A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples
This work addresses solving partial differential equations for computational science and engineering, but it is incremental as it builds on existing methods by integrating neural operators.
The authors tackled solving PDEs by proposing a hybrid iterative method that combines traditional numerical solvers with MIONet neural operators, achieving an excellent acceleration effect in numerical examples like the 1-d and 2-d Poisson equation.
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.