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This research introduces Finslerian graph neural networks as a nonlinear alternative to standard Laplacian-based architectures, enabling the modeling of anisotropic diffusion processes on manifolds. By proving convergence for discrete estimates on point clouds, the study expands GNN capabilities beyond isotropic assumptions into complex geometric regimes relevant to advanced signal processing and physics-informed machine learning.
Finsler Geometry, Graph Neural Networks, and You
arXiv:2606.17185v1 Announce Type: new Abstract: Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.
Abstract:Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.
Machine Learning (cs.LG); Signal Processing (eess.SP); Differential Geometry (math.DG); Machine Learning (stat.ML)
arXiv:2606.17185 [cs.LG]
(or arXiv:2606.17185v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.17185 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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