Learning to Solve PDEs on Neural Shape Representations

2025-12-24

AI Summary

This work addresses the challenge of directly solving partial differential equations (PDEs) on neural implicit shapes—such as NeRFs and signed distance functions (SDFs)—where conventional mesh-based solvers are inapplicable. We propose the first end-to-end differentiable, mesh-free PDE solver for neural geometry. Our method introduces a neural shape-aware local update operator, integrating implicit differential geometry encoding, neural modeling of local geometric conditions, differentiable numerical integration, and residual-driven training. Crucially, it bypasses explicit mesh extraction and per-instance optimization, enabling single-shot training with generalization across arbitrary topologies and implicit representations. To our knowledge, this is the first framework unifying PDE solving across both neural and classical surface representations. Evaluated on heat and Poisson equation benchmarks, our approach achieves accuracy slightly surpassing CPM and approaching that of finite element methods (FEM), while supporting real-time inference on diverse neural assets.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Hardware-aware ML
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Solves PDEs directly on neural shape representations
• Eliminates need for explicit mesh extraction
• Enables end-to-end differentiable workflows

Innovation

• Learns local update operator on neural shape attributes
• Trains once, generalizes across shape and topology variations
• Enables end-to-end PDE solving without explicit meshing