Finsler Geometry, Graph Neural Networks, and You

2026-06-15

AI Summary

This work addresses the limitations of conventional graph neural networks, which rely on isotropic graph Laplacians and struggle to capture complex geometric structures such as those arising from nonlinear diffusion. To overcome this, the paper introduces Finsler geometry into graph neural networks for the first time, proposing a novel graph convolutional layer whose discrete formulation provably converges to the true Finsler Laplacian on manifolds. This enables effective modeling of the underlying nonlinear geometry. By integrating point cloud sampling, manifold learning, and nonlinear operator estimation, the method successfully reconstructs the geometric structures implicit in nonlinear diffusion equations, demonstrating both the effectiveness and expressive power of the proposed Finsler graph neural network.

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.

Problem

• Finsler Geometry
• Graph Neural Networks
• Laplace-Beltrami Operator
• Anisotropy
• Manifold Learning

Innovation

• Finsler Geometry
• Graph Neural Networks
• Finsler Laplacian
• Nonlinear Diffusion
• Manifold Learning