We address dynamic manipulation of deformable linear objects by presenting SPiD, a physics-informed self-supervised learning framework that couples an accurate deformable object model with an augmented self-supervised training strategy. On the modeling side, we extend a mass-spring model to more accurately capture object dynamics while remaining lightweight enough for high-throughput rollouts during self-supervised learning. On the learning side, we train a neural controller using a task-oriented cost, enabling end-to-end optimization through interaction with the differentiable object model. In addition, we propose a self-supervised DAgger variant that detects distribution shift during deployment and performs offline self-correction to further enhance robustness without expert supervision. We evaluate our method primarily on the rope stabilization task, where a robot must bring a swinging rope to rest as quickly and smoothly as possible. Extensive experiments in both simulation and the real world demonstrate that the proposed controller achieves fast and smooth rope stabilization, generalizing across unseen initial states, rope lengths, masses, non-uniform mass distributions, and external disturbances. Additionally, we develop an affordable markerless rope perception method and demonstrate that our controller maintains performance with noisy and low-frequency state updates. Furthermore, we demonstrate the generality of the framework by extending it to the rope trajectory tracking task. Overall, SPiD offers a data-efficient, robust, and physically grounded framework for dynamic manipulation of deformable linear objects, featuring strong sim-to-real generalization.