## 2 Materials and Methods
### Principles and Details of the Algorithm  
Observe that, for \(\mathbf{y}\
ot\in\Omega\), we have \(\Delta\Psi_{\mathbf{y}}(\mathbf{x})=0\) for \(\mathbf{x}\in\Omega\), such that it satisfies the first equation in (2). In this way, for a fixed value of \(\mathbf{y}\
ot\in\overline{\Omega}\), we used the following.  
Main Principle.  
The pair:  
\[([\Psi_{\mathbf{y}}(\mathbf{z}_{1}),\Psi_{\mathbf{y}}(\mathbf{z}_{2}),\ldots, \Psi_{\mathbf{y}}(\mathbf{z}_{N})],[\Psi_{\mathbf{y}}(\mathbf{x}_{1}),\Psi_{ \mathbf{y}}(\mathbf{x}_{2}),\ldots,\Psi_{\mathbf{y}}(\mathbf{x}_{M})]) \tag{3}\]  
delivers a learning data pair (input and output) for Problem 1.  
Likewise, the pair \(([\Psi_{\mathbf{y}}(\mathbf{z}_{1}),\Psi_{\mathbf{y}}(\mathbf{z}_{2}),\ldots, \Psi_{\mathbf{y}}(\mathbf{z}_{N})],[\partial_{v}\Psi_{\mathbf{y}}(\mathbf{z}_ {1}^{*}),\partial_{v}\Psi_{\mathbf{y}}(\mathbf{z}_{2}^{*}),\ldots,\partial_{v }\Psi_{\mathbf{y}}(\mathbf{z}_{M}^{*})])\) delivers a learning data pair for Problem 2.  
Summarizing, the ANN can be trained with the above data for an arbitrary set of points \(\{\mathbf{y}_{j}\}_{j=1}^{K}\subset\mathbb{R}^{2}\setminus\overline{\Omega}\). In this way, we easily obtain a learning dataset consisting of \(K\) input-output pairs.  
Remark 2:
1. _In practice, we did not choose_ \(\mathbf{y}_{j}\in\overline{\Omega}\)_, since in this case,_ \(\Psi\) _will have a singularity here, which results in huge values_ \(\Psi_{\mathbf{y}}(\mathbf{z}_{j})\) _in the vicinity of the boundary._
2. _The present approach (_3_), compared to the conventional one (_1_), defines a much easier procedure:_ * _We can get rid of the first term in (_1_) automatically;_ * _Furthermore, for a stationary problem, we do not need the initial condition in the second term in (_1_)._ _In short, our approach results in a significant reduction of the parameters in the associated ANN. It seems obvious to employ this idea in general, for all stationary boundary values problems, if we know solutions_ \(\Psi_{j}\) _with_ \(D_{\mathbf{x}}\Psi_{j}=0\)_. However, one should be careful here: we can only approximate all possible solutions if we can state some approximation property on the linear hull of the family of functions_ \(\{\Psi_{j}\}\)_._
3. _In other words, our approach can also be considered as a shortcut in the PINNs. As we do not have to compute solution candidates with different right-hand sides_ \(f_{j}\)_, the number of learning data is also reduced, and the ANN can be trained much faster._  
Recall that, in the Dirichlet-to-Neumann problem, the natural norm of the Neumann data is \(\|\cdot\|_{H^{-\frac{1}{2}}(\partial\Omega)}\), which can lead to the slow convergence of the result. In the ANN-based algorithm, this can also lead to the slowing down of the optimization procedure.  
To try an alternative approach for Problem 2, observe that the Neumann boundary data of \(u\) at \(\mathbf{z}_{j}\in\partial\Omega\) can also be estimated using the approximation:  
\[\partial_{v}u(\mathbf{z}_{j}^{*})\approx\frac{u(\mathbf{z}_{j}^{*})-u(\mathbf{ z}_{j}^{*}-sv)}{s} \tag{4}\]for a small value \(s\in\mathbb{R}^{+}\). Then, we have to estimate \(u(\mathbf{z}_{j}^{*}-s\boldsymbol{v})\) and \(u(\mathbf{z}_{j}^{*})\) to obtain the pointwise approximation in (4). A motivation for this approach is the good performance of the estimation of \(u(\mathbf{z}_{j}^{*}-s\boldsymbol{v})\), which is shown later.