Photometric Redshift PDFs via Neural Network Classification for DESI Legacy Imaging Surveys and Pan-STARRS
We present a neural network classification (NNC) method for photometric redshift estimation that produces well-calibrated redshift probability density functions (PDFs). The method discretizes the redshift space into ordered bins and optimizes the Continuous Ranked Probability Score (CRPS), which respects the ordinal nature of redshift and naturally provides uncertainty quantification. Unlike traditional regression approaches that output single point estimates, our method can capture multi-modal posterior distributions arising from color-redshift degeneracies. We apply this method to the DESI Legacy Imaging Surveys Data Release 10 (LSDR10) and Pan-STARRS Data Release 2 (PS1DR2), using an unprecedented spectroscopic training sample from DESI DR1 and SDSS DR19. Our method achieves $\sigma_{\mathrm{NMAD}} = 0.0153$ and $\eta = 0.50\%$ on LSDR10, and $\sigma_{\mathrm{NMAD}} = 0.0222$ and $\eta = 0.34\%$ on PS1DR2 combined with unWISE infrared photometry. The NNC method outperforms Random Forest, XGBoost, and standard neural network regression. We demonstrate that DESI DR1 significantly improves photo-$z$ performance at $z > 1$, while the combination of deep optical photometry and mid-infrared coverage is essential for achieving high precision across the full redshift range. We provide a unified photometric redshift catalog combining LSDR10 and PS1DR2 with a hierarchical model selection strategy based on available photometry. The well-calibrated PDFs produced by our method are valuable for cosmological studies and can be extended to next-generation surveys such as CSST, Euclid, and LSST.