## 5 Numerical Modeling
### The Case of a Single External Source with a Varying Initial Phase  
Below are the results of the numerical simulation of the operation of a neural network with various activation functions (18)-(20) under the influence of a single point source with an amplitude of \(\dot{b}_{1}=\)1, an angle of arrival of \(\theta=-22.5^{\circ}\), and different phases of \(\varphi=0^{\circ},72^{\circ}\), and \(-18.9^{\circ}\); on the left, vertically, are the values of the phases used in modeling the sources \(0^{\circ},72^{\circ}\), and \(-18.9^{\circ}\). For example, Figure 2d shows the plot obtained for the activation function (18) and the source phase \(\varphi=72^{\circ}\). In the graphs of Figure 2a-i, the angular directions in degrees are laid along the horizontal axis (in the grid of angular directions of an excessively complete basic dictionary); along the vertical axis are the values of the modules of the resulting basic components \(|\,X\,|\), obtained as a result of processing using a neural network with the selected activation function. They correspond with the angular directions of the grid, the angles of the basic functions (blue lines), and the modules of the values of the response \(|\,F\,|\), formed in the same angular grid using the standard FFT method (black lines). The position of the point source, considering its amplitude \(\dot{b}_{1}\), is represented in red dots. From the results obtained, it is clear that when the phase value \(\varphi=0^{\circ}\), that is, provided that the phase of the source and the phases of the basis functions (10) coincide, the use of all activation functions leads to a solution with only one non-zero basis coefficient, which corresponds to the amplitude and direction of the source, as shown in Figure 2a-c. However, if a discrepancy occurs between the phases of the source and the bases (18) and (20), it leads to errors in determining the direction and amplitude. In this case, the number of non-zerovalues of the basic coefficients significantly exceeds the number of sources, and among them, there are no corresponding directions to the source. Table 1 shows the number of basic coefficients with a value normalized to a maximum value \(\underset{i}{\max}|x_{i}|\) exceeding \(10^{-3}\). In Figure 3a-i, the horizontal axis shows the time normalized to the duration of one internal calculation cycle \(t\) along the vertical axis (scale on the left), which represents the relative change in the coefficients of the basis functions in one normalized time step, \(\Delta X(t)=\underset{i}{\max}|x_{i}(t-1)-x_{i}(t)|/\underset{i}{\max}|x_{i}(t -1)|,\ t=1,2,\dots,T\) (blue), and the current error in representing the input data by basis components, which is defined as the change in the time of the loss function (6), as shown by the scale on the right (red). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activation Function** & **External Source Phase** & **Coefficients Quantity** \\ \hline (18) & \(0^{\circ}\) & 1 \\ (18) & \(72^{\circ}\) & 42 \\ (18) & \(-18.9^{\circ}\) & 42 \\ (19) & \(0^{\circ}\) & 1 \\ (19) & \(72^{\circ}\) & 1 \\ (19) & \(-18.9^{\circ}\) & 1 \\ (20) & \(0^{\circ}\) & 1 \\ (20) & \(72^{\circ}\) & 1 \\ (20) & \(-18.9^{\circ}\) & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of basis coefficients with a level exceeding \(10^{-3}\), single source. From the results obtained, it follows that the time to establish the values of the coefficients of the basis functions when using the activation functions (18) and (19) weakly depends on the value of the source phase. For the activation functions (18) and (20), it is clear that for some phase values, the process could not converge to a global minimum according to the \(l_{0}\) norm, that is, minimize the number of non-zero basis coefficients, although the loss function (6), for all options, is less than \(10^{-16}\) and practically does not change over time (except for the option Figure 3i.