  One of the main challenges in using deep learning-based methods for
simulating physical systems and solving partial differential equations (PDEs)
is formulating physics-based 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 long-range 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 mesh-refinement.
Inspired by the classical multipole methods, we propose a novel multi-level
graph neural network framework that captures interaction at all ranges with
only linear complexity. Our multi-level formulation is equivalent to
recursively adding inducing points to the kernel matrix, unifying GNNs with
multi-resolution matrix factorization of the kernel. Experiments confirm our
multi-graph network learns discretization-invariant solution operators to PDEs
and can be evaluated in linear time.
