发表机构
Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究如何通过投影环境联络拉普拉斯算子对微分形式进行数据驱动的扩散过程，基于新颖表示构造矩阵值扩散算子，有渐近最优核带宽缩放，推导显式欧拉格式并经数值实验验证，为几何偏微分方程数值逼近奠定基础。
AI 中文摘要
我们开发了一种数据驱动的方法，用于逼近作用于由点云采样的光滑黎曼流形上的微分形式的投影环境联络拉普拉斯算子。所提出的构造将扩散映射和向量扩散映射的经典框架从标量函数和切向量场扩展到任意阶的微分形式。我们的方法基于一种将微分形式表示为通过扩展经典音乐同构得到的交错微分阵列的新颖表示。这种表示使得能够直接从点云数据构造一个矩阵值扩散算子，该算子逼近投影环境联络拉普拉斯算子，而无需网格或单纯复形。所提出的离散化允许从扩散映射继承的渐近最优核带宽缩放，从而比以前的数据驱动的霍奇拉普拉斯逼近具有更严格的收敛保证。在此算子的基础上，我们为微分形式上的热方程推导了一个完全数据驱动的显式欧拉格式，并通过在单位球上的数值实验验证了所提出的方法。实验证实了解析解的预测衰减，并证明了所提出离散化的有效性。所提出的框架为向量扩散映射到任意阶微分形式提供了自然的推广，并为直接从点云数据对几何偏微分方程进行数值逼近奠定了实际基础。
英文摘要
We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds. The proposed construction extends the classical framework of diffusion maps and Vector Diffusion Maps from scalar functions and tangent vector fields to differential forms of arbitrary degree. Our approach is based on a novel representation of differential forms as alternating differential arrays obtained through an extension of the classical musical isomorphism. This representation enables the construction of a matrix-valued diffusion operator that approximates the projected ambient connection Laplacian directly from point cloud data without requiring a mesh or simplicial complex. The proposed discretization admits the asymptotically optimal kernel bandwidth scaling inherited from diffusion maps, leading to sharper convergence guarantees than previous data-driven approximations of the Hodge Laplacian. Building upon this operator, we derive a fully data-driven explicit Euler scheme for the heat equation on differential forms and validate the proposed methodology through numerical experiments on the unit sphere. The experiments confirm the predicted decay of the analytical solution and demonstrate the effectiveness of the proposed discretization. The proposed framework provides a natural generalization of Vector Diffusion Maps to differential forms of arbitrary degree and establishes a practical foundation for the numerical approximation of geometric partial differential equations directly from point cloud data.
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