## 4 Discussion
### Generalization of Results
In the backpropagation stage, the gradients are updated for all layers using regularization, \(\lambda\), and a dynamic learning rate, \(\eta_{I}\), depending on the change in the optimization function. Adaptive parameters such as \(k_{p}\), \(T_{i}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(\alpha\) are optimized, taking into account the error changes. The training process is repeated until the specified criteria, such as the minimum error or the maximum number of epochs, are achieved, and the algorithm'saccuracy is evaluated on test data, demonstrating the neural network's ability to effectively handle nonlinearities and system delays. To conduct the computational experiment, it was noted that the choice of the TV3-117 engine as the research object was based on its widespread use in the Mi-8 MTV helicopter power plant, and the key diagnostic parameter of the gas-generator rotor speed, \(n_{\mathrm{TC}}\) (see Figure 4), recorded by the standard D-2M sensor, was used to form the training dataset (see Table 3). The use of \(z\)-normalization made it possible to standardize the data, improving the neural network training algorithm's convergence (see Table 4), and cluster analysis (see Figure 5) confirmed the representativeness of the training and test datasets. This approach ensures the statistically substantiated reliability of the dataset and error minimization, which are critically important for improving helicopter operation safety. This research demonstrated the effectiveness of the proposed method for automatic parametric optimization for the TV3-117 engine control system. When analyzing transient processes (see Figures 6-9 and Table 5), it provided a twofold reduction in the \(n_{\mathrm{TC}}\) parameter transition time to a level of \(\approx\)1, which confirms its suitability for actual operating conditions. Unlike the search-free method [25; 26; 27; 28], where \(n_{\mathrm{TC}}\) is restored only to a level of \(\approx\) 0.5, the developed method provides higher-quality control, which makes it practically applicable in aviation technology. Sensitivity analysis (\(\xi_{1}\), \(\xi_{2}\), \(\xi_{11}\), \(\xi_{22}\), \(\xi_{12}\)) showed (see Figure 10 and Table 6) that the parameters \(\theta_{1}\) and \(\theta_{2}\) have the most significant influence at the initial time, with \(\theta_{1}\) dominating, and their interaction is insignificant, which simplifies the tuning of the controller. The iteration process confirmed the positive definiteness of the Hessian matrix, and the method's convergence was observed to be faster compared to analogues (1.36-fold reduction in iterations). The application of the developed neural network and training algorithm showed a maximum accuracy of 0.993 at 160 epochs (see Figure 11a), ensuring the error function minimization to \(MSE_{\mathrm{min}}\) = 0.005 (see Figure 11b). These results confirm the reliability and stability of the proposed method for complex objects with significant delays and inertia. The comparative analysis (see Table 7) results of the proposed neural network training algorithm and alternative approaches' efficiency confirmed its superiority. The developed algorithm achieved the mean square error minimum value (\(MSE_{\mathrm{min}}\) = 0.005) for the minimum number of training epochs (160), which is significantly better compared to traditional error backpropagation and other methods (exceeding \(MSE_{\mathrm{min}}\) by 1.86 to 6.02 times and increasing the number of training epochs by 1.38 to 2.38 times). The comparison of neural network architectures showed that the developed network has the highest accuracy (0.993), recall (1.0) and F1-measure (0.995) while taking only 1 min and 12 s to train (see Table 8). Alternative networks, such as RBF, Jordan and Elman, have comparable accuracy but require significantly more time to train (from 2 to 4 min). Hopfield and Hamming networks showed substantially lower accuracy values (0.837 and 0.811, respectively), which limits their use for tasks requiring high accuracy.