  This paper introduces a novel two-stream deep model based on graph
convolutional network (GCN) architecture and feed-forward neural networks
(FFNN) for learning the solution of nonlinear partial differential equations
(PDEs). The model aims at incorporating both graph and grid input
representations using two streams corresponding to GCN and FFNN models,
respectively. Each stream layer receives and processes its own input
representation. As opposed to FFNN which receives a grid-like structure, the
GCN stream layer operates on graph input data where the neighborhood
information is incorporated through the adjacency matrix of the graph. In this
way, the proposed GCN-FFNN model learns from two types of input
representations, i.e. grid and graph data, obtained via the discretization of
the PDE domain. The GCN-FFNN model is trained in two phases. In the first
phase, the model parameters of each stream are trained separately. Both streams
employ the same error function to adjust their parameters by enforcing the
models to satisfy the given PDE as well as its initial and boundary conditions
on grid or graph collocation (training) data. In the second phase, the learned
parameters of two-stream layers are frozen and their learned representation
solutions are fed to fully connected layers whose parameters are learned using
the previously used error function. The learned GCN-FFNN model is tested on
test data located both inside and outside the PDE domain. The obtained
numerical results demonstrate the applicability and efficiency of the proposed
GCN-FFNN model over individual GCN and FFNN models on 1D-Burgers,
1D-Schr\"odinger, 2D-Burgers and 2D-Schr\"odinger equations.
