用随机微分方程重构建模动态3D高斯点云的变形场，提升稳定性。
Stochastic Liquid Deformation Fields: An SDE Generalisation of Closed-Form Continuous-Time Cells for Dynamic 3D Gaussian Splatting
- 将闭式连续时间单元转为带噪声项的随机微分方程，实现连续时间建模
- 在合成数据上性能与确定性模型相当，真实场景中加噪无提升
- 揭示了噪声对液态神经网络有效性的作用边界，适合研究动态3D重建者
可变形3D高斯点云（D-3DGS）通过时变变形场对一组初始3D高斯点进行形变来重建动态场景。将原MLP替换为闭式连续时间（CfC）细胞——一种能解析求解液态时间常数微分方程的液态神经网络——使变形场具备前向计算成本下的连续时间行为。然而，该闭式解仅为噪声驱动系统的确定性极限，丢失了液态网络通常依赖的随机项带来的鲁棒性。本文重新引入小幅度高斯扰动至每个CfC单元的时间门，将确定性场转化为简洁的随机微分方程（SDE）形式。噪声仅用于训练阶段，无需求解器，关闭后即精确退化为原始CfC。在合成的D-NeRF场景中，随机场性能与确定性CfC相当，优于多数情况下的MLP基线；而在真实世界NeRF-DS场景中，确定性版本已最优，加噪无益。研究因此提供了一种清晰的CfC场作为SDE的解读方式，并如实揭示了简单噪声项在何时有效、何时无效。
原文摘要 · Abstract (English)
Deformable 3D Gaussian Splatting (D-3DGS) reconstructs dynamic scenes by deforming a canonical set of 3D Gaussians through a deformation field of frame time. Replacing its MLP with a stack of Closed-form Continuous-time (CfC) cells-a Liquid Neural Network that solves the Liquid Timeconstant ODE in closed form-gives the field continuous-time behaviour at feed-forward cost. That closed form, however, is only the deterministic limit of a noise-driven system, and drops the stochastic term usually credited for the robustness of liquid networks. We put it back: a small Gaussian perturbation is added to the time gate of every CfC cell, turning the deterministic field into a simple stochastic (SDE) one. The noise is used only during training, needs no solver, and reduces exactly to the CfC when switched off. On the synthetic D-NeRF scenes the stochastic field is on par with the deterministic CfC and beats the MLP baseline on most scenes; on the real-world NeRF-DS scenes the deterministic limit is already best and adding noise does not help. The study thus gives both a clean way to read the CfC field as an SDE and an honest account of when a plain noise term helps and when it does not.
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