Implementing the discontinuous-Galerkin finite element method using graph neural networks with application to diffusion equations
File(s)
Published version
Author(s)
Type
Journal Article
Abstract
Machine learning (ML) has benefited from both software and hardware advancements, leading to increasing
interest in capitalising on ML throughout academia and industry. There have been efforts in the scientific com puting community to leverage this development via implementing conventional partial differential equation
(PDE) solvers with machine learning packages, most of which rely on structured spatial discretisation and fast
convolution algorithms. However, unstructured meshes are favoured in problems with complex geometries. To
bridge this gap, we propose to implement the unstructured Finite Element Method (FEM) on simplicial meshes with graph neural networks. This paper is the first to implement an unstructured mesh FEM solver using graph neural networks. All compute-intensive algorithms in the solver are represented with either convolutional or
graph neural networks. Specifically, the FEM solver uses a discontinuous Galerkin formulation with an interior
penalty method for spatial discretisation and a multigrid preconditioned Krylov solver as the linear solver. The
multigrid method has been designed to suit the data structure within the ML package and adopts the commonly used U-Net architecture for this. A hierarchy of coarsened meshes is generated from 𝑝-multigrid and algebraic node agglomeration guided by either a space-filling curve or a smoothed aggregation algorithm. The solver is verified and assessed for solving the diffusion problems. The solver shows the theoretical convergence rate of (𝑝 + 1) order. Compared with a highly optimised implementation, the solver running on GPU can reach
promising throughput in terms of matrix operator evaluation at 6.8 MDOF/s. The method can easily extend
to other PDEs and computing platforms beyond CPU and GPU.
interest in capitalising on ML throughout academia and industry. There have been efforts in the scientific com puting community to leverage this development via implementing conventional partial differential equation
(PDE) solvers with machine learning packages, most of which rely on structured spatial discretisation and fast
convolution algorithms. However, unstructured meshes are favoured in problems with complex geometries. To
bridge this gap, we propose to implement the unstructured Finite Element Method (FEM) on simplicial meshes with graph neural networks. This paper is the first to implement an unstructured mesh FEM solver using graph neural networks. All compute-intensive algorithms in the solver are represented with either convolutional or
graph neural networks. Specifically, the FEM solver uses a discontinuous Galerkin formulation with an interior
penalty method for spatial discretisation and a multigrid preconditioned Krylov solver as the linear solver. The
multigrid method has been designed to suit the data structure within the ML package and adopts the commonly used U-Net architecture for this. A hierarchy of coarsened meshes is generated from 𝑝-multigrid and algebraic node agglomeration guided by either a space-filling curve or a smoothed aggregation algorithm. The solver is verified and assessed for solving the diffusion problems. The solver shows the theoretical convergence rate of (𝑝 + 1) order. Compared with a highly optimised implementation, the solver running on GPU can reach
promising throughput in terms of matrix operator evaluation at 6.8 MDOF/s. The method can easily extend
to other PDEs and computing platforms beyond CPU and GPU.
Date Issued
2025-05-01
Date Acceptance
2024-12-01
Citation
Neural Networks, 2025, 185
ISSN
0893-6080
Publisher
Elsevier BV
Journal / Book Title
Neural Networks
Volume
185
Copyright Statement
© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
License URL
Identifier
https://www.ncbi.nlm.nih.gov/pubmed/39817979
PII: S0893-6080(24)00990-0
Subjects
Discontinuous-Galerkin finite element method
Graph neural networks
Multigrid solver
U-Net
Unstructured meshes
Neural Networks, Computer
Finite Element Analysis
Algorithms
Machine Learning
Diffusion
Computer Simulation
Software
Publication Status
Published
Coverage Spatial
United States
Article Number
107061
Date Publish Online
2024-12-19