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

2025-07-31

AI Summary

To address the slow convergence of preconditioned conjugate gradient (PCG) methods for solving parametric large-scale linear systems, this paper proposes a novel deflation strategy based on operator learning. We introduce DeepONet—previously unexplored in deflation—to learn either near-nullspace basis functions of the underlying PDE operator or directly predict solutions, thereby constructing efficient and generalizable deflation subspaces. A sparsity-pattern pre-specification mechanism is further designed to optimize the structure of the learned deflation operator. The method significantly accelerates PCG convergence across diverse problem classes, including steady-state and time-dependent, scalar and vector-field problems. Crucially, it exhibits strong robustness to parameter variations, geometric deformations, and mesh resolution changes, while demonstrating cross-scale generalization capability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Accelerate PCG convergence for parametric linear systems
• Generate deflation subspaces via DeepONet operator learning
• Enhance convergence with sparsity-aware deflation operators

Innovation

• Deflation subspaces generated via DeepONet operator learning
• Two approaches for assembling deflation operators
• Sparsity pattern strategies to enhance convergence