# Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method

arXiv: https://arxiv.org/abs/2508.00101 · HTML: https://mlsanity.com/paper/2508.00101/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 95 references that are in the map.
- First posted on arXiv 31 Jul 2025.

## Builds on (most cited first, 30 of 33)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [On weight initialization in deep neural networks](https://mlsanity.com/paper/1704.08863/) (2017)
- [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)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [A Deep Conjugate Direction Method for Iteratively Solving Linear Systems](https://mlsanity.com/paper/2205.10763/) (2022)
- [MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods](https://mlsanity.com/paper/2301.11378/) (2023)
- [Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows](https://mlsanity.com/paper/2010.02866/) (2020)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [Learning incomplete factorization preconditioners for GMRES](https://mlsanity.com/paper/2409.08262/) (2024)
- [Accelerating GMRES with Deep Learning in Real-Time](https://mlsanity.com/paper/2103.10975/) (2021)
- [A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms](https://mlsanity.com/paper/2211.12386/) (2022)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [Principled Acceleration of Iterative Numerical Methods Using Machine Learning](https://mlsanity.com/paper/2206.08594/) (2022)
- [Multilevel Objective-Function-Free Optimization with an Application to Neural Networks Training](https://mlsanity.com/paper/2302.07049/) (2023)
- [Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applications](https://mlsanity.com/paper/2406.10997/) (2024)

## Built on by (most cited first, 6 of 6)

- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)
- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers](https://mlsanity.com/paper/2511.08811/) (2025)
- [Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs](https://mlsanity.com/paper/2512.14596/) (2025)
- [McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation](https://mlsanity.com/paper/2606.30495/) (2026)
- [Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulations](https://mlsanity.com/paper/2606.31921/) (2026)

## Abstract (arXiv)

We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network~(DeepONet). To this aim, we introduce two complementary approaches for assembling the deflation operators. The first approach approximates near-null space vectors of the discrete PDE operator using the basis functions learned by the DeepONet. The second approach directly leverages solutions predicted by the DeepONet. To further enhance convergence, we also propose several strategies for prescribing the sparsity pattern of the deflation operator. A comprehensive set of numerical experiments encompassing steady-state, time-dependent, scalar, and vector-valued problems posed on both structured and unstructured geometries is presented and demonstrates the effectiveness of the proposed DeepONet-based deflated PCG method, as well as its generalization across a wide range of model parameters and problem resolutions.