## 4 Discussion and Conclusions  
The mathematical basis of our approach was the approximation property of the shifted fundamental solutions \(\{\Psi_{\mathbf{y}}\}_{j=1}^{k}\) of the free-space Laplacian operator. Our linear ANN-based approach can be considered as an extension of the classical MFS in the following sense:  
* The ANN approach delivers an automatic way to obtain linear mappings \([g(\mathbf{z}_{1}),g(\mathbf{z}_{1}),\dots,g(\mathbf{z}_{N})]\to[u(\mathbf{ x}_{1}),u(\mathbf{x}_{2}),\dots,u(\mathbf{x}_{M})]\) also with different input values. This is the case if, in an implicit discretization of time-dependent problems, more Laplacian problems have to be solved on the same domain. We were faced with such a computational task in the implicit time discretization of problems containing the Laplacian operators:
* The optimal linear combination of the fundamental solution was found using a stochastic optimization method, which, for a large number of unknowns, can result in an efficient approach.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**K** & \(\mathbf{N}=\mathbf{M}\) & \# **of Parameters** & \(\mathbf{l_{2}}\)**-Error** & **max-Error** \\ \hline
4500 & 61 & 3721 & 4.66 & 4.63 \\
9000 & 61 & 3721 & 2.75 & 1.93 \\
10,125 & 89 & 8100 & 3.29 & 2.63 \\
20,250 & 89 & 8100 & 2.48 & 1.93 \\
18,000 & 119 & 14,400 & 1.99 & 1.71 \\
18,000 & 119 & 14,400 & 1.84 & 1.41 \\ \hline
4500 & 61 & 1220 & 15.73 & 10.03 \\
9000 & 61 & 1708 & 7.89 & 6.97 \\
10,125 & 89 & 3738 & 5.95 & 6.24 \\
20,250 & 89 & 4984 & 2.61 & 1.27 \\
18,000 & 119 & 7140 & 2.26 & 1.18 \\
18,000 & 119 & 9520 & 1.81 & 1.35 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational error using the first approach. First half: results for the full-rank case; second half: results for the low-rank case.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**K** & \(\mathbf{N}=\mathbf{M}\) & \begin{tabular}{c} **\# of Parameters** \\ **\end{tabular} & \begin{tabular}{c} **\(\mathbf{l_{2}}\)-Error**, \\ **Pointwise** \\ \end{tabular} & \begin{tabular}{c} **max-Error**, \\ **Pointwise** \\ \end{tabular} & \
\begin{tabular}{c} **1\(\mathbf{l_{2}}\)-Error**, \\ **Neumann** \\ \end{tabular} &
\begin{tabular}{c} **max-Error**, \\ **Neumann** \\ \end{tabular} \\ \hline
4500 & 61 & 3721 & 0.97 & 0.57 & 2.58 & 1.30 \\
9000 & 61 & 3721 & 0.94 & 0.64 & 3.24 & 1.60 \\
10,125 & 89 & 7921 & 0.93 & 0.67 & 2.24 & 1.50 \\
20,250 & 89 & 7921 & 1.02 &  
The ANN-based approach also confirmed that an accurate discretization of the Dirichlet-to-Neumann map should consist of a dense matrix. At the same time, using our approach, its rank can be reduced without a significant increase in the computational error.  
Using the first approach, the low-rank case can deliver the same accuracy using a reduced number of parameters. The same applies to the pointwise values, which can be approximated accurately using our ANN-based approach. At the same time, the Neumann data should be approximated rather immediately using the first approach.  
We finally note that, recently, a number of promising new ANN-based approaches have been developed for solving PDEs numerically. For recent reviews on this topic, see, e.g., [8; 18].  
Conceptualization, F.I.; methodology, F.I.; software, T.E.D.; validation, T.E.D.; formal analysis, F.I.; investigation, F.I. and T.E.D.; resources, T.E.D.; data curation, F.I. and T.E.D.; writing--original draft preparation, F.I.; writing--review and editing, F.I. and T.E.D.; visualization, T.E.D.; supervision, F.I.; project administration, F.I.; funding acquisition, F.I. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Research, Development and Innovation Office within the framework of the Thematic Excellence Program 2021-National Research Sub programme: \"Artificial intelligence, large networks, data security: mathematical foundation and applications\".  
Not applicable.  
The authors declare no conflict of interest.