Data-Driven Diffusion Processes on Differential Forms via the Projected Ambient Connection Laplacian

Abstract: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.

Numerical Analysis (math.NA); Machine Learning (cs.LG)

arXiv:2607.23192 [math.NA]

(or arXiv:2607.23192v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2607.23192 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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