Fig. 7.

Setup of the time-dependent problem (heat equation). (a) The spatio-temporal domain is decomposed into two sub-domains with $\Gamma$ as the interface between FEM and NN domain. (b–c) Formulation of the coupling framework for the heat equation. The framework first solves for $u_{FEM}^{n+1,k+1}$ from FEM with updated information at the interface $\tilde{u}$ at $k$ th step. Then, the computed derivative at $\Gamma$ is transmitted into DeepONet, which predicts the spatial derivative in ${\Omega }_{NN}$ (c). The solution in ${\Omega }_{NN}$ is computed based on a time-stepping scheme, followed by a relaxation update (b). The initialization at $k=0$ and $n=0$ is described in the main text.