Algorithm 1 Coupling Method. ${\Gamma }_1≔\partial {\Omega }_I$, ${\Gamma }_2≔\partial {\Omega }_{II}$. If non-overlapping, ${\Gamma }_1={\Gamma }_2$, else ${\Gamma }_1\ne {\Gamma }_2$.

Initialization: Set model I with $u\left(x_1\right)=0$ and model II with $u\left(x_1\right)=0$

Main Loop:

for $n=0:n_{\mathit{max}}-1$ do

Model I (FEM):

• Receive the interface information $h^n\left(x_1\right)$ from Model II $\left(x_1\in {\Gamma }_1\right)$.

• Solve for $u_I^{n+1}$ from Model I.

• Calculate $u_I^{n+1}\left(x_2\right)$ or ${\frac{\partial u_I^{n+1}}{\partial x}|}_{x_2}$ and pass it to Model II $\left(x_2\in {\Gamma }_2\right)$.

Model II (NN):

• Receive the interface information $u_I^{n+1}\left(x_2\right)$ or ${\frac{\partial u_I^{n+1}}{\partial x}|}_{x_2}$ from Model I $\left(x_2\in {\Gamma }_2\right)$.

• Solve for $u_{II}^{n+1}/T_{II}^{n+1}$ from Model II.

• Calculate $h_{II}^{n+1}\left(x_1\right)=R_1{u}_{II}^{n+1}\left(x_1\right)+R_2{T}_{II}^{n+1}\left(x_1\right)$.

• Calculate ${\tilde{h}}^{n+1}\left(x_1\right)=(1-\theta )h_{II}^{n+1}\left(x_1\right)+\theta h_I^{n+1}\left(x_1\right)$ and pass it to Model I $\left(x_1\in {\Gamma }_1\right)$.

If converged, stop;

end for