The propagation of oceanic internal tides (IT) is influenced by their interactions with balanced vortical ocean flows. Satellite measurements are powerful tools to observe ocean flows globally but lack the temporal resolution to distinguish between balanced and IT flows using harmonic filtering. This presents a challenge for inferring ocean circulation and the propagation of ITs. We take two approaches to determine the strength of their interactions and the resulting impact on measuring ocean flows. The first is an idealized study where we model a plane wave interacting with an isolated (cyclo)geostrophic vortex using shallow water simulations. We vary the Rossby number, Ro, the Burger number, Bu, and the size of the vortex compared to the wave wavelength, K. We measure the percentage of wave energy scattering S for each simulation and find S ~ (FrK)^2 when S < 20, where Fr = Ro/Bu^(0.5) is the Froude number. When S>20, S plateaus. In the second approach, we train a deep learning model called U-Net on Boussinesq simulation data, which models a mode-1 IT propagating through quasi-geostrophic flows. The inputs to the model are combinations of the sea surface measurements for velocity U, temperature T, and height H, and the output is the wave component of the height field. U is the most valuable input, followed by H, and T hardly has any predictive power in isolation. When all three fields are used in combination as input, we obtain close to perfect performance. Lastly, we explore two alternative deep learning models to compare their strengths and weaknesses, namely, a cGAN with a novel spectral loss function, and a so-called deformation model. They do not show obvious improvements over the U-Net. These models give insight into which sea surface variables are most useful for disentangling ITs and balanced flows, and thus where resources should be allocated to ultimately measure ocean circulation.