Phase-field models have become an efficient and flexible solution for simulating the evolution of microstructure and phase separation at the mesoscale, whose applications can be easily extended by coupling to other physical systems to simulate multi-physics phenomenon. The physical-informed neural network is a recent promising development in scientific computing by penalizing physical equations as part of the loss function so that the network can predict the physics and infer system parameters from data. In this work, we present the first application of PINNs in the context of solving phase field equations, namely Allen-Cahn equation (AC), Cahn-Hilliard (CH) equation and their coupling with Navier-Stokes (NS) equations. For the high-order CH equation, we solve the system by decoupling the high order equation into a system of equations with residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. The trained network predicts the multi-physics field and infers the phase parameters for the AC-NS and the CH-NS problem with a good agreement. The performance of PINNs in solving phase-field equations facilitates its applications in further applications in material sciences, clot formation, fracture modeling, etc.