GaussPDE: Graph-Based Partial Differential Equation-Driven Rendering for 3D Gaussian Splatting
Authors: Haoyuan Yue, Fengyuan Ye, Ziyin Li
Organizations: Department of Physics, Westlake University, Hangzhou, China · The Chinese University of Hong Kong, Shenzhen, China · Department of Accounting, Xi’an Jiaotong-Liverpool University, Suzhou, China
We present GaussPDE, a framework that injects physically structured partial differential equation (PDE) dynamics into pretrained 3D Gaussian scenes without mesh extraction, voxelization, or retraining. Our key observation is that PDE rendering requires not only accurate appearance, but also a reliable discrete computational domain. We therefore first introduce camera-aware regularization during 3DGS reconstruction to suppress camera-near floaters and oversized primitives that would create unstable graph topology. We then construct an active Gaussian graph using covariance-aware distances and opacity, appearance, and boundary-aware conductance, enabling mass-weighted graph Laplacian PDE evolution directly over Gaussian primitives. The evolving scalar PDE state is coupled back to rendering by modifying the direct-current spherical harmonic color coefficients while preserving geometry, opacity, and view-dependent rendering behavior. Experiments on real and synthetic scenes show that GaussPDE produces stable, controllable, and spatially coherent dynamic visualizations, with reduced cross-boundary leakage compared with baselines.
Figures & tables
Figure 1: GaussPDE pipeline. Camera-aware reconstruction produces Gaussian primitives, from which a sparse conductance graph is constructed. PDE states evolve on this graph and update Gaussian appearance for dynamic novel-view rendering. SAM denotes the Segment Anything Model [ 8 ] .
Figure 2: PDE-driven scalar-field visualization on NeRF-Synthetic, LLFF, and DTU. The evolving field modifies Gaussian appearance while the reconstructed geometry remains fixed.
Leakage (%) ↓ at g/h=
Mean leakage
Mean reference
Method
0.35
0.50
0.70
1.00
(%) ↓
error ↓
KNN
37.33
37.13
37.13
26.46
34.51
0.3451
RBF
38.00
37.78
36.76
25.85
34.60
0.3460
LBO [ 22 ]
47.73
<0.01
<0.01
<0.01
11.93
0.1197
GaussPDE
<0.01
<0.01
<0.01
<0.01
<0.01
0.0013
Table 1: Controlled diffusion on disconnected sheets with identical appearance. Leakage and grid-reference error are evaluated at matched in-sheet diffusion extent. Means average the four separation ratios.
Figure 3: Field visualization with GaussPDE, KNN, and RBF. The top row is ground truth, which contains static reference RGB images. The remaining rows illustrate how graph construction changes the rendered propagation pattern.
Figure 4: CamReg ablation on room and trex from LLFF. The upper row shows static reconstruction; the lower row shows PDE-rendered appearance. Each scene is displayed with and without CamReg.