Learning to Solve PDEs on Neural Shape Representations

TL;DR

This paper introduces a mesh-free, neural operator-based method for solving surface PDEs directly on neural shape representations, enabling end-to-end workflows without explicit meshing or per-instance optimization.

Contribution

The authors propose a novel local update operator conditioned on neural shape attributes that generalizes across shapes and topologies, solving surface PDEs directly within neural representations.

Findings

Outperforms CPM slightly on benchmarks

Approximates FEM accuracy closely

First end-to-end pipeline for neural and classical surface PDEs

Abstract

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, mesh-free 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 equation and Poisson solve on sphere) and real neural assets across different representations, our method slightly outperforms CPM while remaining reasonably close to FEM, and, to our knowledge, delivers the first end-to-end pipeline that solves surface PDEs on both neural and classical surface representations. Code will be released on acceptance.