École Nationale Supérieure de
Techniques Avancées

Abstract

Solving partial differential equations (PDEs) on surfaces underpins denoising, fairing, harmonic interpolation, and shape analysis, yet prevailing solvers assume meshes while modern 3D assets increasingly live as neural implicit surfaces. This mismatch leaves no suitable method to solve surface PDEs within the neural domain, forcing brittle mesh extraction or per-instance residual training. We present a novel, mesh-free formulation that learns a local update operator conditioned on implicit shape attributes and applies a single grid-to-grid step in a narrow band to advance the surface PDE solution. The operator integrates naturally with neural fields, is trained once on representative patches, and generalizes across shapes and topologies, enabling accurate, fast inference without meshing or per-instance optimization while preserving differentiability. Across analytic benchmarks (sphere and torus) and real neural assets, our approach achieves competitive accuracy and runtime compared to established baselines, and, to our knowledge, delivers the first end-to-end pipeline that solves and uses surface PDEs directly on neural implicit representations.