Melt viscosity is regarded as a key quality indicator of the polymer melt in polymer extrusion processes. However, limitations such as disturbances to the melt flow and measurement delays of the existing in-line and side-stream rheometers prevent the monitoring and controlling of this key parameter in real time. Soft sensors can be employed to monitor physical parameters that are difficult to measure using hardware sensing instruments. This study presents a grey-box soft sensing solution to predict the melt viscosity in real time, which combines physics-based knowledge with machine learning. A fine-tuned physics-based mathematical model is used to make melt viscosity predictions, and a deep neural network is employed to compensate for its prediction errors. The proposed soft sensor model reported a normalised root mean square error of 2.2[Formula: see text]10