Multipole Graph Neural Operator for Parametric Partial Differential Equations
Abstract
One of the main challenges in using deep learningbased methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physicsbased data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore longrange interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under meshrefinement. Inspired by the classical multipole methods, we propose a novel multilevel graph neural network framework that captures interaction at all ranges with only linear complexity. Our multilevel formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multiresolution matrix factorization of the kernel. Experiments confirm our multigraph network learns discretizationinvariant solution operators to PDEs and can be evaluated in linear time.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.09535
 Bibcode:
 2020arXiv200609535L
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Statistics  Machine Learning