<p>Accurate precipitation nowcasting is crucial for mitigating the impacts of extreme weather, especially as climate change increases their frequency and severity. Traditional methods, such as numerical weather prediction and radar extrapolation, face limitations in short-term and high-resolution forecasting. Recently, while deep learning approaches have advanced nowcasting by learning spatiotemporal patterns from radar data, they often suffer from blurry results due to uncertainty predictions and limited physical consistency. To deal with these challenges, we propose ThoR, a Motion-Dependent Physics-Informed Deep Learning Framework with Constraint-Centric Theory of Functional Connections for Rainfall Nowcasting. ThoR integrates attention-centric spatio-temporal modeling with explicit physical constraints derived from partial differential equations (PDEs) for forward simulation, which employs a cascaded-branch architecture that integrates an attention-driven generator with an unsupervised, lead-time-conditioned module for motion field extraction. Physical consistency is enforced by weighted embedding the advection-diffusion equation directly into the optimization objective, establishing a Theory of Functional Connections (TFC) framework tailored for precipitation nowcasting. Extensive experiments on real-world radar datasets demonstrate that ThoR consistently outperforms existing methods in deterministic metrics, particularly at longer lead times and during extreme weather events, highlighting the potential of physics-informed deep learning for operational nowcasting.</p>