  Despite the great promise of the physics-informed neural networks (PINNs) in
solving forward and inverse problems, several technical challenges are present
as roadblocks for more complex and realistic applications. First, most existing
PINNs are based on point-wise formulation with fully-connected networks to
learn continuous functions, which suffer from poor scalability and hard
boundary enforcement. Second, the infinite search space over-complicates the
non-convex optimization for network training. Third, although the convolutional
neural network (CNN)-based discrete learning can significantly improve training
efficiency, CNNs struggle to handle irregular geometries with unstructured
meshes. To properly address these challenges, we present a novel discrete PINN
framework based on graph convolutional network (GCN) and variational structure
of PDE to solve forward and inverse partial differential equations (PDEs) in a
unified manner. The use of a piecewise polynomial basis can reduce the
dimension of search space and facilitate training and convergence. Without the
need of tuning penalty parameters in classic PINNs, the proposed method can
strictly impose boundary conditions and assimilate sparse data in both forward
and inverse settings. The flexibility of GCNs is leveraged for irregular
geometries with unstructured meshes. The effectiveness and merit of the
proposed method are demonstrated over a variety of forward and inverse
computational mechanics problems governed by both linear and nonlinear PDEs.
