Phase-field models are essential in materials design and optimization by enabling accurate simulation of microstructure evolution, particularly for grain growth and spinodal decomposition. Despite their strong physical fidelity, phase-field models suffer from high computational cost, which limits their applicability to large-scale or long-time simulations. To address this limitation, data-driven machine learning approaches have been explored as surrogate models. However, these methods typically require a large amount of phase-field simulation data and often lack physical consistency when extrapolated beyond the training domain. Thus, in this work, a Physics-Informed Neural Network (PINN) framework is developed to simulate grain growth and spinodal decomposition without using any training data, relying solely on the governing partial differential equations (PDEs), namely the Allen-Cahn and Cahn-Hilliard equations. The proposed framework employs Fourier feature-based gated neural network with adaptive skip connections. In addition, to enable stable learning of long-time microstructure evolution, a time-marching scheme is adopted, in which the temporal domain is decomposed into overlapping windows and the network is sequentially trained using the predicted solution at the final time of each window as the initial condition for the next. This strategy enforces temporal causality and allows the PINN to accurately capture long-term microstructure evolution. Consequently, for grain growth, phase-field simulation results for systems containing three and five grains are accurately reproduced by the proposed PINN. Diffusive phenomena at grain boundaries and interactions between neighboring grains are captured in physically consistent manner. For spinodal decomposition, despite the increased complexity of the governing equation, the proposed PINN successfully reproduces the coarsening stage by accurately capturing the temporal evolution of composition distributions. Overall, the proposed framework offers a method to reduce dependence on data and to enable physically consistent extrapolation beyond training domain, thereby providing a foundation for efficient materials design and optimization using artificial intelligence.