This research explores recent advancements in solving Partial Differential Equations (PDEs) through a fusion of data-driven methods and Physics-Informed Neural Networks (PINNs). Traditional numerical techniques encounter challenges with complex systems in modeling diverse physical phenomena. The study focuses on applying PINNs to efficiently solve elliptic PDEs, crucial equations in scientific and engineering disciplines. Elliptic PDEs, known for their steady-state nature, pose challenges for traditional methods due to high computational costs over complex domains. The research utilizes a comprehensive dataset, synthetically generated for PINN training and accuracy validation. The study introduces a novel approach to constructing a PINN that not only efficiently solves elliptic PDEs but also quantifies prediction uncertainties. This dual capability is crucial for applications where decision-making relies on understanding the reliability of model outputs. The results demonstrate the potential of PINNs to revolutionize elliptic PDE solving, offering a fast, accurate, and physically consistent method that inherently accounts for uncertainty, advancing computational science.