  Physics-informed neural networks (PINNs) are capable of finding the solution
for a given boundary value problem. We employ several ideas from the finite
element method (FEM) to enhance the performance of existing PINNs in
engineering problems. The main contribution of the current work is to promote
using the spatial gradient of the primary variable as an output from separated
neural networks. Later on, the strong form which has a higher order of
derivatives is applied to the spatial gradients of the primary variable as the
physical constraint. In addition, the so-called energy form of the problem is
applied to the primary variable as an additional constraint for training. The
proposed approach only required up to first-order derivatives to construct the
physical loss functions. We discuss why this point is beneficial through
various comparisons between different models. The mixed formulation-based PINNs
and FE methods share some similarities. While the former minimizes the PDE and
its energy form at given collocation points utilizing a complex nonlinear
interpolation through a neural network, the latter does the same at element
nodes with the help of shape functions. We focus on heterogeneous solids to
show the capability of deep learning for predicting the solution in a complex
environment under different boundary conditions. The performance of the
proposed PINN model is checked against the solution from FEM on two prototype
problems: elasticity and the Poisson equation (steady-state diffusion problem).
We concluded that by properly designing the network architecture in PINN, the
deep learning model has the potential to solve the unknowns in a heterogeneous
domain without any available initial data from other sources. Finally,
discussions are provided on the combination of PINN and FEM for a fast and
accurate design of composite materials in future developments.
