mlsanityData to 14 Oct 2026

Principled Acceleration of Iterative Numerical Methods Using Machine Learning

arXiv 2206.08594 ↗First posted 17 Jun 2022math.NA cs.LG cs.NAcited by 6

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Abstract (arXiv)

Iterative methods are ubiquitous in large-scale scientific computing applications, and a number of approaches based on meta-learning have been recently proposed to accelerate them. However, a systematic study of these approaches and how they differ from meta-learning is lacking. In this paper, we propose a framework to analyze such learning-based acceleration approaches, where one can immediately identify a departure from classical meta-learning. We show that this departure may lead to arbitrary deterioration of model performance. Based on our analysis, we introduce a novel training method for learning-based acceleration of iterative methods. Furthermore, we theoretically prove that the proposed method improves upon the existing methods, and demonstrate its significant advantage and versatility through various numerical applications.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 26 of its 51 references that are in the map. First posted on arXiv 17 Jun 2022.

Builds on

26 of 51 references.

References · 26

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  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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14 more
  1. 2019SciPy 1.0--Fundamental Algorithms for Scientific Computing in Pythoncited by 49
  2. 2021Meta-Auto-Decoder for Solving Parametric Partial Differential Equationscited by 36
  3. 2021A novel meta-learning initialization method for physics-informed neural networkscited by 35
  4. 2022Multigrid-augmented deep learning preconditioners for the Helmholtz equationcited by 25
  5. 2019Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian meshcited by 22
  6. 2021Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learningcited by 21
  7. 2019Meta-learning Pseudo-differential Operators with Deep Neural Networkscited by 17
  8. 2020A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasoundcited by 12
  9. 2022A Deep Conjugate Direction Method for Iteratively Solving Linear Systemscited by 12
  10. 2021Neural Fixed-Point Acceleration for Convex Optimizationcited by 10
  11. 2021Using neural networks to solve the 2D Poisson equation for electric field computation in plasma fluid simulationscited by 10
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  13. 2021Accelerating GMRES with Deep Learning in Real-Timecited by 8
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 2; 2025: 2; 2026: 1.

6 papers.

  1. 2025Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Methodcited by 6
  2. 2026PRISM: Distribution-free Adaptive Computation of Matrix Functions for Accelerating Neural Network Trainingcited by 4
  3. 2025Learning to accelerate distributed ADMM using graph neural networkscited by 3
  4. 2024Mitigating distribution shift in machine learning-augmented hybrid simulationcited by 2
  5. 2024Accelerating Legacy Numerical Solvers by Non-intrusive Gradient-based Meta-solvingcited by 2
  6. 2023Learning to Relax: Setting Solver Parameters Across a Sequence of Linear System Instancescited by 0