GaussPDE: Graph-Based Partial Differential Equation-Driven Rendering for 3D Gaussian Splatting

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.