标题： 利用算子学习加速预条件共轭梯度法的收敛 显示英文标题

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

摘要： 我们提出了一种新的降阶策略，以加速预条件共轭梯度（PCG）方法在求解参数化大规模线性方程组时的收敛速度。 与依赖于特征向量近似或回收Krylov子空间的传统降阶技术不同，我们使用操作符学习生成降阶子空间，具体为深度操作符网络（DeepONet）。 为此，我们引入了两种互补的方法来构建降阶算子。 第一种方法利用由DeepONet学习到的基函数来近似离散PDE算子的近零空间向量。 第二种方法直接利用由DeepONet预测的解。 为了进一步提高收敛性，我们还提出了几种用于指定降阶算子稀疏模式的策略。 我们展示了一系列涵盖稳态、时间相关、标量和矢量值问题的数值实验，这些问题定义在结构化和非结构化几何上，并证明了所提出的基于DeepONet的降阶PCG方法的有效性，以及其在广泛模型参数和问题分辨率下的泛化能力。 显示英文摘要

摘要： 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.