Abstract: Achieving superior polymeric components through additive manufacturing (AM) relies on precise control of rheology. One key rheological property particularly relevant to AM is melt viscosity ($\eta$). Melt viscosity is influenced by polymer chemistry, molecular weight ($M_w$), polydispersity, induced shear rate ($\dot{\gamma }$), and processing temperature ($T$). The relationship of $\eta$ with $M_w$, $\dot{\gamma }$, and $T$ may be captured by parameterized equations. Several physical experiments are required to fit the parameters, so predicting $\eta$ of a new polymer material in unexplored physical domains is a laborious process. Here, we develop a Physics-Enforced Neural Network (PENN) model that predicts the empirical parameters and encodes the aforementioned equations to calculate $\eta$ as a function of polymer chemistry, $M_w$, polydispersity, $\dot{\gamma }$, and $T$. We benchmark our PENN against physics-unaware Artificial Neural Network (ANN) and Gaussian Process Regression (GPR) models. Finally, we demonstrate that the PENN offers superior values of $\eta$ when extrapolating to unseen values of $M_w$, $\dot{\gamma }$, and $T$ for sparsely seen polymers.