## 2 Proposed Data-Driven Approaches
### Hidden Layer Structure  
The neural network possesses the capacity to adapt to either a single or multiple layer configuration. The addition of nodes in the initial layer, or the inclusion of an extra layer, improves the flexibility of the neural network. It is feasible to integrate an unlimited number of nodes into either layer. The nodes in the second layer exhibit a dependency on the \(x\) variables. The operational capabilities of the initial layer nodes are contingent upon the second layer nodes. The dependent variables, also known as \(y\) variables, are defined by the functions performed by the nodes in the first layer. This research paper examines the process of function approximation through the use of feedforward artificial neural networks, specifically focusing on the restriction of connections just between adjacent layers. In the subsequent section, we present a formal exposition of the notion of a neural network and the associated nomenclature. Let \(L\) be an element of the set of natural numbers, and let \(l_{0},\dots,\ l_{L}\in\mathbb{N}\) also be elements of the set of \(\sigma:\mathbb{R}\longrightarrow\mathbb{R}\) be an activation function and define the parameter space in Equation (3) [22]:  
\[\Theta=\underset{L\in\mathbb{N}_{0},\dots,\underset{L}{\cup}}{\underset{L\in \mathbb{N}_{k}=1}{\cup}}\overset{L}{\left(\mathbb{R}^{k_{k}\times l_{k-1}} \times\mathbb{R}^{l_{k}}\right)} \tag{3}\]  
for any \(\theta\in\Theta\), it is defined \(\theta_{k}:=(W_{k},\ b_{k})\), where \(W_{k}\) and \(b_{k}\) are matrices. Additionally, it is defined \(\mathrm{A}_{k}\) as a function that maps from \(\mathbb{R}^{l_{k-1}}\longrightarrow\mathbb{R}^{l_{k}}:x\mapsto W_{k}x+b_{k}:x \longmapsto W_{k}x+b_{k}\), where \(x\) is the input and the output is given by \(\mathrm{A}_{k}\). The formula is defined for all values of \(k\) such that \(1\leq k\leq L\). Furthermore, it denotes \(\Psi_{\theta}\) as a function that maps from \(\mathbb{R}^{l_{0}}\rightarrow\mathbb{R}^{l_{L}}\), \(x\longmapsto\Psi_{\theta}(x)\) in Equation (4):  
\[\psi_{\theta}(x)=\left\{ \begin{array}{l}A_{1}x\\ (A_{L}\circ\sigma\circ A_{L-1}\circ\sigma\circ\dots\sigma\circ A_{1})(x)\end{array} \right. \tag{4}\]  
where \(\sigma\) is applied elementwise. The term \(\Psi_{\theta}\) is used to denote the instantiation of the neural network linked to the parameter \(\theta\), which consists of \(L\) layers and has widths \((l_{0},\ l_{1},\ \dots,\ l_{L})\). The initial \(L-1\) layers are commonly denoted as hidden layers. For each integer \(k\) such that \(1\leq k\leq L\), it is defined that the width of the layer \(k\) as \(l_{k}\). Additionally, the weights and biases associated with layer \(k\) are denoted as \(W_{k}\) and \(b_{k}\), respectively. The width of \(\Psi_{\theta}\) is formally defined as the value among the set of lengths \(\max(l_{0},\ l_{1},\ \dots,\ l_{L})\). When the value of \(L=2\), it is appropriate to refer to \(\Psi_{\theta}\) as a deep neural network. Therefore, it can be observed that a shallow neural network consists of a single hidden layer, while deep neural networks are characterised by the presence of two or more hidden layers. This paper will primarily examine neural networks that employ the hyperbolic tangent as an activation function. The hyperbolic tangent function, denoted as \(\sigma(x)\), is defined as  
\[\sigma(x):=\tanh(x)=\frac{e^{2x}-1}{e^{2x}+1} \tag{5}\]  
for \(x\in\mathbb{R}\). The networks shall be denoted as tanh neural networks. While it is possible to apply concepts to various smooth activation functions, concentrating on a specific activation function enables the establishment of precise and explicit bounds without compromising the clarity of the arguments [23]. It is worth mentioning that the findings are specifically applicable to the sigmoid or logistic activation function, which may be understood as a transformed version of the hyperbolic tangent function with shifts and scaling.