# The Method of Restoring Lost Information from Sensors Based on Auto-Associative Neural Networks
## 4 Results
The loss function dynamics comparison of the data set validation and training made it possible to assess the model generalization ability. Reducing the loss function on the validation and training data sets is also a pattern in the data. At the same time, the loss function on the validation data set also decreases, which indicates the overfitting effect elimination. 3. The loss function dynamics analysis also helped determine whether adjustments to training parameters such as training rates or regularization coefficients were required. Since the loss function is reduced on the training and validation data sets, changes to the training parameters may not be necessary to improve the ANN training process. The work additionally assessed the Kullback-Leibler divergence [49] (Figure 11), which shows the information divergence (relative entropy) of two probability distributions and is calculated as follows:  
\[D_{KL}(P\|Q)=\sum_{i=0}^{N}P(x_{i})\cdot\log\left(\frac{P(x_{i})}{Q(x_{i})} \right), \tag{14}\]  
Figure 10: The loss function dynamics diagram at the model training and validation stages. Figure 9: The auto-associative neural network training feature diagram: (**a**) permissible generalizing ability; (**b**) retraining [45]. where \(P(x_{i})\) is the original distribution and \(Q(x_{i})\) is the approximating distribution. As can be seen in Figure 11, the maximum value of the deviation between two probabilistic distributions of the TV3-117 TE parameter values does not exceed 0.025 (2.5%) and does not exceed the maximum value of the loss function at the model validation and training stages, which is also 0.025 (2.5%); then, we can judge the acceptable divergence of probability distributions of the TV3-117 TE parameters. To simulate the failure of one of the standard sensors of the TV3-117 TE, a virtual installation was created in the LabVIEW software package, which simulated a system with three sensors: gas generator rotor r.p.m. (D-2M), free turbine rotor speed sensor (D-1M), and gas temperature in the compressor turbine front (14 dual thermocouples T-102) (Figure 12). Using the developed virtual installation, the LabVIEW software package simulated a single sensor failure (break), as well as the lost information restoration using an ANN. The lost information restoration diagram by the ANN in the failure event of one of the standard sensors of the TV3-117 TE (using the example sensor of the gas temperature in the compressor turbine front) is shown in Figure 13a; the lost information restoration process diagram from the onboard system for monitoring and diagnosing sensors of the \"optimal\" ANN in a single failure case of the sensors (their disconnection) is shown in Figure 13b where the \"white line\" represents the parameter value, the \"green line\" represents the restored signal value using the ANN, and the \"red line\" represents the parameter value obtained from the sensor. The approach disadvantage is proposed in Figure 13. There is a high error of value restoration in an onboard system failure case for monitoring and diagnosing one sensor: the restoration error is 4.5..50% in the case of an interruption and up to 5.5% in the gradual failure case. Another method of lost information restoration from sensors based on an ANN is the buffer (memory stack) used that stores the last correct value in the sensor failure case (Figure 13b). Experimental studies using the \"optimal\" ANN have shown that the lost information restoration error in the case of a single failure is 1.5..2.0% and, in the gradual failure case, does not exceed 2.5%. Additional error reduction is achieved by using a failure detection logic that identifies the failure and isolates the failed sensor. Figure 11: Kullback–Leibler divergence diagram. Figure 12: Virtual installation in the LabVIEW software package, simulating a system with three sensors.