Finsler Geometry, Graph Neural Networks, and You

• URL: http://arxiv.org/abs/2606.17185v1
• Date: Mon, 15 Jun 2026 18:24:08 GMT
• Title: Finsler Geometry, Graph Neural Networks, and You
• Abstract summary: We consider estimates of the Finsler Laplacian on point clouds sampled from a manifold.<n>We show that these discrete estimates converge to the true operator on the manifold.<n>We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.
• Score: 29.261683120389886
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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