Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers
  for non-stationary and nonlinear simulations on arbitrary meshes
Tobias Würth Niklas Freymuth Clemens Zimmerling Gerhard Neumann Luise Kärger
Engineering components must meet increasing technological demands in ever
shorter development cycles. To face these challenges, a holistic approach is
essential that allows for the concurrent development of part design, material
system and manufacturing process. Current approaches employ numerical
simulations, which however quickly becomes computation-intensive, especially
for iterative optimization. Data-driven machine learning methods can be used to
replace time- and resource-intensive numerical simulations. In particular,
MeshGraphNets (MGNs) have shown promising results. They enable fast and
accurate predictions on unseen mesh geometries while being fully differentiable
for optimization. However, these models rely on large amounts of expensive
training data, such as numerical simulations. Physics-informed neural networks
(PINNs) offer an opportunity to train neural networks with partial differential
equations instead of labeled data, but have not been extended yet to handle
time-dependent simulations of arbitrary meshes. This work introduces PI-MGNs, a
hybrid approach that combines PINNs and MGNs to quickly and accurately solve
non-stationary and nonlinear partial differential equations (PDEs) on arbitrary
meshes. The method is exemplified for thermal process simulations of unseen
parts with inhomogeneous material distribution. Further results show that the
model scales well to large and complex meshes, although it is trained on small
generic meshes only.