A deep learning algorithm for black hole spin estimation using hot-spot secondary images

arXiv:2608.18208 · astro-ph.HE, astro-ph.IM · Submitted 2026-08-18 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: I'm Vera.

Jocelyn: And I'm Jocelyn, here with Subrahmanyan.

Vera: We're looking at the paper: "A deep learning algorithm for black hole spin estimation using hot-spot secondary images".

Jocelyn: The gist:

Vera: First, who's behind it and why it matters.

Title and authors: Jocelyn: Now, moving to the improvements they suggest, they’ve categorized their experiments into five levels of increasing complexity, starting with Experiment I for "Orbit-aggregated PA " and going up to Experiment V for "Non-Equatorial Orbits."

Vera: That progression from simple to complex scenarios is a good way to test the model’s robustness across different physical setups.

Subrahmanyan: The paper specifically points out that the method can handle realistic observational errors by employing two strategies. The first approach perturbs the input data during training by adding a predetermined noise to the inputs, and then they use DL models trained on noise-free inputs for uncertainty estimation through Monte Carlo chains.

Jocelyn: That’s smart because real data is never perfect, and having a way to quantify that error is essential when we start applying this to things like the EHT or ngEHT arrays.

Vera: And they show that their performance metrics, like mean absolute error MAE, standard deviation sigma, and coefficient of determination R squared, are assessed across different observational arrays such as EHT, ngEHT, and BHEX.

Subrahmanyan: They also test the framework using virtual observations to check it against expected constraints from those different observational arrays before they even look at real data.

The paper's summary: Jocelyn: So to wrap up this paper on "A deep learning algorithm for black hole spin estimation using hot-spot secondary images," they’ve shown that this DL approach provides robust and accurate estimates for the black hole spin a* and inclination i by linking those key parameters to the observed angle differences between primary and secondary images PA.

Vera: It shows remarkable accuracy in calculating spin and inclination from the majority of the observational tests they perform, which is what makes this work appealing.

Subrahmanyan: Connecting those geometric observables to a* means we get a direct probe into the spacetime around Sagittarius A*, which is a significant step for constraining black hole physics and structure formation.

Jocelyn: The improvements they listed, especially handling observational noise with those two training strategies, show that this isn't just a theoretical exercise; it’s designed to work with noisy real data.

Vera: It’s about taking the complex relationship between hot spots and geometry and making it solvable through deep learning for observational astronomy.

Subrahmanyan: For me, what this means is we can start using those angle differences from future interferometric arrays to directly map out the spin distribution of supermassive black holes across the galaxy, which is a big picture thing.

Jocelyn: It’s a solid tool for moving from raw angular measurements to physical parameters like spin and inclination.

Vera: We'll keep an eye on how this STIHOS framework performs when it gets fed actual data from the EHT and other arrays.

Subrahmanyan: This paper, "A deep learning algorithm for black hole spin estimation using hot-spot secondary images," provides a concrete path forward for using these secondary images as powerful diagnostics.

The paper's improvements: Jocelyn: So, to wrap up this paper, "A deep learning algorithm for black hole spin estimation using hot-spot secondary images," they’ve shown that this deep learning approach gives us robust and accurate estimates for the black hole spin a* and inclination i by linking those key parameters to the angle differences between primary and secondary images, PA.

Vera: It shows remarkable accuracy in calculating spin and inclination from the majority of the observational tests they perform, which is what makes this work really appealing.

Jocelyn: Next week, we’re looking at how those same image techniques are being used to study the structure of galactic nuclei in a completely different way.

Conclusion: Jocelyn: So we’ve been talking about how this deep learning algorithm for black hole spin estimation using hot-spot secondary images works, and it really shows how robust those angle differences between primary and secondary images can be for figuring out the black hole's spin and inclination.

Vera: Exactly. The researchers used a framework called STIHOS, which essentially takes all those geometric inputs—like the radius of the hot spot r hs and its orbital period T —and feeds them into a neural network to predict the actual spin a* and inclination i.

Subrahmanyan: It’s interesting because they model those hot spots as purely synchrotron emission from a thermal electron population, defined by fixed plasma parameters like the electron density n e, the electron temperature theta e, and the magnetic field strength B.

