mlsanityData to 14 Oct 2026

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

arXiv 2508.00101 ↗First posted 31 Jul 2025math.NA cs.LG cs.NAcited by 6

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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.

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

33 of 95 references.

References · 33

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21 more
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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 4; 2026: 2.

6 papers.

  1. 2025NOWS: Neural Operator Warm Starts for Accelerating Iterative Solverscited by 9
  2. 2025Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learningcited by 7
  3. 2025A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solverscited by 5
  4. 2025Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEscited by 3
  5. 2026McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equationcited by 0
  6. 2026Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulationscited by 0