# Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches

arXiv: https://arxiv.org/abs/2307.07695 · HTML: https://mlsanity.com/paper/2307.07695/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 15 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Domain Adaptation for Time Series Under Feature and Label Shifts](https://mlsanity.com/paper/2302.03133/) (2023)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Encoding Time-Series Explanations through Self-Supervised Model Behavior Consistency](https://mlsanity.com/paper/2306.02109/) (2023)
- [MedDiff: Generating Electronic Health Records using Accelerated Denoising Diffusion Model](https://mlsanity.com/paper/2302.04355/) (2023)
- [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)
- [AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows](https://mlsanity.com/paper/2206.02102/) (2022)
- [Reducing Parallel Communication in Algebraic Multigrid through Sparsification](https://mlsanity.com/paper/1512.04629/) (2015)
- [Learning Relaxation for Multigrid](https://mlsanity.com/paper/2207.11255/) (2022)
- [Reducing Communication in Algebraic Multigrid with Multi-step Node Aware Communication](https://mlsanity.com/paper/1904.05838/) (2019)

## Built on by (most cited first, 2 of 2)

- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)

## Abstract (arXiv)

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.