Estimating $T_2$ relaxation time distributions from multi-echo $T_2$-weighted MRI ($T_2W$) data can provide valuable biomarkers for assessing inflammation, demyelination, edema, and cartilage composition in various pathologies, including neurodegenerative disorders, osteoarthritis, and tumors. Deep neural network (DNN) based methods have been proposed to address the complex inverse problem of estimating $T_2$ distributions from MRI data, but they are not yet robust enough for clinical data with low Signal-to-Noise ratio (SNR) and are highly sensitive to distribution shifts such as variations in echo-times (TE) used during acquisition. Consequently, their application is hindered in clinical practice and large-scale multi-institutional trials with heterogeneous acquisition protocols. We propose a physically-primed DNN approach, called $P_2T_2$, that incorporates the signal decay forward model in addition to the MRI signal into the DNN architecture to improve the accuracy and robustness of $T_2$ distribution estimation. We evaluated our $P_2T_2$ model in comparison to both DNN-based methods and classical methods for $T_2$ distribution estimation using 1D and 2D numerical simulations along with clinical data. Our model improved the baseline model's accuracy for low SNR levels ($SNR<80$) which are common in the clinical setting. Further, our model achieved a $\sim$35\% improvement in robustness against distribution shifts in the acquisition process compared to previously proposed DNN models. Finally, Our $P_2T_2$ model produces the most detailed Myelin-Water fraction maps compared to baseline approaches when applied to real human MRI data. Our $P_2T_2$ model offers a reliable and precise means of estimating $T_2$ distributions from MRI data and shows promise for use in large-scale multi-institutional trials with heterogeneous acquisition protocols.