VAR-PZnn: A machine-learning framework for AGN photometric redshifts using color and variability-based features

Photometric redshift estimation for active galactic nuclei (AGNs) remains a fundamental challenge for current and upcoming large-scale photometric surveys. Traditional spectral energy distribution (SED) fitting suffers from color-redshift degeneracies, particularly for AGNs whose power-law continua hide the strong spectral features required to anchor redshift estimates. While AGN variability provides additional constraining power, existing frameworks require multi-band light curves that are not always available. This work presents VAR-PZnn, a fully connected mixture density network that integrates 26 variability features extracted from ZTF g-band light curves with optical photometry from Pan-STARRS1, mid-infrared (MIR) photometry from CatWISE, and, for a subsample, NIR photometry from UKIDSS. The model is trained and tested on 72,728 spectroscopically confirmed AGNs/QSOs spanning 0.01 < z < 4.5 and g-band magnitudes from 17 to 21.5. For the main sample, we achieve \sigma_{NMAD} = 0.058 and an outlier fraction of \eta = 8.2%, which reduces to 5.4% when the 10% of sources with the highest predicted uncertainty are excluded. An ablation study demonstrates that MIR photometry provides the dominant constraint for photo-z accuracy, while variability features serve as a secondary refiner. Using UKIDSS NIR data as a proxy for future synergies between LSST and space-based missions like Euclid and Roman, we obtain \eta = 13.3% without MIR data and \eta = 4.6% when MIR is available. We benchmark against Low-Resolution Templates (LRT) SED fitting (\eta = 28.7%) and the VAR-PZ framework; applying single-band VAR-PZ priors worsens LRT performance to \eta = 39.4% due to single-band light-curve degeneracies, confirmed via simulations (\eta = 27.6% to 28.1%). This framework provides a scalable approach for the Legacy Survey of Space and Time (LSST).