As emerging concepts, leveraging physical information with machine learning to construct hybrid models enables a scientific data-driven paradigm for the investigation of complex kinetic issues in the field of polymer science. Herein, the model of physics-informed neural networks (PINNs), integrating advantages of neural networks with physics-based knowledge, is extended to predict the spatiotemporal patterns of phase-separated microstructures of multicomponent polymers. It is demonstrated that the PINN-based data-driven model can achieve the forward, backward and bidirectional predictions of spatiotemporally phase-separated patterns of homopolymer blends. All predicted patterns reveal a high accuracy and robustness of PINN-based data-driven model by leveraging a small amount of training data. Particularly, the PINN-based method can be harnessed to infer the latent physical law about the growth kinetics of phase-separated microstructures. In addition, PINN-based data-driven model can also be generalized to forecast the spatiotemporal patterns of microphase-separated nanostructures of block copolymers and infer their growth kinetics. Our study opens the possibility of learning non-equilibrium phenomena of complex polymer systems from only a few snapshots of microstructural evolution.