Jocelyn: They set that hot spot radius to one point five times the mass of the black hole with a strict cut-off, and these spots travel in circular orbits that can lie in any plane parallel to the equatorial plane.

Vera: The input features for their model include observables like r hs, its orbital period T, and those sampled angle differences PA(t K), where t K is defined by integer multiples of PA divided by ten times the orbital period.

Subrahmanyan: The optimization they use minimizes a loss function that's essentially the squared difference between the true parameter values and what their neural network predicts for those inputs, which they call L = y - f (x) squared.

Jocelyn: They also talk about how they calculate uncertainty in two different ways: either by perturbing the input data during training or by using a Monte Carlo chain when they're actually making predictions.

Vera: That uncertainty calculation is important because we need to know how much error bar comes with those spin and inclination estimates when we actually look at real data from arrays like EHT or ngEHT.

Subrahmanyan: Connecting these geometric observables to a* means we get a direct probe into the spacetime around Sagittarius A*, which is a significant step for constraining black hole physics and structure formation.

Vera: It’s about taking the complex relationship between hot spots and geometry and making it solvable through deep learning for observational astronomy.

Jocelyn: It’s a solid tool for moving from raw angular measurements to physical parameters like spin and inclination.

Vera: We'll keep an eye on how this STIHOS framework performs when it gets fed actual data from the EHT and other arrays.

Jocelyn: So that’s what we’ve seen here today with this work on black hole spin. Next week, we're looking at how those same image techniques are being used to study the structure of galactic nuclei in a completely different way.

Instituto de Astrofísica de Andalucía-CSIC · Department of Applied Mathematics and Computer Science, Technical University of Denmark

astro-ph.HE, astro-ph.IM

Submitted: 2026-08-18

Updated: 2026-10-19

Code: https://github.com/RamiAlBel/SMBH-HotSpot-Orbits

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist: A deep learning algorithm for black hole spin estimation using hot-spot secondary images provides robust and accurate estimates for black hole spin and inclination by linking key parameters

Key concepts

Hot Spot Secondary Images: These are specific secondary images of emission from a hot spot near the black hole. The angle difference ($\Delta P_A$) between the primary and secondary image is used as a key observable. This angular shift carries information about the spacetime around the black hole, which is linked to its spin parameter ($a^*$).
STIHOS (Space-Time Inference via Hotspot Observations): STIHOS is a deep learning framework designed to solve an inverse problem. It takes observational data—like the hot spot radius and angle differences—as input and learns the underlying physical parameters, such as black hole spin ($a^*$) and inclination ($i$), by comparing them against detailed general-relativistic simulations.
Black Hole Spin Parameter ($a^*$): The spin parameter $a^*$ is a dimensionless quantity that describes how much angular momentum the black hole possesses relative to its mass and charge. It is a fundamental property of the black hole's spacetime, and this paper aims to estimate it precisely using secondary image analysis.
General-Relativistic Radiative Transfer Code (ipole): This is a specialized numerical code used to simulate how radiation (light) travels through curved spacetime near a black hole. The simulations within ipole are used to generate the 'ground truth' data against which the deep learning model is trained, ensuring the algorithm learns physically accurate relationships.

Summary

The gist: A deep learning algorithm for black hole spin estimation using hot-spot secondary images provides robust and accurate estimates for black hole spin and inclination by linking key parameters to observed angle differences between primary and secondary images.

Context and Motivation

Sagittarius A star exhibits frequent flaring activity associated with a hot spot, which can be linked to black hole inclination and spin through the observed angle difference between the primary and secondary image (∆PA). This approach utilizes null geodesic theory to extract information about the spacetime and the dimensionless spin parameter, a∗ = Jc/GM2. The study focuses on secondary images of localized emission rather than photon rings.

Methods

The research employs a Deep Learning (DL) framework called STIHOS (Space-Time Inference via Hotspot Observations) to solve the inverse problem of hot-spot simulations.

