用霍奇分解保持拓扑结构，提升物理场方程求解的精度与效率。
Topology-Preserving Neural Operator Learning via Hodge Decomposition
- 基于霍奇理论将拓扑与几何分量分离，构建可学习的结构保持子空间。
- 在几何图上实现更高保真度的物理守恒量，精度显著优于基线方法。
- 适合需要精确模拟物理规律的科学计算场景，如流体、电磁仿真。
本文从函数空间视角研究定义在几何网格上的物理场方程的解算符。我们发现，霍奇正交性通过将不可学习的拓扑自由度与可学习的几何动力学分离，从根本上缓解了谱干扰，实现了仅限于结构保持子空间的加法逼近。基于霍奇理论与算子分裂，我们推导出一种原理性的算子级分解。结果是一个混合欧拉-拉格朗日架构，带有我们称为霍奇谱对偶（HSD）的代数层次归纳偏置。在该框架中，我们使用离散微分形式捕捉主导拓扑的成分，并利用正交辅助环境空间表示复杂的局部动力学。我们的方法在几何图上实现了更优的准确率与效率，同时增强了对物理不变量的保真度。代码已公开于 https://github.com/ContinuumCoder/Hodge-Spectral-Duality。
原文摘要 · Abstract (English)
In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。