  We consider a mesh-based approach for training a neural network to produce
field predictions of solutions to parametric partial differential equations
(PDEs). This approach contrasts current approaches for "neural PDE solvers"
that employ collocation-based methods to make point-wise predictions of
solutions to PDEs. This approach has the advantage of naturally enforcing
different boundary conditions as well as ease of invoking well-developed PDE
theory -- including analysis of numerical stability and convergence -- to
obtain capacity bounds for our proposed neural networks in discretized domains.
We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkin
loss function based on the Finite Element Method (FEM) on a parametric elliptic
PDE. The weighted Galerkin loss (FEM loss) is similar to an energy functional
that produces improved solutions, satisfies a priori mesh convergence, and can
model Dirichlet and Neumann boundary conditions. We prove theoretically, and
illustrate with experiments, convergence results analogous to mesh convergence
analysis deployed in finite element solutions to PDEs. These results suggest
that a mesh-based neural network approach serves as a promising approach for
solving parametric PDEs with theoretical bounds.
