在神经形状表示上学习求解偏微分方程
Learning to Solve PDEs on Neural Shape Representations
提出一种无需网格、直接在神经形状表示上求解表面偏微分方程的可微局部更新算子，可替代传统网格提取+残差训练流程，为几何处理、物理仿真与神经渲染提供端到端可微工具。
原文依据 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.
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论在形状上求解偏微分方程是许多形状分析与工程任务的基础；然而，主流的偏微分方程求解器在多边形/三角形网格上运行，而现代三维资产越来越多地以神经表示的形式存在。这种不匹配导致没有合适的方法能在神经域内直接求解表面偏微分方程，迫使进行显式网格提取或逐实例残差训练，阻碍了端到端工作流。我们提出了一种新颖的、无网格的公式，学习以神经（局部）形状属性为条件的局部更新算子，使得表面偏微分方程能够直接在（神经）数据所在之处求解。该算子自然地与主流的神经表面表示集成，在单个代表性形状上训练一次，即可泛化到形状和拓扑变化，实现准确、快速的推理，无需显式网格化或逐实例优化，同时保持可微性。在解析基准（球面上的热扩散和泊松方程）以及多种形状和神经表面表示上，我们的方法达到了与经典求解器相当的精度，同时实现了跨神经和传统表面表示的统一端到端流程。源代码和项目页面：https://welschinger.github.io/Learning-to-Solve-PDEs-on-Neural-Shape-Representations/。
英文原文摘要 1370 字符
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 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/.