  Deep learning approaches for partial differential equations (PDEs) have
received much attention in recent years due to their mesh-freeness and
computational efficiency. However, most of the works so far have concentrated
on time-dependent nonlinear differential equations. In this work, we analyze
potential issues with the well-known Physic Informed Neural Network for
differential equations with little constraints on the boundary (i.e., the
constraints are only on a few points). This analysis motivates us to introduce
a novel technique called FinNet, for solving differential equations by
incorporating finite difference into deep learning. Even though we use a mesh
during training, the prediction phase is mesh-free. We illustrate the
effectiveness of our method through experiments on solving various equations,
which shows that FinNet can solve PDEs with low error rates and may work even
when PINNs cannot.
