  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.
