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.

翻译：基于图拉普拉斯算子的图神经网络架构近似于拉普拉斯-贝尔特拉米算子，从而限制了其在各向同性算子中的应用。作为拉普拉斯-贝尔特拉米算子的非线性替代方案，我们考虑对从流形采样的点云上的芬斯勒拉普拉斯算子进行估计。我们证明，随着样本点数量的增加，这些离散估计会收敛到流形上的真实算子。此外，我们展示了该算子可以表示为图神经网络层，并据此定义了一类受限于表达芬斯勒几何的芬斯勒图神经网络家族。我们证明，在实践中，芬斯勒图神经网络能够恢复非线性扩散方程背后的几何结构。