In this study, a novel physics-informed neural network (PINN) is proposed to allow efficient training with improved accuracy. PINNs typically constrain their training loss function with differential equations to ensure outputs obey underlying physics. These differential operators are typically computed via automatic differentiation (AD), but this can fail with insufficient collocation points. Hence, the idea of coupling both AD and numerical differentiation (ND) is employed. The proposed coupled-automatic-numerical differentiation scheme (can-PINN) strongly links collocation points, thus enabling efficient training while being more accurate than simply using ND. As a demonstration, two instantiations of can-PINN were derived for the incompressible Navier-Stokes equations and applied to modeling of lid-driven flow in a cavity. Results show that can-PINNs can achieve very good accuracy even when the corresponding AD-based PINN fails.