This work presents a novel method for augmented reality in liver surgery. Leveraging advanced reconstruction and segmentation techniques, our approach transfers diverse intraoperative data to Euclidean space, using vessel centerlines as a common representation. Hence, overcoming limitations in operation room equipment and clinician expertise. In addition, we capitalize on the graph-like structure of intraoperative vascular trees to ensure a robust and automatic initial registration. Our approach combines physics-based modeling accuracy with neural network speed for real-time elastic registration. The proposed method is trained on data synthesized from real human acquisitions and validated against a biomechanical model. Our approach achieves a mean target registration error less than 0.6 mm for different synthesized intraoperative cases. Moreover, it demonstrates robustness over different modalities and segmentation errors.