Implementing the discontinuous-Galerkin finite element method using graph neural networks with application to...

Area of Science:

• Computational Science
• Numerical Analysis
• Machine Learning

Background:

• Machine learning (ML) advancements drive interest in academia and industry.
• Existing ML solvers for partial differential equations (PDEs) often rely on structured meshes, limiting applications with complex geometries.
• Unstructured meshes are crucial for complex geometric problems.

Purpose of the Study:

• To implement an unstructured mesh Finite Element Method (FEM) solver using graph neural networks (GNNs).
• To bridge the gap between ML capabilities and the need for unstructured mesh solvers in scientific computing.

Main Methods:

• The proposed FEM solver utilizes graph neural networks for compute-intensive algorithms.
• It employs a discontinuous Galerkin formulation with an interior penalty method for spatial discretization.
• A multigrid preconditioned Krylov solver, adapted with a U-Net architecture, serves as the linear solver.

Main Results:

• The solver demonstrates theoretical convergence rates of (p+1) order for diffusion problems.
• On GPUs, it achieves a throughput of 6.8 MDOF/s for matrix operator evaluation.
• The GNN-based FEM solver shows promising performance compared to optimized implementations.

Conclusions:

• This work presents the first GNN-based implementation of an unstructured mesh FEM solver.
• The method is adaptable to various PDEs and computing platforms (CPU, GPU).
• This approach offers a viable path for leveraging ML in complex scientific computing problems.