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We consider a mesh-based approach for training a neural network to producefield 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 ofsolutions to PDEs. This approach has the advantage of naturally enforcingdifferent boundary conditions as well as ease of invoking well-developed PDEtheory -- including analysis of numerical stability and convergence -- toobtain capacity bounds for our proposed neural networks in discretized domains.We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkinloss function based on the Finite Element Method (FEM) on a parametric ellipticPDE. The weighted Galerkin loss (FEM loss) is similar to an energy functionalthat produces improved solutions, satisfies a priori mesh convergence, and canmodel Dirichlet and Neumann boundary conditions. We prove theoretically, andillustrate with experiments, convergence results analogous to mesh convergenceanalysis deployed in finite element solutions to PDEs. These results suggestthat a mesh-based neural network approach serves as a promising approach forsolving parametric PDEs with theoretical bounds.

NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs