Accurate registration of vascular shapes is essential for comparing anatomical geometries, extracting reliable measurements, and generating realistic models in cardiovascular research. Conventional surface registration methods often face limitations in efficiency, scalability, and generalization across shape cohorts. In this work, we present AD–SVFD, a deep learning framework that simultaneously performs deformable registration of vascular geometries to a pre–defined reference anatomy and enables the synthesis of new shapes. AD–SVFD represents each geometry as a point cloud and models ambient deformations as solutions at unit time of ordinary differential equations (ODEs), whose time–independent right–hand sides are parameterized by neural networks. Registration is optimized by minimizing the Chamfer distance between deformed and reference geometries, while shape generation is achieved by integrating the ODE backward in time from sampled latent codes. A distinctive auto–decoder architecture associates each anatomy with a low–dimensional embedding, jointly optimized with the network parameters during training, and fine–tuned at inference, reducing computational overhead. Numerical experiments on healthy aortic anatomies demonstrate the capability of AD–SVFD to yield accurate approximations at competitive computational costs. Compared to existing approaches, our model offers an efficient, unified framework for processing multiple shapes and robustly generating plausible geometries.