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

TL;DR

NeuFENet introduces a mesh-based neural network approach for solving parametric PDEs, leveraging finite element theory to ensure stability, convergence, and boundary condition enforcement, with proven theoretical bounds and practical effectiveness.

Contribution

This work presents NeuFENet, a novel mesh-based neural PDE solver that incorporates finite element principles to provide theoretical guarantees and improved solution quality.

Findings

NeuFENet achieves mesh convergence similar to traditional FEM.

The approach effectively models Dirichlet and Neumann boundary conditions.

Theoretical bounds support the stability and convergence of the neural solutions.

Abstract

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.