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$\texttt{Jipole}$: A Differentiable $\texttt{ipole}$-based Code for Radiative Transfer in Curved Spacetimes

Pedro Naethe Motta, Ben S. Prather, Alejandro Cárdenas-Avendaño

TL;DR

Jipole addresses the growing cost of forward-model image libraries in black-hole imaging by providing a differentiable radiative-transfer code built on ipole, enabling direct gradients of the observed intensity with respect to physical parameters like spin $a$ and inclination $\theta_o$. It delivers a fully differentiable pipeline using automatic differentiation to propagate parameter sensitivities through both geodesic integration and radiative transfer, demonstrated via backward ray tracing and conjugate-gradient parameter fitting. Validation against ipole shows machine-precision agreement for intensities, while AD vs FD derivative comparisons reveal expected, method-dependent differences near caustics. The authors demonstrate robust parameter recovery under ideal, blurred, and noisy conditions, highlighting the potential of AD-based methods to accelerate robust, high-fidelity model-data comparisons in current and future black-hole imaging efforts, with plans to extend to GRMHD inputs, polarization, slow-light, and Bayesian inference frameworks.

Abstract

Recent imaging of supermassive black holes by the Event Horizon Telescope (EHT) has relied on exhaustive parameter-space searches, matching observations to large, precomputed libraries of theoretical models. As observational data become increasingly precise, the limitations of this computationally expensive approach grow more acute, creating a pressing need for more efficient methods. In this work, we present $\texttt{Jipole}$, an automatically differentiable (AD), $\texttt{ipole}$-based code for radiative transfer in curved spacetimes, designed to compute image gradients with respect to underlying model parameters. These gradients quantify how parameter changes-such as the black hole's spin or the observer's inclination-affect the image, enabling more efficient parameter estimation and reducing the number of required images. We validate $\texttt{Jipole}$ against $\texttt{ipole}$ in two analytical tests and then compare pixel-wise intensity derivatives from AD with those from finite-difference methods. We then demonstrate the utility of these gradients by performing parameter recovery for an analytical model using a conjugate gradient optimizer in three increasingly complex cases for the injected image: ideal, blurred, and blurred with added noise. In most cases, high-accuracy fits are obtained in only a few optimization steps, failing only in cases with extremely low signal-to-noise ratios and large blurring size kernels. These results highlight the potential of AD-based methods to accelerate robust, high-fidelity model-data comparisons in current and future black hole imaging efforts.

$\texttt{Jipole}$: A Differentiable $\texttt{ipole}$-based Code for Radiative Transfer in Curved Spacetimes

TL;DR

Jipole addresses the growing cost of forward-model image libraries in black-hole imaging by providing a differentiable radiative-transfer code built on ipole, enabling direct gradients of the observed intensity with respect to physical parameters like spin and inclination . It delivers a fully differentiable pipeline using automatic differentiation to propagate parameter sensitivities through both geodesic integration and radiative transfer, demonstrated via backward ray tracing and conjugate-gradient parameter fitting. Validation against ipole shows machine-precision agreement for intensities, while AD vs FD derivative comparisons reveal expected, method-dependent differences near caustics. The authors demonstrate robust parameter recovery under ideal, blurred, and noisy conditions, highlighting the potential of AD-based methods to accelerate robust, high-fidelity model-data comparisons in current and future black-hole imaging efforts, with plans to extend to GRMHD inputs, polarization, slow-light, and Bayesian inference frameworks.

