# The Method of Restoring Lost Information from Sensors Based on Auto-Associative Neural Networks
## 3 Materials and Methods
In the presented work context, which considers the ANN use to monitor and restore information about the complex nonlinear technical object state, improving the ANN training algorithm is critical. The training algorithm directly affects the network's ability to adequately respond to changes in data, including sensor failures, which is important to ensure the object's safety and stability. Thus, improving the ANN training algorithm includes several aspects:  
1. Improved adaptability, which implies the algorithm's ability to adapt to changes in object operating conditions, including operating modes changes and possible abnormal situations;
2. Improving generalization ability, which means that the network needs to undergo training not just on data collected during tests but also on possessing the ability to generalize its acquired insights to adapt to unforeseen scenarios encountered in real-world operations;
3. Accounting for process dynamics, which requires the training algorithm to take into account the engine state dynamics and the corresponding dynamics data coming from the sensors;
4. Improving noise and error tolerance, which requires the network's ability to operate effectively even in the data noise or errors presence in sensor measurements;
5. Training time optimization, refers to reducing the training time required by a network to improve its efficiency and practical applicability. Figure 4: Functional control diagram of a complex nonlinear technical object based on an auto-associative neural network ([9]). Thus, the approach to improving the training algorithm should be comprehensive, including both theoretical developments and practical experiments with various training methods and network architectures. This approach will help ensure optimal operation in the system for information monitoring and restoration of the object condition in a wide range of operating conditions. Using the basic ANN training algorithm, a modified method for training is proposed by adding regularization to the loss function and updating the weights to achieve the following goals:  
Goal 1. Prevent overfitting. Regularization helps reduce the model overfitting risk where it overfits the training data and loses its ability to generalize new data. This is especially important for small data volumes or complex models. Goal 2. Improve generalization ability. We try to create a more robust and generalizable model by adding regularization to the loss function. The model not only performs well on training data but can also produce good results on new data. Goal 3. Control the model complexity. Regularization helps control the model complexity by the weights and their growth restraining. This allows for simpler and more interpretable models, which is often a desirable property in practical applications. Goal 4. Improve training robustness: Adding regularization can help improve the training process stability by reducing the model sensitivity to changes in training parameters or input data. Below is a step-by-step description of the proposed modified ANN training algorithm with the regularization addition to the loss function and updating the weights. Step 1. The ANN weights are the initialized data with random values. Let us denote the weight matrix \(W^{(l)}\) between layer \(l\) and \(l+1\). Step 2. Next, the input data \(x\) is fed to the input layer, and activation \(a^{(0)}=x\) is carried out. For each layer \(l\), we can write:  
\[z^{(l+1)}=a^{(l)}.W^{(l)}+b^{(l)}, \tag{3}\]  
Figure 1:0  
where \(f\) is the activation function and \(b^{(l)}\) is the displacement vector on layer \(l\). Step 3. At this stage, the resulting output \(a^{(L)}\) is compared with the input data \(x\) using the loss function \(L\) according to the expression:  
Figure 1:1  
where \(n\) is the input data elements number and \(\lambda_{1}\) and \(\lambda_{2}\) are the regularization coefficients for \(L_{1}\) and \(L_{2}\), respectively. Step 4.