A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples
Jun Xi Hu, Pengzhan Jin · Mathematics of Computation · 2025
We propose a hybrid iterative method based on multiple-input operator network for partial differential equations, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyzes 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.