## 2 Materials and Methods
### Problem Statement and Mathematical Background  
The focus of the present study is the ANN-based approximation of some mathematical problems based on the Laplacian equation:  
\[ \begin{cases}\Delta u(\mathbf{x})=0&\mathbf{x}\in\Omega\\ u(\mathbf{x})=g(\mathbf{x})&\mathbf{x}\in\partial\Omega,\end{cases}  \tag{2}\]  
where \(\Omega\subset\mathbb{R}^{d}\) denotes the computational domain, \(u:\Omega\rightarrow\mathbb{R}\) is an unknown function, and \(g:\partial\Omega\rightarrow\mathbb{R}\) is given. The corresponding theory (see, e.g., [13]) ensures that this problem is well-posed: for any \(g\in H^{\frac{1}{2}}(\partial\Omega)\), it has a unique solution \(u\in H^{1}(\Omega)\) depending continuously on \(g\). Here, it was assumed that \(\Omega\) is bounded and has a Lipschitz boundary, and we used the classical Sobolev spaces:  
\[H^{1}(\Omega)=\{u\in L_{2}(\Omega):\
abla u\in[L_{2}(\Omega)]^{d}\}\quad\text {with}\quad\|u\|_{H^{1}(\Omega)}=\left(\|u\|_{L_{2}(\Omega)}^{2}+\|\
abla u\|_ {[L_{2}(\Omega)]^{d}}^{2}\right)^{\frac{1}{2}}\]  
and  
\[H^{\frac{1}{2}}(\partial\Omega)=\{u|_{\partial\Omega}:u\in H^{1}(\Omega)\} \quad\text{with}\quad\|u\|_{H^{\frac{1}{2}}(\partial\Omega)}=\inf\{\|u\|_{H^{ 1}(\Omega)}:\bar{u}|_{\partial\Omega}=u\}.\]  
In practice, however, \(g\) is not given in all points of \(\partial\Omega\), and possibly, \(u\) is sought only in some interior points. Accordingly, as an introductory problem, we discuss the following case:  
**Problem 1**. : _For given values of \(g(\mathbf{z}_{1}),g(\mathbf{z}_{2}),\ldots,g(\mathbf{z}_{N})\) with \(\{\mathbf{z}_{j}\}_{j=1}^{N}\subset\partial\Omega\), we should determine \(u(\mathbf{x}_{1}),u(\mathbf{x}_{2}),\ldots,u(\mathbf{x}_{M})\) with \(\{\mathbf{x}_{j}\}_{j=1}^{M}\subset\Omega\)._  
Our objective was to reduce the computational costs and approximate \(u(\mathbf{x}_{1}),u(\mathbf{x}_{2}),\ldots,\)\(u(\mathbf{x}_{M})\) without computing a \"complete\" numerical solution \(u_{h}\approx u\) of (2). A related problem is computing the corresponding Dirichlet-to-Neumann map. **Problem 2**. : _For given values of \(g(\mathbf{z}_{1}),g(\mathbf{z}_{2}),\ldots,g(\mathbf{z}_{N})\) with \(\{\mathbf{z}_{j}\}_{j=1}^{N}\subset\partial\Omega\), we should determine \(\partial_{v}u(\mathbf{z}_{1}^{*}),\partial_{v}u(\mathbf{z}_{2}^{*}),\ldots, \partial_{v}u(\mathbf{z}_{M}^{*})\) with \(\{\mathbf{z}_{j}^{*}\}_{j=1}^{M}\subset\partial\Omega\)._  
The associated \"full\" mapping corresponding to (2) is defined as  
\[\text{DtN}:H^{\frac{1}{2}}(\partial\Omega)\to H^{-\frac{1}{2}}(\partial \Omega),\ \text{DtN}(g)=\partial_{v}u,\]  
where \(u\in H^{1}(\Omega)\) is the solution of (2), \(\
u\) denotes the (space-dependent) outward normal on \(\partial\Omega\), and \(H^{-\frac{1}{2}}(\partial\Omega)\) denotes the dual-space of \(H^{\frac{1}{2}}(\partial\Omega)\). This is called the Dirichlet-to-Neumann map. Of course, to develop an efficient procedure, we intended to perform it without approximating \(u\) in the entire domain. In a related mathematical model, the function \(g\) corresponds to an applied potential (using electrodes) at the surface of some organ and \(\partial_{v}u\) corresponds to the generated current on the boundary. Here, instead of the Laplacian, we used a general elliptic operator \(\
abla\cdot(\mu\
abla)\), where the space-dependent function \(\mu:\Omega\rightarrow\mathbb{R}\) describes the permittivity. In this way, the Dirichlet-to-Neumann map can be measured, and in a real-life inverse problem (in EIT), the corresponding permittivity has to be determined. **Remark 1**. : _Note that neither Problem 1 nor Problem 2 are well-posed. Therefore, we cannot perform a conventional error analysis._Furthermore, an objective of our approach was to include and use knowledge from the PDE theory and the conventional numerical methods in the ANN-based algorithm.