Neutral particle transport is an important phenomenon in many applications, mainly regarding radiative heat and neutron transport. The fundamental modeling is given by the linear Boltzmann equation, a first-order integro-differential equation. The ANN-MoC method is a novel variant of the classical application of the method of characteristics (MoC) to the solution of transport problems. It couples an artificial neural network (ANN) to perform estimations of the incident density of particles. The training of the ANN is embedded into a type of source iteration scheme. The flexibility of the training scheme allows the method to be applied to data-driven (direct or inverse) problems, where data is known at collocation points on the boundary or in the domain. The solution processing yields an ANN able to estimate the density at any point of the computational domain. Applications of the novel method are presented to a manufactured solution problem in 1D geometry. The results highlight the potentialities of the novel method to data-driven transport problems.