# Finsler Geometry, Graph Neural Networks, and You Source: [Finsler Geometry, Graph Neural Networks, and You]() Collection index: [llms.txt]() Data updated: 2026-06-18T05:00:06.089Z Authors: T. Mitchell Roddenberry, Richard G. Baraniuk Published: 2026-06-15T18:24:08.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Approximating Finsler heat flow on Finsler Heat Flow Graph (train)](): Approximating Finsler heat flow; Finsler Heat Flow Graph (train); MSE: 1.6. - [Approximating Finsler heat flow on Finsler Heat Flow (test)](): Approximating Finsler heat flow; Finsler Heat Flow (test); MSE: 1.1. ## Original sources and implementation [Original paper]() [PDF]()