The Neural Network Finite Element (NNFE) method was first described in Sacks et al. 2022. In brief, the NNFE method replaces the constant parameters defined in traditional finite elements with functions that are parameterized by boundary conditions (BCs). In doing so, the solution can be learned over the range of BCs trained over. The issue with this approach is the formation of the matrix becomes intractable for multiple instances, so the parameters of the network are trained through standard gradient methods where the FE residual is the loss. By using the residual to represent the loss, there is no data required and a inate error metric built into the method.