  This article presents a new methodology called deep Theory of Functional
Connections (TFC) that estimates the solutions of partial differential
equations (PDEs) by combining neural networks with TFC. TFC is used to
transform PDEs with boundary conditions into unconstrained optimization
problems by embedding the boundary conditions into a "constrained expression."
In this work, a neural network is chosen as the free function, and used to
solve the now unconstrained optimization problem. The loss function is taken as
the square of the residual of the PDE. Then, the neural network is trained in
an unsupervised manner to solve the unconstrained optimization problem. This
methodology has two major differences when compared with popular methods used
to estimate the solutions of PDEs. First, this methodology does not need to
discretize the domain into a grid, rather, this methodology randomly samples
points from the domain during the training phase. Second, after training, this
methodology represents a closed form, analytical, differentiable approximation
of the solution throughout the entire training domain. In contrast, other
popular methods require interpolation if the estimated solution is desired at
points that do not lie on the discretized grid. The deep TFC method for
estimating the solution of PDEs is demonstrated on four problems with a variety
of boundary conditions.
