LSTM-MDNz: Estimating Quasar Photometric Redshifts with an LSTM-augmented Mixture Density Network

Quasar photometric redshifts are essential for studying cosmology and large-scale structures. However, their complex spectral energy distributions cause significant redshift–color degeneracy, limiting the accuracy of traditional methods. To overcome this, we introduce long short-term memory (LSTM)-MDNz, a novel end-to-end deep learning model combining LSTM networks with mixture density networks (MDNs). The model directly uses multiband photometric fluxes and associated errors as wavelength-ordered sequential inputs, eliminating the need for manual feature engineering while enabling simultaneous point estimation and probability density function (pdf) prediction of quasar redshifts. We integrate data from four major sky surveys—Sloan Digital Sky Survey (SDSS), DESI-Legacy Imaging Surveys, Wide-field Infrared Survey Explorer (WISE), and Galaxy Evolution Explorer—to assemble a sample of over 550,000 spectroscopically confirmed quasars (0 ≤ zspec ≤ 5) across 14 ultraviolet to infrared bands for model training and testing. Experimental results show that using all 14 bands yields optimal performance, with a normalized median absolute deviation (σNMAD) of 0.037 and an outlier rate (fout) of 3.5% on the test set. These values represent reductions of 29% and 56%, respectively, compared to the commonly adopted SDSS+WISE band set. Probability integral transform and continuous ranked probability score analyses confirm that the predicted pdfs align closely with the true redshift distribution. Band-ablation experiments further highlight the essential role of ultraviolet and infrared data in alleviating color degeneracy and reducing systematic bias. This study demonstrates the effectiveness of multiband fusion in improving quasar photo-z accuracy and offers a ready-to-use estimation framework for future surveys like LSST, CSST, and Euclid.