# Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms

arXiv: https://arxiv.org/abs/2603.17641 · HTML: https://mlsanity.com/paper/2603.17641/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 61 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## Builds on (most cited first, 22 of 22)

- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [DENSER: Deep Evolutionary Network Structured Representation](https://mlsanity.com/paper/1801.01563/) (2018)
- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)
- [Graph Neural Networks and Applied Linear Algebra](https://mlsanity.com/paper/2310.14084/) (2023)
- [UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs](https://mlsanity.com/paper/2408.04846/) (2024)
- [waLBerla: A block-structured high-performance framework for multiphysics simulations](https://mlsanity.com/paper/1909.13772/) (2019)
- [Toward Performance-Portable PETSc for GPU-based Exascale Systems](https://mlsanity.com/paper/2011.00715/) (2020)
- [Differentiable programming across the PDE and Machine Learning barrier](https://mlsanity.com/paper/2409.06085/) (2024)
- [The DUNE Framework: Basic Concepts and Recent Developments](https://mlsanity.com/paper/1909.13672/) (2019)
- [Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming](https://mlsanity.com/paper/2204.12846/) (2022)
- [Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan Bidiagonalization](https://mlsanity.com/paper/2001.10424/) (2020)
- [Textbook efficiency: massively parallel matrix-free multigrid for the Stokes system](https://mlsanity.com/paper/2010.13513/) (2020)
- [Fundamental Data Structures for Matrix-Free Finite Elements on Hybrid Tetrahedral Grids](https://mlsanity.com/paper/2308.01792/) (2023)
- [Code Generation and Performance Engineering for Matrix-Free Finite Element Methods on Hybrid Tetrahedral Grids](https://mlsanity.com/paper/2404.08371/) (2024)
- [Towards Automated Algebraic Multigrid Preconditioner Design Using Genetic Programming for Large-Scale Laser Beam Welding Simulations](https://mlsanity.com/paper/2412.08186/) (2024)
- [Large-scale Thermo-Mechanical Simulation of Laser Beam Welding Using High-Performance Computing: A Qualitative Reproduction of Experimental Results](https://mlsanity.com/paper/2503.09345/) (2025)

## Abstract (arXiv)

Although multigrid is asymptotically optimal for solving many important partial differential equations, its efficiency relies heavily on the careful selection of the individual algorithmic components. In contrast to recent approaches that can optimize certain multigrid components using deep learning techniques, we adopt a complementary strategy, employing evolutionary algorithms to construct efficient multigrid cycles from proven algorithmic building blocks. Here, we will present its application to generate efficient algebraic multigrid methods with so-called flexible cycling, that is, level-specific smoothing sequences and non-recursive cycling patterns. The search space with such non-standard cycles is intractable to navigate manually, and is generated using genetic programming (GP) guided by context-free grammars. Numerical experiments with the linear algebra library, hypre, demonstrate the potential of these non-standard GP cycles to improve multigrid performance both as a solver and a preconditioner.