超越任意几何：神经PDE算子中的拓扑泛化

中文摘要 AI 辅助

接受任意网格的神经算子通常被视为几何泛化的，但未见过的域拓扑同时改变了PDE算子的不变子空间和衰减子空间。我们使用Hodge热流作为这一区别的可控视角，并引入TopoBox-3D，其中隧道和空洞变化Betti支撑，而精确的Hodge分解将调和核与正谱分离。在六种架构中，隐式推断拓扑的模型在45个模型-任务拓扑OOD单元中的37个中遭受过度的匹配退化，但改变调和维度的案例在平均上并未受到更强的惩罚。主导困难反而是谱方面的：初始Rayleigh商是误差最稳定的预测因子，谱展宽为边和面余链增加了信息。最引人注目的是，受控探针表明，显式的关联和调和坐标并不能产生最佳核恒等精度；尽管如此，TNO在所有六个具有非平凡调和支撑的任务中，在混合输入非调和精度上排名第一。这些结果共同确立了拓扑作为超越任意几何兼容性的一个独特泛化轴，并表明其影响延伸至整个Hodge谱，而非局限于调和核。更广泛地说，它们表明全局、低频的结构先验可能有助于在更快衰减的互补分量中组织预测，为神经算子如何在拓扑和几何上泛化提供了新视角。

英文摘要

Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying subspaces of a PDE operator. We use Hodge heat flow as a controlled lens on this distinction and introduce TopoBox-3D, where tunnels and cavities vary Betti support while the exact Hodge decomposition separates the harmonic kernel from the positive spectrum. Across six architectures, models that infer topology implicitly suffer excess matched degradation in 37 of 45 model--task topology-OOD cells, yet cases that change harmonic dimension are not more strongly penalized on average. The dominant difficulty is instead spectral: the initial Rayleigh quotient is the most stable predictor of error, and spectral broadening adds information for edge and face cochains. Most strikingly, controlled probes show that explicit incidence and harmonic coordinates do not yield the best kernel-identity accuracy; nevertheless, TNO ranks first in mixed-input nonharmonic accuracy on all six tasks with nontrivial harmonic support. Together, these results establish topology as a distinct generalization axis beyond arbitrary-geometry compatibility and show that its influence extends across the Hodge spectrum rather than remaining confined to the harmonic kernel. More broadly, they suggest that global, low-frequency structural priors may help organize predictions in the faster-decaying complementary component, offering a new perspective on how neural operators may generalize across topology as well as geometry.

发表机构

• Tsinghua University（清华大学）

机构由 AI 辅助整理，请以论文原文为准。