We introduce a new class of hybrid preconditioners for solving parametric linear systems of equations. The proposed preconditioners are constructed by hybridizing the deep operator network, namely DeepONet, with standard iterative methods. Exploiting the spectral bias, DeepONet-based components are harnessed to address low-frequency error components, while conventional iterative methods are employed to mitigate high-frequency error components. Our preconditioning framework comprises two distinct hybridization approaches: direct preconditioning (DP) and trunk basis (TB) approaches. In the DP approach, DeepONet is used to approximate an action of an inverse operator to a vector during each preconditioning step. In contrast, the TB approach extracts basis functions from the trained DeepONet to construct a map to a smaller subspace, in which the low-frequency component of the error can be effectively eliminated. Our numerical results demonstrate that utilizing the TB approach enhances the convergence of Krylov methods by a large margin compared to standard non-hybrid preconditioning strategies. Moreover, the proposed hybrid preconditioners exhibit robustness across a wide range of model parameters and problem resolutions.

翻译：本文提出一类新的混合预处理方法，用于求解参数化线性方程组。所提预处理算子通过将深度算子网络（即DeepONet）与标准迭代方法进行混合构造。利用谱偏差特性，基于DeepONet的组件被用于处理低频误差分量，而传统迭代方法则用于缓解高频误差分量。我们的预处理框架包含两种不同的混合策略：直接预处理（DP）方法和基函数（TB）方法。在DP方法中，DeepONet用于在每个预处理步骤中近似逆算子对向量的作用；而TB方法则从训练好的DeepONet中提取基函数，构建到低维子空间的映射，在该子空间中误差的低频分量可被有效消除。数值结果表明，与标准非混合预处理策略相比，采用TB方法能大幅提升Krylov方法的收敛性能。此外，所提出的混合预处理方法在广泛的模型参数和问题分辨率下均展现出鲁棒性。