# Finsler Geometry, Graph Neural Networks, and You

> Source: <https://arxiv.org/abs/2606.17185>
> Published: 2026-06-17 04:00:00+00:00

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.