The accurate prediction of laminar-turbulent transition is critically important for many aeronautical applications. Transition models that perform satisfactorily at low-Mach numbers were shown not to be suitable for supersonic and hypersonic flows as the transition mechanisms are different and substantially more complex. As a first step towards the development of a hypersonic transition model, a methodology for deriving amplification factor transport equations based on neural networks that are trained with results from linear stability theory computations is proposed. As an example, an incompressible amplification factor transport equation is developed and integrated into the three-equation 2019 Coder transition model (AIAA-2019-0039). The new neural network model is validated for an incompressible zero-pressure-gradient flat-plate boundary layer with a Tollmien-Schlichting primary instability. The model is also validated for an S809 airfoil and a linear low-pressure turbine cascade that develop laminar separation bubbles. Finally, to demonstrate that the model is also suited for three-dimensional problems, the flow through a linear cascade with endwall effects is considered.