## 5 Numerical Modeling
### A Large Number of Sources  
The following is the result of an analysis of the influence of the number of sources, which varied from 7 to 16, on the results of the MFNN neural network for complex data with the activation function (19), which showed the best result. In Figure 7a,c,e, on the graphs, the horizontal axis shows the angular direction in degrees, and the vertical axis shows the values of the modules of the coefficients of the basis components \(X\) obtained at different times. They were obtained as a result of processing using a neural network with activation function (19), corresponding to the angular directions of the grid of angles of basis functions, as shown along the horizontal axis (blue lines), and the modules of the radiation pattern formed using FFT (black lines). The position of point sources, considering amplitudes \(\dot{b}_{i}\) and angles \(\theta_{i}\), is represented by red dots. In Figure 7a-f, the results are shown for the case of seven external sources, with levels \(\dot{b}_{1-7}\) = [1.0, 1.0, 0.5, 2.0, 0.7, 1.0, 1.0] and angular directions \(\theta_{1-7}\) = [\(-42.19^{\circ}\), \(-33.75^{\circ}\), \(-22.50^{\circ}\), \(-18.98^{\circ}\), \(-11.25^{\circ}\), \(-5.63^{\circ}\), \(-2.11^{\circ}\)] for the next three normalized time intervals \(T\) = [0-1000 ((a), (b)), 500-5000 ((c), (d)), 140,500-148,000 ((e), (f))], which show how the \(X\) coefficients and loss functions change (6). It can be seen that the \(X\) coefficients group quickly enough around the directions of the sources, and the values of the coefficients outside these zones drop below \(10^{-8}\) (Figure 7a). If attention is paid to the graphs (Figure 7b,d,e), you can see that loss function (6) changes unevenly in jumps, while the changes in the basic coefficients remain generally monotonic. Subsequently, the process of changing the coefficients \(X\) slows down significantly, but their grouping continues in such a way that each direction to the source corresponds to its own area of significant coefficients, separated by zero coefficients from the rest (Figure 7c,e). By the time \(T\) = 148,000, there are only 16 significant basis coefficients of \(X\), and only 10 of them have a value greater than \(10^{-1}\). It turns out that in fact, the sums of the coefficients in these areas are equal to the levels of the corresponding sources, and the weighted average values of the angular directions in these areas are very close to the directions of the sources. The directions and levels determined in this way are displayed in Figure 7a,c,e with red dotted lines, which show the coincidence with the directions and levels of external sources. This feature makes it possible to significantly reduce the time required to find a solution to the system of differential Equation (14). Almost upon completion of the interval \(T=\) [0-1000] (Figure 7a), a result similar to that obtained in the interval \(T=\) [140,500-148,000] (Figure 7e) is achieved. The graphs shown in Figure 8a,b are similar to those shown in Figure 7a, except the number of external sources for Figure 8a was 15: \(\dot{b}_{1-15}=\) [1.0, 1.0, 0.5, 2.0, 0.7, 1.0, 1.0, 1.0, 0.5, 1.0, 0.7, 0.8, 0.5, 1.0], and \(\dot{\theta}_{1-15}=\) [\(-42.19^{\circ}\), \(-33.75^{\circ}\), \(-22.50^{\circ}\), \(-18.98^{\circ}\), \(-11.25^{\circ}\), \(-5.63^{\circ}\), \(-2.11^{\circ}\), \(5.63^{\circ}\), \(9.14^{\circ}\), \(16.88^{\circ}\), \(19.69^{\circ}\), \(28.13^{\circ}\), \(31.64^{\circ}\), \(39.36^{\circ}\), \(47.81^{\circ}\)]; for Figure 8b--16, another external source was added: \(\dot{b}_{16}=\)1.0, \(\dot{\theta}_{16}=\) 53.44\({}^{\circ}\). When exposed to 15 external sources (Figure 8a), the basis coefficients are concentrated around the directions to the sources, as noted above.