## 2 Materials and Methods
### Artificial Neural Network  
One of the most commonly used tools to describe dependencies in geotechnics was standard regression. Recently, non-standard methods, including an artificial neural network and a fuzzy logic regression, have also been increasingly used as approximation tools. Moreover, the most common neural networks [46] used for function approximation (nonlinear regression) are multi-layer perceptron (MLP), radial basis function (RBF) networks and support vector machines (SVM). The feedforward networks consist of a series of layers (Figure 2). The first layer has a connection from the network input. Each subsequent hidden layer has a connection from the previous layer. The final output layer produces the network's output.  
The training network process requires a set of examples of proper network adjust (relations of parameter inputs and target outputs). The process of training a neural network involves changing the values of the weights and biases of the network to optimize network performance, as defined by the  
Figure 1: Selected cross-section of the local subsoil based on CPTM probes at the Rzeszow site. CPTM: cone penetration test.  
Figure 2: Diagram of the artificial neural network.  
network performance function. The default performance function for feedforward networks is mean square error.  
For training multilayer feedforward networks, any standard numerical optimization algorithm can be used to optimize the performance function, but there are a few that have shown excellent performance for neural network training. These optimization methods use either the gradient of the network performance with respect to the network weights or the Jacobian of the network errors with respect to the weights [47].  
The Levenberg-Marquardt [48; 49] method generally is the fastest training method. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton [50] backpropagation method is also quite fast. This algorithm requires more computation in each iteration and more storage than the conjugate gradient methods, although it generally converges in fewer iterations. Both of these methods tend to be less efficient for large networks (with thousands of weights), since they require more memory and more computation time. The Bayesian regularization [51] algorithm requires more time, but can result in good generalization for difficult, small or noisy data sets.  
The architecture of the networks used can be described as \(I\)-\(H\)-\(O\), where \(I\) is the number of inputs, \(H\) is the number of neurons in the hidden layer and \(O\) is the number in the output layer. Mostly, the definition of the network uses several types of transfer (activation) functions: log-sigmoid, hyperbolic tangent sigmoid, radial basis and some linear.  
All neural network computation was performed using the Neural Network Toolbox for Matlab [52]. In all the considered examples, a shallow multilayer feedforward network with one hidden layer was applied. In input of nets, we take into account two parameters, or one of them, the organic content \(LOI_{T}\) and the natural water content \(w\). The output of nets was soil unit weight of organic soils \(\gamma_{t}\). In the calculations, five to eight neurons were used in the hidden layers. The Levenberg-Marquardt method was used in the training process. We used a log-sigmoid transfer function in the hidden layer and a linear function in the output layer.