\begin{equation} \label{residual}
\begin{aligned}
r_{b}(u_{NN}, \mathbf{x}, t)&=u_{NN}-u, & \forall(\mathbf{x}, t) \in \partial \Omega \times(0, T] \nonumber \\
r_{0}(u_{NN}, \mathbf{x}, t)&=u_{NN}-u_0, & \forall(\mathbf{x}, t) \in \Omega \times\{t=0\}. \nonumber 
\end{aligned}\end{equation}
