Traditionally, modeling single-screw extrusion is based on Newtonian flow theory, and yields analytical solutions for the conveying characteristics of metering zones, giving first insight into flow patterns of the polymer melts flow. Numerical methods are required to solve the flow field of non-Newtonian polymer melts. Due to the increase of computational power, numerical simulations became an important tool for solving these shear-thinning flows. Nevertheless, fully three-dimensional analyses still require intensive expert knowledge, time for model preparation, and computational resources. With this research (Part A), we present a hybrid modeling approach to predict the pumping capability and viscous dissipation in single-screw extrusion by combining analytical, numerical, and data-based modeling techniques. A comprehensive numerical design study was carried out, varying the identified independent influencing parameters within a very wide range of application. This data set served as input for the symbolic regression based on genetic programming. Analytical approximation equations for the pumping capability and viscous dissipation were developed in dimensionless representation. As result of the three-dimensional modeling approach the shear-thinning fluid behavior, the effect of the cross-channel flow, and the influence of the flight flanks are included. Fast, stable, and accurate predictions of flow rate and viscous dissipation are possible, without further numerical simulations. Furthermore, the models are suitable for predicting the axial melt-temperature development, non-isothermal extrusion characteristics, required drive power, and application to injection molding machines is enabled. Due to the simple algebraic structure they can easily be implemented in any expert system like screw simulation routines, digital twins, soft sensors, and model-based control. In Part B, these novel melt-conveying models were validated against experimental extrusion data.