Artificial neural networks (ANN) (Schalkoff, 1997; Yegnanarayana, 2009)
is a biologically inspired machine learning method that allows
recognizing the pattern of a given data set. This technique is broadly
employed in civil and petroleum engineerings (Flood and Kartam, 1994;
Yeh, 1998; Mohaghegh, 2000) to deal with the hazardous of the
materials' properties and to predict new data from historically collected
data. Basically, known data are used to design and traine an ANN
network that is then used to predict new data sets. Mathematically, an
ANN relates input and output data by following equation systems: *Y=b
+ ω * f(B + W * X)*, where *X* and *Y* are the vectors
of input and output data, respectively; *f* an activation
function; *ω* and *W* the weight matrix of the network; *b*
and *B* the bias vectors. By mean of training a network, the
number of hidden layers, the activation function, the weight and the
bias are calibrated to minimize the mean square error (MSE) between
the computed output and the measured data. One the network is trained,
it can be used to predict the output *Y* of new data set.

Let consider a network with a single hidden layer (one hidden layer is
enough to deal with a large range of problems) and a single output
value for each sample. In this exemple, the sigmoid activation
function, *f(x)=1/(1+exp(-λx))*, is choosen and the stepest
method (Fletcher and Powell, 1963) is employed to speedup the
convergence of the MSE to zero during the training procedure.

Schalkoff, R. J. (1997). Artificial neural networks (Vol. 1). New York: McGraw-Hill.

Yegnanarayana, B. (2009). Artificial neural networks. PHI Learning Pvt. Ltd..

Flood, I., & Kartam, N. (1994). Neural networks in civil engineering. I: Principles and understanding. Journal of computing in civil engineering, 8(2), 131-148.

Yeh, I. C. (1998). Modeling of strength of high-performance concrete using artificial neural networks. Cement and Concrete research, 28(12), 1797-1808.

Mohaghegh, S. (2000). Virtual-intelligence applications in petroleum engineering: Part 1—Artificial neural networks. Journal of Petroleum Technology, 52(09), 64-73.

Fletcher, R., & Powell, M. J. (1963). A rapidly convergent descent method for minimization. The computer journal, 6(2), 163-168.

Trained weights and bias: