LagrangeGS：动态3D高斯溅射上的非保守拉格朗日系统
LagrangeGS: Non-Conservative Lagrangian System on Dynamic 3D Gaussian Splatting
AI总结：
LagrangeGS将动态3DGS建模为非保守拉格朗日系统，通过近似速度-海森矩阵、限制非保守力与时间无关、引入局部刚性对齐，解决了现有方法的三大问题，实现稳定外推、时间可逆性及物理编辑。
AI中文摘要：
动态3D高斯溅射（3DGS）可实现时变场景的照片级真实感重建，近期的物理感知扩展方法通过显式预测速度场提升了外推能力，但这些扩展仅将向量场拟合到视觉变形，未满足拉格朗日力学要求，导致三大问题：（i）物理上不一致的轨迹；（ii）缺乏时间可逆性；（iii）长期外推过程中出现几何崩溃。本文提出LagrangeGS，将动态3DGS建模为非保守拉格朗日系统，该拉格朗日公式从根本上解决了问题（i）。直接将通用拉格朗日神经网络（LNN）应用于动态3DGS需对数百万高斯粒子进行大尺度速度-海森矩阵求逆，为克服这一计算瓶颈，我们将速度-海森矩阵近似为单位矩阵，解耦粒子动态以提升计算可行性。针对问题（ii），我们限制非保守力显式与时间无关，实现一致的反向积分。最后，为解决问题（iii），我们引入局部刚性对齐对粒子轨迹进行正则化。在动态场景基准上的大量评估表明，LagrangeGS可实现稳定的长期外推、一致的时间可逆性，以及无需重新训练的基于反事实物理的编辑。
英文摘要：
Dynamic 3D Gaussian Splatting (3DGS) achieves photorealistic reconstruction of time-varying scenes, and recent physics-aware extensions improve extrapolation by explicitly predicting velocity fields. However, these extensions merely fit vector fields to visual deformations without satisfying Lagrangian mechanics, leading to three major issues: (i) physically inconsistent trajectories, (ii) lack of time-reversibility, and (iii) geometric collapse during long-term extrapolation. In this paper, we propose LagrangeGS, which formulates dynamic 3DGS as a non-conservative Lagrangian system. While this Lagrangian formulation fundamentally solves (i), a direct application of general LNNs to dynamic 3DGS requires a large velocity-Hessian inversion for millions of Gaussian particles. To overcome this computational bottleneck, we approximate the velocity-Hessian as an identity matrix, decoupling particle dynamics for computational tractability. For (ii), we restrict the non-conservative forces to be explicitly time independent, enabling consistent backward integration. Finally, to address (iii), we introduce local rigid alignment that regularizes particle trajectories. Extensive evaluations on dynamic scene benchmarks demonstrate that LagrangeGS enables stable long-term extrapolation, consistent time reversal, and counterfactual physics-based editing without retraining.
