A Convolutional Neural Network for the Recovery of Transfer Functions From Velocity-Resolved Reverberation Mapping Data

TL;DR

This work tackles the ill-posed problem of recovering the velocity-delay transfer function $\Psi(\nu,\tau)$ in reverberation mapping, which constrains the broad-line region around supermassive black holes. It introduces a custom (D)CNN that learns a deconvolution operator mapping continuum and line lightcurves to $\Psi$, trained on synthetic DRW-driven data convolved with diverse transfer functions and augmented with noise and gaps. An ensemble of models provides robust 1D and 2D reconstructions, showing accuracy comparable to analytic MEMEcho, resilience to missing data, and the ability to adapt to new continua via transfer learning. The approach offers a scalable path to extract BLR structure from upcoming RM datasets and could be extended to RM problems in disks and tori, albeit with limitations in interpretability and the need for realistic training sets.

Abstract

One of the hallmarks of active galactic nuclei are that they are highly variable with time. In watching the spectra vary it has been observed that the emission-lines often appear to "reverberate" -- that is they vary in response to continuum variations assumed to originate close to the black hole. This critical observation underlies the reverberation mapping technique, an elegant physics experiment that has allowed us to characterize the environment around many supermassive black holes in nearby active galactic nuclei. Recent observations are of such quality that the response can be measured as a function of velocity across the emission-line, and in doing so we can construct velocity-delay maps that show the structure and physics of the gas in the broad-line region better than any other measurement to date. Unfortunately constructing such maps requires a deconvolution, and given that the data are often noisy and with gaps such deconvolutions are non-trivial. Here we present a novel deconvolution method for the recovery of velocity-delay maps using a custom convolutional neural network architecture, showcasing that such methods have great promise for the deconvolution of reverberation mapping data products. While we have designed this new method with the BLR in mind, in principle this technique could be applied to any reverberation deconvolution problem, including in the accretion disk and torus.

Figures (11)

• Figure 1: Top: Sample 1D transfer functions for each "basis" function used in training the model. Middle: Sample lightcurves resulting from the convolution of the corresponding transfer function from the top panel with the continuum. Bottom: The DRW continuum lightcurve convoled with the top panel to generate the middle panel.
• Figure 2: Top grid: Sample recoveries of each of the six "basis" transfer functions shown in Figure \ref{['fig:1Dtrain']}. Bottom: Distribution of errors for all of the model predictions in the validation set.
• Figure 3: Comparing how the novel (D)CNN and existing analytic MEMEcho techniques perform when the training data quality are degraded Left: Recoveries of a more complicated 1D transfer function that is the result of the combination of several of the "basis" functions shown in Figure \ref{['fig:1Dtrain']} and Figure \ref{['fig:1DSampleResults']}. The top panel shows the (D)CNN recovery with error envelope in blue, the MEMEcho recovery with error envelope in purple, and the ground truth as the dashed green line. The bottom panel shows the residuals with the same colors. Middle: Sample recoveries of the same transfer function with the same colors as the left panel, but this time from data with random observational gaps equaling 10% of the total number of data points. Right: Sample recoveries of the same transfer function with the same colors as the left panel, but this time from data with random observational gaps equaling 50% of the total number of data points.
• Figure 4: The distribution of errors for the sample ensemble of 2D models. Full examples of each model transfer function and lightcurve inputs are in Figure \ref{['fig:2DSamples']} in the \ref{['appendix']}, and a sample inversion for a complicated hybrid BLR is shown in Figure \ref{['fig:2Dcombined']}. Vertical lines showcase the median error of each sub-population for each of the basis functions.
• Figure 5: a: Sample full 2D transfer function of a complicated combined synthetic BLR with both a "blob" component superimposed on a more traditional virial "disk-wind" component. b: (D)CNN model recovery of the full 2D transfer function. c: Residuals (prediction in top middle panel - ground truth shown in top left panel). Regions where the model overpredicts are shown in red, underpredicted regions are shown in blue, and areas in white are where the model prediction is within $\pm0.1\%$ of the maximum ground truth value. d: The 2D uncertainty in the ensemble of predictions for this lightcurve. e: The 1D transfer function (2D maps integrated over velocity, see Figure \ref{['fig:1DSampleResults']}) showing the mean model prediction in blue (with ribbon corresponding to uncertainty), the ground truth in green, and the residuals in red. f: The 1D line profile (2D maps integrated over delay) again showing the mean model prediction in blue (with ribbon corresponding to the uncertainty), the ground truth in green, and the residuals in red. g: Mean loss curves for the ensemble of models, with the validation shown in blue and the training set shown in red. Note that the models stop learning after $\sim$ 30-40 epochs.