** Hot-spot modeling is performed within the general-relativistic radiative transfer code ipole. The hot spots produce purely synchrotron emission from a thermal electron population and are defined by fixed plasma parameters ne, θe, and B. Their radius is set to 1.5M with a strict cut-off. They travel in circular orbits that lie in any plane parallel to the equatorial plane, for non-equatorial angular offsets with θz ∈ [−45, 45]. The remaining free parameters include the black hole spin a∗, the hot-spot center rhs, the orbital velocity via the parameter omegacoef, and the system’s inclination angle i. The simulations are run with resolution 256×256. Each simulation is capped at a full orbital period, while all have a two-minute temporal cadence. The DL model architecture uses a Multi-Layer Perceptron (MLP)-based architecture for parameter estimation across all experiments. Input features consist of observables, including the hot-spot radius rhs, its orbital period T, and primary-secondary angles ∆PA(tK) sampled on tK = N∆PAT/10 for integer N∆PA ∈ [0, 10). Optimization minimizes the loss function given by L = y − fΘ(x)2. Uncertainty is computed using two strategies: perturbing input data during training or utilizing a Monte Carlo chain during inference. Experiments are categorized into five categories of increasing complexity, such as Experiment I: “Orbit-aggregated ∆PA” and Experiment V: “Non-Equatorial Orbits”. The test metrics include the mean absolute error (MAE), the standard deviation (σ), and the coefficient of determination (R2). Uncertainty under observational noise is assessed by comparing performance across different arrays like EHT, ngEHT, and BHEX. Virtual observations are used to test the framework against expected constraints from different observational arrays. The results are summarized in Table 3 and visualized in Figure 10. The final summary is provided in Section 5. The software used includes Python, ipole, and STIHOS (available in GitHub-stihos). The paper's references are listed at the end. The acknowledgements thank M. Foschi, M. Wielgus and the whole group of J. L. Gomez at IAA for useful discussions. The funding is acknowledged from the European Union’s Horizon Europe research and innovation program under grant agreement No. 101093934 (RADIOBLOCKS). The support also comes from the COMA cluster at Radboud University, where a large part of the simulations was performed. The authors have contributed equally in this work. The paper was received on August 20, 2026. The manuscript number is aa lens2. The authors are A. I. Yfantis and R. Al-Belmpeisi. The authors have contributed equally in this work. The received date is August 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The paper is published in Astronomy & Astrophysics manuscript no. aa lens2 ©ESO 2026. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026. The key words are Black hole physics – Galaxy: center – Gravitational lensing: strong – Methods: numerical – Techniques: high angular resolution. The authors have contributed equally in this work. The received date is August 20, 2026.

Improvements for AI systems

Bold header: Precise Black Hole Parameter Estimation for Lensed Emission

This improved system can provide robust estimates for black hole spin a∗ and inclination i by leveraging the angle difference between primary and secondary images (∆PA), as demonstrated by STIHOS, which shows remarkable accuracy in calculating spin and inclination from the majority of the observational tests we perform.

Bold header: Robust Inference Under Observational Noise

The system can handle realistic observational errors by employing two strategies: the first approach perturbs the input data during training by adding a predetermined noise to the inputs and the second error-handling strategy utilizes DL models trained on noise-free inputs, allowing for uncertainty estimation through Monte Carlo chains.

Bold header: Generalization to Non-Equatorial Orbits

The improved model can be extended beyond equatorial assumptions by incorporating non-equatorial hot spots, though this introduces a very close degeneracy between θz, i and a∗, which the system can characterize by providing predictions for the axial hot-spot distance z.

Abstract

Sagittarius A* exhibits frequent flaring activity across the electromagnetic spectrum that is often associated with a localized region of strong emission known as a hot spot. We aim to train a deep learning model to provide a link between key parameters of this phenomenon - hot-spot emission radius, and black hole inclination and spin - to the observed angle difference between the primary and secondary image (ΔPA) that present and future interferometric arrays could resolve. Using the general relativistic radiative transfer code IPOLE, we generated a library of about100.000 models with varying system parameters and computed the position angle difference on the image plane between the primary and secondary images of the hot spot. We explore equatorial and non-equatorial circular orbits and evaluate our models against approximate observational constraints, including partial-orbit visibility and observational errors. Our algorithm STIHOS shows remarkable accuracy in calculating spin and inclination from the majority of the observational tests we perform (σ a*=0.04,,σ i=2), even in extreme conditions where only half of the orbit is visible. The off-equatorial estimation provides softer constraints in the absence of prior information. Our results demonstrate the importance of hot-spot observations for spacetime estimations. Given the increasing efforts to detect the photon-ring, our framework could prove valuable in interpreting the first observations of lensed emission.