Machine-learning models that relate local atomic structure to the thermodynamic state of metallic glasses typically assess physical consistency after training rather than enforcing it during learning. Here, we develop a multi-task physics-informed neural network (PINN) that predicts temperature directly from Voronoi-motif population histograms while incorporating autograd-derived gradient constraints representing physically motivated relationships between structural motifs, quench rate, and temperature. The model simultaneously classifies each configuration as liquid, supercooled/transition, or glass through an auxiliary classification head. The classification task achieves 96.6% test accuracy with a macro-F1 score of 0.95, while the regression head yields a mean absolute error of 23.4 K on a trajectory-disjoint held-out test set. Five-fold trajectory-grouped cross-validation, deep-ensemble predictions, and Monte Carlo dropout are used to assess model robustness and predictive uncertainty. Sensitivity analyses demonstrate that physics constraints can be incorporated over a broad range of loss weights without compromising predictive accuracy, while substantially improving compliance with the prescribed physical trends. Benchmarking against conventional machine-learning regressors further demonstrates competitive predictive performance. SHAP analysis across the cross-validation ensemble identifies the coupled near-icosahedral motif family, particularly the full icosahedral motif and its single-atom-perturbed counterpart, as the dominant structural fingerprints of the glassy state. These results demonstrate that physically motivated constraints can be embedded directly into motif-based structure-property models, providing a pathway from post-hoc interpretability toward physically constrained machine learning for metallic glasses.