Reducing operator complexity in Algebraic Multigrid with Machine
  Learning Approaches
Ru Huang Kai Chang Huan He Ruipeng Li Yuanzhe Xi
We propose a data-driven and machine-learning-based approach to compute
non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods,
addressing the well-known issue of increasing operator complexity. Guided by
the AMG theory on spectrally equivalent coarse-grid operators, we have
developed novel ML algorithms that utilize neural networks (NNs) combined with
smooth test vectors from multigrid eigenvalue problems. The proposed method
demonstrates promise in reducing the complexity of coarse-grid operators while
maintaining overall AMG convergence for solving parametric partial differential
equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian
and linear elasticity problems are provided to showcase the performance and
compare with existing methods for computing non-Galerkin coarse-grid operators.