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Microstructure-sensitive material design has become popular among materials engineering researchers in the last decade because it allows the control of material performance through the design of microstructures. In this study, the microstructure is defined by an orientation distribution function (ODF). A physics-informed machine learning approach is integrated into microstructure design to improve the accuracy, computational efficiency, and explainability of microstructure-sensitive design. When data generation is costly and numerical models need to follow certain physical laws, machine learning models that are domain-aware perform more efficiently than conventional machine learning models. Therefore, a new paradigm called Physics-Informed Neural Network (PINN) is introduced in the literature. This study applies the PINN to microstructure-sensitive modeling and inverse design to explore the material behavior under deformation processing. In particular, we demonstrate the application of PINN to small-data problems driven by a crystal plasticity model that needs to satisfy the physics-based design constraints of the microstructural orientation space. For the first problem, we predict the microstructural texture evolution of Copper during a tensile deformation process as a function of initial texturing and strain rate. The second problem aims to calibrate the crystal plasticity parameters of Ti-7Al alloy by solving an inverse design problem to match PINN-predicted final texture prediction and the experimental data.