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PreCorrector Takes the Lead: How It Stacks Up Against Other Neural Preconditioning Methods
- The article discusses a novel approach, PreCorrector, in preconditioner construction, utilizing neural networks to outperform classical numerical preconditioners.
- Authors mention the challenge of selecting effective preconditioners, as the choice depends on specific problems and requires theoretical and numerical understanding.
- Li et al. introduced a method using GNN and a new loss function to approximate matrix factorization for efficient preconditioning.
- FCG-NO Rudikov et al. combined neural operators with conjugate gradient for PDE solving, proving to be computationally efficient.
- Kopanicáková and Karniadakis proposed hybrid preconditioners combining DeepONet with iterative methods for parametric equations solving.
- The HINTS method by Zhang et al. combines relaxation methods with DeepONet for solving differential equations effectively and accurately.
- PreCorrector aims to reduce κ(A) by developing a universal transformation for sparse matrices, showing superiority over classical preconditioners on complex datasets.
- Future work includes theoretical analysis of the loss function, variations of the target objective, and extending PreCorrector to different sparse matrices.
- The article provides references to related works on iterative methods, neural solvers, and preconditioning strategies for linear systems.
- Appendix details training data, additional experiments with ICt(5) preconditioner, and information about correction coefficient α.
- The paper is available on arxiv under CC by 4.0 Deed license, facilitating attribution and sharing under international standards.
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