Can neural networks fix multigrid's fundamental tradeoff?

Solving enormous sparse linear systems is foundational to scientific computing, but algebraic multigrid (AMG) solvers face a hard choice: sparse operators are fast but converge slowly; dense ones converge well but waste computation. RAPNet, a graph neural network trained level-by-level, learns to generate sparse coarse operators that sidestep this tradeoff. The method runs only during setup, keeping the solve phase efficient, and generalizes from small subgraphs to million-node problems. Testing on PDE discretizations and graph Laplacians shows 10–50% speedups over classical alternatives, with particular value for repeated-solve tasks like eigenproblems and inverse problems.