  Continuum mechanics simulators, numerically solving one or more partial
differential equations, are essential tools in many areas of science and
engineering, but their performance often limits application in practice. Recent
modern machine learning approaches have demonstrated their ability to
accelerate spatio-temporal predictions, although, with only moderate accuracy
in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph
neural network model for learning to infer unsteady continuum mechanics.
MultiScaleGNN represents the physical domain as an unstructured set of nodes,
and it constructs one or more graphs, each of them encoding different scales of
spatial resolution. Successive learnt message passing between these graphs
improves the ability of GNNs to capture and forecast the system state in
problems encompassing a range of length scales. Using graph representations,
MultiScaleGNN can impose periodic boundary conditions as an inductive bias on
the edges in the graphs, and achieve independence to the nodes' positions. We
demonstrate this method on advection problems and incompressible fluid
dynamics. Our results show that the proposed model can generalise from uniform
advection fields to high-gradient fields on complex domains at test time and
infer long-term Navier-Stokes solutions within a range of Reynolds numbers.
Simulations obtained with MultiScaleGNN are between two and four orders of
magnitude faster than the ones on which it was trained.
