  In this work, we present a Boundary Oriented Graph Embedding (BOGE) approach
for the Graph Neural Network (GNN) to serve as a general surrogate model for
regressing physical fields and solving boundary value problems. Providing
shortcuts for both boundary elements and local neighbor elements, the BOGE
approach can embed structured mesh elements into the graph and performs an
efficient regression on large-scale triangular-mesh-based FEA results, which
cannot be realized by other machine-learning-based surrogate methods. Focusing
on the cantilever beam problem, our BOGE approach cannot only fit the
distribution of stress fields but also regresses the topological optimization
results, which show its potential of realizing abstract decision-making design
process. The BOGE approach with 3-layer DeepGCN model \textcolor{blue}{achieves
the regression with MSE of 0.011706 (2.41\% MAPE) for stress field prediction
and 0.002735 MSE (with 1.58\% elements having error larger than 0.01) for
topological optimization.} The overall concept of the BOGE approach paves the
way for a general and efficient deep-learning-based FEA simulator that will
benefit both industry and design-related areas.
