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：我们提出了一类新型混合预条件器，用于求解参数化线性方程组。所提出的预条件器通过将深度算子网络（即DeepONet）与标准迭代方法相混合构建而成。利用频谱偏差特性，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抑制高频误差分量。我们的预条件框架包含两种不同的混合方法：直接预条件方法和主干基方法。在直接预条件方法中，DeepONet被用于在每个预条件步骤中近似算子逆对向量的作用。相比之下，主干基方法从训练好的DeepONet中提取基函数，以构建到更小子空间的映射，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，与标准的非混合预条件策略相比，采用主干基方法能大幅提升Krylov方法的收敛性。此外，所提出的混合预条件器在广泛的模型参数和问题分辨率下均表现出鲁棒性。