Learning to Solve PDEs on Neural Shape Representations

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows. We present a novel, meshfree formulation that learns a local update operator conditioned on neural (local) shape attributes, enabling surface PDEs to be solved directly where the (neural) data lives. The operator integrates naturally with prevalent neural surface representations, is trained once on a single representative shape, and generalizes across shape and topology variations, enabling accurate, fast inference without explicit meshing or per-instance optimization while preserving differentiability. Across analytic benchmarks (heat diffusion and Poisson equations on the sphere) and on diverse shapes and neural surface representations, our method achieves accuracy comparable to classical solvers while enabling a unified, end-to-end pipeline across neural and traditional surface representations. Our source code and project page: https://welschinger.github.io/Learning-to-Solve-PDEs-on-Neural-Shape-Representations/.