The Surface Water and Ocean Topography (SWOT) satellite is expected to observe the sea surface height (SSH) down to scales of ∼10-15 kilometers. While SWOT will reveal submesoscale SSH patterns that have never before been observed on global scales, how to extract the corresponding velocity fields and underlying dynamics from this data presents a new challenge. At these soon-to-be-observed scales, geostrophic balance is not sufficiently accurate, and the SSH will contain strong signals from inertial gravity waves — two problems that make estimating surface velocities non-trivial. Here we show that a data-driven approach can be used to estimate the surface flow, particularly the kinematic signatures of smaller scales flows, from SSH observations, and that it performs significantly better than directly using the geostrophic relationship. We use a Convolution Neural Network (CNN) trained on submesoscale-permitting high-resolution simulations to test the possibility of reconstructing surface vorticity, strain, and divergence from snapshots of SSH. By evaluating success using pointwise accuracy and vorticity-strain joint distributions, we show that the CNN works well when inertial gravity wave amplitudes are weak. When the wave amplitudes are strong, the model may produce distorted results; however, an appropriate choice of loss function can help filter waves from the divergence field, making divergence a surprisingly reliable field to reconstruct in this case. We also show that when applying the CNN model to realistic simulations, pretraining a CNN model with simpler simulation data improves the performance and convergence, indicating a possible path forward for estimating real flow statistics with limited observations.