Global surface snowfall rate estimation is crucial for hydrological and meteorological applications but is still a challenging task. We present a novel approach to comprehensively consider passive microwave, infrared and physical constraints using deep neural networks with attention module for retrieving surface snowfall rate, namely PCSSR-DNNWA. PCSSR-DNNWA outperforms traditional approaches in predicting surface snowfall rate with CC ~ 0.75, ME ~ -0.03 mm/h, and RMSE ~ 0.21 mm/h. In addition, we found that graupel water path (GWP) is of vital importance with largest contributions in retrieving surface snowfall rate. Integrating the physical constraints, PCSSR-DNNWA paves a new avenue for retrieving satellite-borne surface snowfall rate by intelligently considering the varying importance of the multiple predictors, resulting in increased accuracy, interpretability, and computational efficiency.