We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.

翻译：我们提出一种基于MIONet的偏微分方程混合迭代方法，该方法结合了传统数值迭代求解器与近期强大的神经算子机器学习方法，并进一步系统分析了其理论性质，包括收敛条件、谱行为以及收敛速率，这些分析基于离散化和模型推理的误差。我们针对常用的平滑器，即Richardson（阻尼Jacobi）和高斯-赛德尔，给出了理论结果。我们给出了混合方法关于模型修正周期的收敛速率上界，该上界表明存在一个使混合迭代收敛最快的最小值点。几个数值算例，包括一维（二维）泊松方程的混合Richardson（高斯-赛德尔）迭代，验证了我们的理论结果，并反映出卓越的加速效果。作为一种无网格加速方法，它在实际应用中具有巨大潜力。