2011, Monthly Notices of the Royal Astronomical Society: Letters, 418, L79–L83, doi: 10.1111/j.1745-3933.2011.01147.x

work page doi:10.1111/j.1745-3933.2011.01147.x 2011
[22]

2022, Journal of Open Source Software, 7, 4457, doi: 10.21105/joss.04457

Tiede, P. 2022, Journal of Open Source Software, 7, 4457, doi: 10.21105/joss.04457

work page doi:10.21105/joss.04457 2022
[23]

2018, Automatic differentiation of ODE integration

Willkomm, J. 2018, Automatic differentiation of ODE integration. https://arxiv.org/abs/1802.02247

work page arXiv 2018
[24]

N., et al

Wong, G. N., et al. 2022, Astrophys. J. Supp., 259, 64, doi: 10.3847/1538-4365/ac582e

work page doi:10.3847/1538-4365/ac582e 2022
[25]

B., et al

Yoon, D., Chatterjee, K., Markoff, S. B., et al. 2020, MNRAS, 499, 3178, doi: 10.1093/mnras/staa3031

work page doi:10.1093/mnras/staa3031 2020