用深度算子网络加速求解大规模线性方程组的收敛速度。
Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method
- 用DeepONet学习近零空间基函数，构建新型预处理共轭梯度法的降维子空间。
- 在结构与非结构网格上，对稳态和时变问题均实现快速收敛，泛化能力强。
- 适合需反复求解参数化方程组的科学计算场景，如物理模拟与优化。
针对参数化大规模线性方程组求解，本文提出一种基于算子学习的新降维策略，以加速预处理共轭梯度（PCG）方法的收敛。不同于依赖特征向量近似或循环Krylov子空间的传统方法，本工作利用深度算子网络（DeepONet）生成降维子空间。提出两种互补方法：第一种通过DeepONet学习到的基函数逼近离散偏微分算子的近零空间向量；第二种直接使用DeepONet预测的解作为降维依据。为进一步提升性能，还设计了多种稀疏模式配置策略以优化降维算子结构。通过涵盖稳态、时变、标量与矢量问题的广泛数值实验，验证了该方法在结构与非结构几何上的有效性，并展现出对模型参数与问题分辨率的强泛化能力。
原文摘要 · Abstract (English)
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.
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