带边界流形上的加权霍奇拉普拉斯算子
Weighted Hodge Laplacians on Manifolds with Boundary
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中文总结 AI 辅助
该研究针对经典霍奇拉普拉斯算子无法适配流形上异质局部特征数据的局限，提出带边界流形的加权霍奇拉普拉斯算子框架，经蛋白质柔性分析实验验证其有效性，可用于分析流形数据的拓扑与几何特征。
中文摘要 AI 辅助
微分流形上霍奇拉普拉斯算子的谱蕴含丰富的拓扑与几何信息，因此是分析流形上数据的有力工具。然而，经典的未加权形式在研究具有不同局部特征的数据时能力受限。为解决这一局限，我们在理论与计算层面提出了带边界流形上的加权霍奇拉普拉斯算子框架，通过在流形上引入一个权重函数实现。在适当的边界条件下，我们构建了对应的加权德拉姆-霍奇理论，其中加权霍奇拉普拉斯算子的核与加权调和空间一致，且与基础流形的德拉姆上同调保持同构。加权霍奇拉普拉斯算子的调和谱捕获全局拓扑信息，而非调和谱则编码由权重诱导的局部几何性质。该框架因此能够研究不同权重下流形上数据的拓扑与几何特征，还可通过选择强调感兴趣区域的权重来突出局部结构。我们通过蛋白质柔性分析的原理验证实验证明了所提方法的有效性，结果显示其具有应用前景。
英文摘要
The spectrum of the Hodge Laplacian on differential manifolds encodes rich topological and geometric information and thus provides a powerful tool for analyzing data on manifolds. However, the classical unweighted formulation is restricted in its ability to study data with varying local features. To address this limitation, we propose a weighted Hodge Laplacian framework for manifolds with boundary, both in theory and in computation, by incorporating a weight function on the manifold. Under appropriate boundary conditions, we formulate the corresponding weighted de Rham-Hodge theory, in which the kernel of the weighted Hodge Laplacian coincides with the weighted harmonic space, and remains isomorphic to the de Rham cohomology of the underlying manifold. The harmonic spectrum of the weighted Hodge Laplacian captures the global topological information, while its non-harmonic spectrum encodes the local geometric property induced by the weight. The proposed framework therefore enables the study of topological and geometric features of data on manifolds across varying weights, and in addition, allows local structure to be highlighted by choosing weights that emphasize regions of interest. We demonstrate the effectiveness of the proposed method through proof-of-principle experiments in protein flexibility analysis, and the results show its promise.
