Computerized fluid dynamics models of particle deposition in the human airways are used to characterize deposition patterns that enable the study of lung diseases like asthma and chronic obstructive pulmonary disease (COPD). Despite this fact, the influence of patient-specific geometry on the deposition efficiency and patterns is not well documented nor modeled. In part, this is due to the complexity of simulating the full Computational Fluid Dynamics (CFD) solution in patient-specific airway geometries, a factor that becomes a major hurdle for patient-specific studies given the complexity of the geometry of human lungs and their related airflow. In this paper, we present an approximation method based on neural networks to the Navier-Stokes equations that govern airway flow in a Physiologically Realistic Bifurcation (PRB) model for the conducting region of a single generation human airway branch. The flow distribution and deposition of tobacco particles have been simulated for the inspiratory regime using ANSYS Fluent and a neural network has been trained to regress the mean velocity and mass flow components. Our results show that the approximation works well under the modeled assumptions and the serial application of this model to a two-generation airway geometry provides reasonable approximations.