Abstract

Recent imaging of supermassive black holes by the Event Horizon Telescope (EHT) has relied on exhaustive parameter-space searches, matching observations to large, precomputed libraries of theoretical models. As observational data become increasingly precise, the limitations of this computationally expensive approach grow more acute, creating a pressing need for more efficient methods. In this work, we present , an automatically differentiable (AD), -based code for radiative transfer in curved spacetimes, designed to compute image gradients with respect to underlying model parameters. These gradients quantify how parameter changes-such as the black hole's spin or the observer's inclination-affect the image, enabling more efficient parameter estimation and reducing the number of required images. We validate against in two analytical tests and then compare pixel-wise intensity derivatives from AD with those from finite-difference methods. We then demonstrate the utility of these gradients by performing parameter recovery for an analytical model using a conjugate gradient optimizer in three increasingly complex cases for the injected image: ideal, blurred, and blurred with added noise. In most cases, high-accuracy fits are obtained in only a few optimization steps, failing only in cases with extremely low signal-to-noise ratios and large blurring size kernels. These results highlight the potential of AD-based methods to accelerate robust, high-fidelity model-data comparisons in current and future black hole imaging efforts.

Paper Structure

This paper contains 16 sections, 28 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Derivative Evolution
  4. Numerical Methods
  5. Geodesics Integration
  6. Intensity Integration
  7. The Conjugate Gradient Method
  8. Validation of Jipole
  9. Thin Disk Comparison
  10. Analytical Parameterized Matter Distributions
  11. Automatic Differentiation vs Finite Differences
  12. Data Analysis Application
  13. Inferring the spacetime parameters in Model 5
  14. Inferring the spacetime parameters in Model 5 with Added Blurring
  15. Inferring the spacetime parameters in Model 5 with Added Blurring and Noise
  16. ...and 1 more sections

Figures (11)

  • Figure 1: Image intensity map generated by Jipole (left column) and ipole (middle column) for the thin disk model described in Section \ref{['sec:thin_disk_comparison']}. The right column illustrates the absolute difference between the two images. Total flux density values (see Eq. \ref{['eq:flux_density']}) are overlaid as text on the intensity maps in the left and middle columns, and the NMSE (Eq. \ref{['eq:NMSE']}) is overlaid on the absolute difference map in the right column.
  • Figure 2: Intensity images generated by Jipole (left column) and ipole (middle column), alongside their absolute difference (right column), for the analytical Model 5 presented in Gold_2020. The total flux density ($F_{\rm tot}$) for each image is annotated in the top-left corner of the Jipole and ipole panels, while the NMSE (Eq. \ref{['eq:NMSE']}) on the top-left corner of the right panel.
  • Figure 3: Image comparison of the intensity differential $dI/da$ for each pixel generated by Jipole. Panel (a): $dI/da$ computed using the FD method, shown in symmetric logarithmic scale. Panel (b): $dI/da$ computed using AD, shown in symmetric logarithmic scale. Panel (c): Absolute difference between the FD and AD results, in logarithmic scale; the NMSE (Eq. \ref{['eq:NMSE']}) is indicated in the top-right corner. Panel (d): Relative difference between the FD and AD results, in logarithmic scale. These intensity differentials display the characteristic lensing behavior of black holes: nested rings that are successively rotated and demagnified.
  • Figure 4: Image comparison of $dI/d\theta_o$ for each pixel generated by Jipole. Panel (a): $dI/d\theta_o$ computed using the FD method, shown in symmetric logarithmic scale. Panel (b): $dI/d\theta_o$ computed using AD, shown in symmetric logarithmic scale. Panel (c): Absolute difference between the FD and AD results, in logarithmic scale; the NMSE (Eq. \ref{['eq:NMSE']}) is indicated in the top-right corner. Panel (d): Relative difference between the FD and AD results, in logarithmic scale.
  • Figure 5: Convergence of the value of the observer’s inclination fitting using conjugate gradient optimization for the analytical Model 5. Panel (a): Evolution of the inclination angle $\theta_o$ during iterative fitting, starting from two initial conditions—$\theta_o = 80^\circ$ (blue) and $\theta_o = 5^\circ$ (red)—and converging toward the reference value of $60^\circ$ (dashed black line). Panel (b): The MSE during the $\theta_o$ fitting, showing convergence to the tolerance threshold of $4 \times 10^{-17}$ (dashed black line).
  • ...and 6 more figures