Fig. 4.

Setup of the Poisson equation. (a) A1 × 1 unit square is decomposed into two overlapping regions, ${\Omega }_{NN}$ (light blue) and ${\Omega }_{FEM}$ (meshed). ${\Omega }_{FEM}$ is bounded by ${\Gamma }_1$ and ${\Gamma }_0$, whereas ${\Omega }_{NN}$ is a 0.3 × 0.3 square in the center with boundary ${\Gamma }_2$. (b) DeepONet takes ${u_{FEM}|}_{{\Gamma }_2}$ as input and yields the solution in ${\Omega }_{NN}$, denoted as ${u_{NN}|}_{{\Omega }_{NN}}$. (c) Formulation of an overlapping D-D method. Given the boundary condition on ${\Gamma }_0$, the coupling framework iteratively updates the corresponding boundary condition on ${\Gamma }_1$. The relaxation parameter $\theta$ is either fixed or updated based on the Aitken’s rule [73].