  Physics-informed neural networks (PINNs) have successfully addressed various
computational physics problems based on partial differential equations (PDEs).
However, while tackling issues related to irregularities like singularities and
oscillations, trained solutions usually suffer low accuracy. In addition, most
current works only offer the trained solution for predetermined input
parameters. If any change occurs in input parameters, transfer learning or
retraining is required, and traditional numerical techniques also need an
independent simulation. In this work, a physics-driven GraphSAGE approach
(PD-GraphSAGE) based on the Galerkin method and piecewise polynomial nodal
basis functions is presented to solve computational problems governed by
irregular PDEs and to develop parametric PDE surrogate models. This approach
employs graph representations of physical domains, thereby reducing the demands
for evaluated points due to local refinement. A distance-related edge feature
and a feature mapping strategy are devised to help training and convergence for
singularity and oscillation situations, respectively. The merits of the
proposed method are demonstrated through a couple of cases. Moreover, the
robust PDE surrogate model for heat conduction problems parameterized by the
Gaussian random field source is successfully established, which not only
provides the solution accurately but is several times faster than the finite
element method in our experiments.
