Next generation neutrino oscillation experiments are expected to measure the remaining oscillation parameters with very good precision. They will have unprecedented capabilities to search for new physics that modify oscillations. DUNE, with its broad band beam, good particle identification, and relatively high energies will provide an excellent environment to search for new physics. If deviations from the standard three-flavor oscillation picture are seen however, it is crucial to know which new physics scenario is found so that it can be verified elsewhere and theoretically understood. We investigate several benchmark new physics scenarios by looking at existing long-baseline accelerator neutrino data from NOvA and T2K and determine at what sensitivity DUNE can differentiate among them. We consider sterile neutrinos and both vector and scalar non-standard neutrino interactions, all with new complex phases, the latter of which could conceivably provide absolute neutrino mass scale information. We find that, in many interesting cases, DUNE will have good model discrimination. We also perform a new fit to NOvA and T2K data with scalar NSI.