Multi-resolution Graph Neural Networks for PDE Approximation.

Deep Learning algorithms have recently received a growing interest to learn from examples of existing solutions and some accurate approximations of the solution of complex physical problems, in particular relying on Graph Neural Networks applied on a mesh of the domain at hand. On the other hand, state-of-the-art deep approaches of image processing use different resolutions to better handle the different scales of the images, thanks to pooling and up-scaling operations. But no such operators can be easily defined for Graph Convolutional Neural Networks (GCNN). This paper defines such operators based on meshes of different granularities. Multi-resolution GCNNs can then be defined. We propose the MGMI approach, as well as an architecture based on the famed U-Net. These approaches are experimentally validated on a diffusion problem, compared with projected CNN approach and the experiments witness their efficiency, as well as their generalization capabilities.