Inverse statistical mechanics is a powerful optimization methodology that has been widely applied to design optimal isotropic pair interactions that robustly yield a broad spectrum of target many-particle configurations or physical properties. In this work, we generalize inverse techniques to design experimentally realizable spherical colloidal particles with optimized "patchy" anisotropic interactions for a wide class of targeted low-coordinated two-dimensional crystal structures that are defect-free. Our target crystals include square, honeycomb, kagome, and parallelogrammic crystals. The square, honeycomb, and kagome crystals possess desirable photonic, phononic, and magnetic properties, which are useful for a wide range of applications. We demonstrate that these target configurations can be robustly achieved with relatively few defects at sufficiently low temperatures. Our findings provide experimentalists with the optimal parameters to synthesize these crystals with patchy colloids under standard laboratory conditions.