Finslerian GNNs: a new layer for anisotropic diffusion on point clouds

Standard graph neural networks built on the graph Laplacian only approximate the isotropic Laplace-Beltrami operator, limiting them to symmetric, uniform diffusion processes. In a new preprint, researchers from Rice University propose a fundamentally different approach: Finslerian graph neural networks (FinslerGNN) based on estimates of the Finsler Laplacian. Unlike the classic Laplacian, the Finsler Laplacian captures anisotropic (direction-dependent) diffusion, allowing GNNs to model complex phenomena like heat flow in non-uniform media.

The core contribution is a rigorous convergence proof: as the number of point samples from a manifold increases, discrete estimates of the Finsler Laplacian converge to the true continuous operator. Importantly, the authors show this operator can be realized as a graph neural network layer, creating a family of architectures that are constrained to express Finsler geometry. In practical experiments, FinslerGNN successfully recovers the geometry underlying nonlinear diffusion equations, marking a step toward more expressive geometric deep learning models that can handle anisotropic signals on manifolds.