用芬斯勒几何改进图神经网络，让其能捕捉非均匀扩散过程。
Finsler Geometry, Graph Neural Networks, and You
- 基于点云估计芬斯勒拉普拉斯算子，逼近流形上的真实算子。
- 离散估计随采样点数增加而收敛，理论保证可靠。
- 新模型可还原非线性扩散的几何结构，适合复杂空间建模。
基于图拉普拉斯的图神经网络架构仅能近似各向同性的算子，受限于拉普拉斯-贝尔特拉米算子。本文考虑在从流形采样的点云上估计芬斯勒拉普拉斯算子，作为其非线性替代。我们证明：当点样本数量增加时，这些离散估计收敛于流形上的真实算子。此外，该算子可表示为图神经网络层，由此定义一类受芬斯勒几何约束的芬斯勒图神经网络。实验表明，这类网络能有效恢复实际中非线性扩散方程背后的几何结构。
原文摘要 · Abstract (English)
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.
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