# Application of Artificial Neural Networks to Predict Insulation Properties of Lightweight Concrete
## 2 Materials and Methods
### Applications of ANN
Based on the theorem that for a sigmoid activation function, any continuous function can be uniformly approximated, with any accuracy by means of a feedforward neural network containing two hidden layers the ANN with two hidden layers was trained by multiple feeding of the training dataset to the ANN's input, until the stabilization of the total mean square error of the network at a very low level while avoiding the phenomenon of ANN overfitting [30,35]. After the completed process of the ANN training, a test data set was given to the network input in order to assess the quality of a given network architecture. The relative error of ANN was calculated as the sum of the squares of differences between the laboratory results and those, predicted by the network. The network architecture was selected on the basis of the [35,37] theorem, there is an activation function \(f\) of class \(C^{\infty}\), strictly increasing, sigmoidal, with the following properties:  
For any function \(F\!\in\!C[0,1]^{n}\) and \(\varepsilon>0\), there are constants \(d_{i}\), \(c_{ij}\), \(\theta_{ij}\), \(\gamma_{i}\) and vectors \(\mathbf{w}_{ij}\in\mathbb{R}^{n}\), for which \(\left|F(\mathbf{x})-\sum_{i=1}^{4n+3}d_{i}\,f\left(\sum_{j=1}^{2n+1}c_{ij}\,f \left(\mathbf{w}_{ij}\cdot\mathbf{x}_{i}+\theta_{ij}\right)+\gamma_{i}\right)\right|< \varepsilon\) for all \(\mathbf{x}\in[0,1]^{n}\). In other words, for this specific (sigmoidal) activation function, any continuous function on a unit sample in \(\mathbb{R}^{n}\) can be uniformly approximated with any accuracy by a neural network containing two hidden layers. Given that the universal approximation theorems do not provide information on the number of neurons in the hidden layer needed to provide the desired accuracy, the ANN was first tested with two hidden layers containing a relatively large number of neurons in the hidden layers. After the learning process was completed, the weights of connections between neurons were analyzed and units with very low connection weights were eliminated. On the test set, the quality of each thus obtained neural network was checked and the optimal network was selected for prediction. The SNNSv4.2 (Stuttgart Neural Network Simulator) software used at work has no limitations on the multi-layer neural network architecture with backpropagation. Taking into account the fact that the input data set had only 15 elements and the assumption that the tested properties of LWC should be continuous functions, the network was limited to a network with two hidden layers. In order to obtain the optimal number of neurons in each hidden layer, the same method was used for a given ANN architecture, performing repetitions for each of the ten training and test data sets. The optimal ANN architecture was obtained as 2-11-15-1 during the prediction of density, i.e., two input neurons with shares of GEGA 2 mm and GEGA 4 mm, eleven neurons in the first hidden layer, fifteen neurons in the second hidden layer and one output neuron as density. In the case of properties such as porosity and compressive strength the best results were achieved by the ANN with the same architecture, while in the case of predicting thermal conductivity the optimal architecture of the 2-7-3-1 type was established.