What modelling approaches are used to predict the aurora?
One model is the OVATION auroral precipitation model.

OVATION stands for Oval  Variation,  Assessment,  Tracking,  Intensity, and  Online  Nowcasting.

The NOAA Space Weather Prediction Center uses the OVATION Prime model for their 30-minute aurora forecast. They use a modified version of the model for their 3-day forecast. It was devised by Newell et al. (2010) at Johns Hopkins University. 

You can read all about it in Machol et al. (2013). The paper is open access and contains several equations and a description of the method. They explain:


  The OVATION Prime model is derived from electron and proton
  flux measurements from the SSJ4 detectors on the DMSP satellites.


SSJ4 sensors, according to one source:


  [...consist] of an array of four cylindrical, curved plate, electrostatic analyzers configured into two pairs; one pair each to measure electrons and ions. Each pair covers the energy range from 30 eV to 30 keV in 20 channels spaced at equal intervals in energy on a logarithmic scale. 


References

Machol, J. L., J. C. Green, R. J. Redmon, R. A. Viereck, and P. T. Newell (2012), Evaluation of OVATION Prime as a forecast model for visible aurorae, Space Weather, 10, S03005, doi:10.1029/2011SW000746.

Newell, P. T., T. Sotirelis, and S. Wing (2010a), Seasonal variations in diffuse, monoenergetic, and broadband aurora, J. Geophys. Res., 115, A03216, doi:10.1029/2009JA014805.