## 3 Parameter Selection  
A significant problem, which influences the neural network training speed and convergence of the neural network response to the target function, is the choice of method for optimizing the network parameters, the activation function of neurons, and the choice of a criterion for evaluating the result (the loss minimization function). Modern optimization methods in linear and nonlinear regression and approximation problems use two main methods for minimizing the loss function: the stochastic gradient descent method [42] with its varieties and the accelerated Nesterov gradient method (the pulse accumulation method) [43]. Each ensures that the neural network training procedure quickly converges to the target value (target function).  
The [PERSON] optimization method [44] was used in this paper, combining the ideas of momentum accumulation and adaptive learning rate for each parameter. In addition to momentum accumulation, the algorithm considered the average non-centered variance of the gradients, which reduced the number of iterations required to converge the algorithm while improving the neural network's prediction accuracy. The Adam algorithm is distinguished by using exponential moving averages for gradients and their squares, controlled by the parameters \(\beta_{1}\) (for the gradient) and \(\beta_{2}\) (for the gradient square). Selecting these averages determines the speed and accuracy of algorithm convergence. To solve the problem empirically, the values \(\beta_{1}=0.99\) and \(\beta_{2}=0.999\) were selected at the initial learning rate \(\eta=10^{-3}\).  
Continuous differentiability is a general requirement for an activation function, which comes from the requirements of algorithms based on calculation of the loss function gradient. The selection of the activation function type is typically influenced by the nature of the problem to be solved, the input and output data types, and the desired outcome of the neural network model. A linear activation function, for example, involves solving a linearly separable problem that does not require the hidden layer of the neural network. Hidden layers in a neural network indicate that a nonlinear separable problem is being solved. Nonlinear activation functions, such as sigmoidal functions, ReLU (Rectified Linear Unit) and its numerous varieties (Leaky ReLU, PReLU, RReLU), exponential functions, and radial basis functions, are required for such problems. Because of its advantages, such as monotonicity, boundedness, and the presence of a section close to linear, the sigmoidal activation function is convenient to use for the problem formulated here:  
\[f(x,\alpha)=\frac{1}{1+e^{-ax}},\ \ \ \forall\ \alpha>0, \tag{2}\]  
Figure 2: Computational model of a three-layer Rosenblatt’s perceptron.  
where the parameter \(\alpha=0.5\) corresponds to a large length of the nearly linear function section. In Equation (2), the parameter \(\alpha\) is responsible for the slope of the linear section near the ordinate 0. At \(\alpha=0\), the sigmoid function turns into a line \(y=0.5\); for \(\alpha=+\infty\), it turns into a threshold function. The choice of the parameter \(\alpha=0.5\), or another value close to 0, leads to an increase in the length of the \"linear\" section of the function, as a result of which the sigmoid function behaves like a linear function (in the working section), while maintaining its nonlinearity.  
In turn, the mapping from the associative layer space to the response layer space is linear, meaning that the value of the response layer neuron is the weighted sum of the values of the associative layer neurons.  
The Euclidean norm between the target vector and the neural network's response vector is the way to tell if the network is getting closer to the target value:  
\[d(\mathbf{p},\mathbf{q})=\sqrt{\sum_{i=0}^{K-1}(p_{i}-q_{i})^{2}}\;, \tag{3}\]  
where \(\mathbf{p}\) is the target vector, \(\mathbf{q}\) is the response vector of the neural network, and \(K\) is the dimension of the target vector \(\mathbf{p}\).