Curvature-aware Graph Attention for PDEs on Manifolds

Deep models have recently achieved remarkable performances in solving partial differential equations (PDEs). The previous methods are mostly focused on PDEs arising in Euclidean spaces with less emphasis on the general manifolds with rich geometry. Several proposals attempt to account for the geometry by exploiting the spatial coordinates but overlook the underlying intrinsic geometry of manifolds. In this paper, we propose a Curvature-aware Graph Attention for PDEs on manifolds by exploring the important intrinsic geometric quantities such as curvature and discrete gradient operator. It is realized via parallel transport and tensor field on manifolds. To accelerate computation, we present three curvature-oriented graph embedding approaches and derive closed-form parallel transport equations, and a subtree partition method is also developed to promote parameter-sharing. Our proposed curvature-aware attention can be used as a replacement for vanilla attention, and experiments show that it significantly improves the performance of the existing methods for solving PDEs on manifolds. Our code is available at https://github.com/Supradax/CurvGT.

We use graph neural networks (GNNs) to solve partial differential equations (PDEs) on surfaces. Currently proposed methods only focus on PDEs in Euclidean spaces. PDEs defined on curved spaces also matter in many regions. It is expected that the introduction of surface geometry to neural networks can enhance their performance. To this end, we observe that the message passing on GNN can be naturally extended to be tensor operations, a generalization of matrix operations. This is because tensors are naturally defined on surfaces. But there is also a gap between discretized surface meshes and smooth surfaces. We further propose a fast computation method by locally embedding the surface onto constant curvature surfaces. As the parameters and feature vectors vary according to the curvature there, our proposed GNN can thereby be aware of the curvature geometry.