This paper presents a hybrid framework, Neural-Enhanced Multigrid (NEMG), integrating neural networks into classical multigrid solvers for PDEs. NEMG leverages U-Net architecture with weight sharing to learn problem-dependent correction operators, maintaining convergence guarantees and achieving significant speedups over traditional methods.
Key findings
NEMG integrates neural networks as learnable coarse-grid correction operators in multigrid solvers.
Theoretical foundations prove that learned corrections preserve multigrid iteration's fixed-point properties.
NEMG achieves 3-8x speedup over classical geometric multigrid while maintaining convergence guarantees.
U-Net architecture with cross-level weight sharing enables generalization across grid resolutions.
Limitations & open questions
The current implementation focuses on structured grids, limiting its applicability to unstructured problems.
Further research is needed to explore the scalability of NEMG to three-dimensional PDE problems.