  This work proposes a Physics-informed Neural Network framework with Graph
Embedding (GPINN) to perform PINN in graph, i.e. topological space instead of
traditional Euclidean space, for improved problem-solving efficiency. The
method integrates topological data into the neural network's computations,
which significantly boosts the performance of the Physics-Informed Neural
Network (PINN). The graph embedding technique infuses extra dimensions into the
input space to encapsulate the spatial characteristics of a graph while
preserving the properties of the original space. The selection of these extra
dimensions is guided by the Fiedler vector, offering an optimised pathologic
notation of the graph. Two case studies are conducted, which demonstrate
significant improvement in the performance of GPINN in comparison to
traditional PINN, particularly in its superior ability to capture physical
features of the solution.
