  Data over non-Euclidean manifolds, often discretized as surface meshes,
naturally arise in computer graphics and biological and physical systems. In
particular, solutions to partial differential equations (PDEs) over manifolds
depend critically on the underlying geometry. While graph neural networks have
been successfully applied to PDEs, they do not incorporate surface geometry and
do not consider local gauge symmetries of the manifold. Alternatively, recent
works on gauge equivariant convolutional and attentional architectures on
meshes leverage the underlying geometry but underperform in modeling surface
PDEs with complex nonlinear dynamics. To address these issues, we introduce a
new gauge equivariant architecture using nonlinear message passing. Our novel
architecture achieves higher performance than either convolutional or
attentional networks on domains with highly complex and nonlinear dynamics.
However, similar to the non-mesh case, design trade-offs favor convolutional,
attentional, or message passing networks for different tasks; we investigate in
which circumstances our message passing method provides the most benefit.
