  Mainstream numerical Partial Differential Equation (PDE) solvers require
discretizing the physical domain using a mesh. Mesh movement methods aim to
improve the accuracy of the numerical solution by increasing mesh resolution
where the solution is not well-resolved, whilst reducing unnecessary resolution
elsewhere. However, mesh movement methods, such as the Monge-Ampere method,
require the solution of auxiliary equations, which can be extremely expensive
especially when the mesh is adapted frequently. In this paper, we propose to
our best knowledge the first learning-based end-to-end mesh movement framework
for PDE solvers. Key requirements of learning-based mesh movement methods are
alleviating mesh tangling, boundary consistency, and generalization to mesh
with different resolutions. To achieve these goals, we introduce the neural
spline model and the graph attention network (GAT) into our models
respectively. While the Neural-Spline based model provides more flexibility for
large deformation, the GAT based model can handle domains with more complicated
shapes and is better at performing delicate local deformation. We validate our
methods on stationary and time-dependent, linear and non-linear equations, as
well as regularly and irregularly shaped domains. Compared to the traditional
Monge-Ampere method, our approach can greatly accelerate the mesh adaptation
process, whilst achieving comparable numerical error reduction.
