Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite-Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite-basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operators. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosities, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.