Finsler Geometry Enhances Graph Neural Networks for Nonlinear Diffusion

Traditional Graph Neural Networks (GNNs) often rely on the graph Laplacian, which approximates the Laplace-Beltrami operator, thereby restricting their applicability to isotropic operations. This new research proposes an innovative approach by integrating Finsler geometry into GNN architectures to overcome this limitation. The core idea involves using estimates of the Finsler Laplacian on point clouds sampled from a manifold. The authors prove that these discrete estimates accurately converge to the true Finsler Laplacian operator as the number of samples increases. This mathematical foundation allows for the development of a new type of GNN layer that inherently expresses Finsler geometry. By defining a family of "Finslerian graph neural networks," the research demonstrates that these models can effectively recover the underlying geometry of nonlinear diffusion equations in practical applications. This advancement opens new avenues for GNNs to handle more complex and anisotropic data structures.