Application of Artificial Neural Networks in Geoscience and Petroleum Industry

Abstract

It has been shown that artificial neural networks (ANNs), as a method of artificial intelligence, have the potential to increase the ability of problem solving to geoscience and petroleum industry problems particularly in case of limited availability or lack of input data. ANN application has become widespread in engineering including geoscience and petroleum engineering because it has shown to be able to produce reasonable outputs for inputs it has not learned how to deal with. In this chapter, the following subjects are covered: artificial neural networks basics (neurons, activation function, ANN structure), feed-forward ANN, backpropagation and learning (perceptrons and backpropagation, multilayer ANNs and backpropagation algorithm), data processing by ANN (training, over-fitting, testing, validation), ANN and statistical parameters, an applied example of ANN, and applications of ANN in geoscience and petroleum Engineering.

Appendix: Important Statistical Parameters

The corresponding relations of a few important statistical parameters to compare performance and accuracy of different neural network models are given as follows:

• 1. Average percent relative error (APE):

  This error is defined as the relative deviation from the measured data.

  $$ {\text{APE}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} E_{\text{i}} $$

  (23)
  $$ E_{i} = \left[ {\frac{{P_{\text{m}} - P_{\text{e}} }}{{P_{\text{m}} }}} \right]_{i} \quad i = 1,{ 2},{ 3}, \ldots , n, $$

  (24)

• 2. Average Absolute Percent Relative Error (AAPE):

  This error gives an idea of absolute relative deviation of estimated outputs from the measured or expected output data.

  $$ {\text{AAPE}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left| {e_{i} } \right| $$

  (25)
  $$ e_{i} = \left[ {P_{\text{m}} - P_{\text{e}} } \right]_{i} $$

  (26)

• 3. Mean squared error (MSE):

  This error is corresponding to the expected value of the squared error loss.

  $$ {\text{MSE}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} e_{i}^{2} $$

  (27)

• 4. Average root-mean-square error (ARMSE):

  This error is an indeed measure of scatter or lack of accuracy of the estimated data.

  $$ {\text{ARMSE}} = \sqrt {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} e_{i}^{2} } $$

  (28)

• 5. Standard deviation (SD):

  This error shows the dispersion of the values from the average value or mean.

  $$ {\text{SD}} = \left[ {\frac{{n\mathop \sum \nolimits_{i = 1}^{n} E_{i}^{2} - (\mathop \sum \nolimits_{i = 1}^{n} E_{i} )^{2} }}{{n^{2} }}} \right]^{\frac{1}{2}} $$

  (29)

• 6. Variance or V, $\sigma^{2}$:

  This error is the square of the standard deviation.

  $$ \sigma^{2} = \frac{{\sum {\left( {X - M} \right)^{2} } }}{N} $$

  (30)

• 7. Correlation coefficient or Pearson coefficient (R):

  It represents the degree of success in reduction of the standard deviation (SD). It is normally used as a measure of the extent of the linear dependence between two variables. The nearer R is to 1, the better the convergence and ANN performance is.

  $$ R = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left[ {\left( {P_{{{\text{m}},i}} - P_{{m,{\text{av}}}} } \right) \times \left( {P_{{{\text{e}},i}} - P_{{e,{\text{av}}}} } \right)} \right]}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{n} \left[ {\left( {P_{{{\text{m}},i}} - P_{{{\text{m}},{\text{av}}}} } \right)^{2} } \right] \times \mathop \sum \nolimits_{i = 1}^{n} \left( {P_{{{\text{e}},i}} - P_{\text{e,av}} } \right)^{2} } }} $$

  (31)
  $$ P_{\text{av}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} P_{i} $$

  (32)

• 8. Squared Pearson coefficient: R ²