Reservoir properties from well logs using neural networks 3

Today, I finish reading the dissertation ‘Reservoir Properties from Well Logs Using Neural Networks’ Chapter 2.

Summary

Neural networks can be used to perform the two basic tasks of pattern recognition and function approximation.

Pattern recognition

An unsupervised network

1.     An unsupervised network for feature extraction, namely transforming the input patterns into an intermediate smaller dimensional space

2.     A supervised network for classification: map the intermediate patterns into one of the classes in an r-dimensional space where r is the number of classes to be distinguished.

A supervised network

It performs the task of feature extraction and classification both based on the information it has extracted from the training data.

Function approximation

Functional relationship:

Given a set of examples to map the function:

Approximate the unknown :  for all x

Bias variance dilemma

Overtraining:

A small bias but a large variance

Cross-validation approach:

Early stopping method of training

In this study we use the overtraining approach for predicting porosity and water saturation, cross validation approach for predicting permeability and a soft overtraining approach for lithofacies identification.

A MLP network:

Advantages: It does not require any assumptions. It exhibits a great degree of robustness or fault tolerance because of built-in redundancy. It has a strong capability for function approximation. Previous knowledge of the relationship between input and output is not necessary. The MLP can adapt its synaptic weights to changes in the surrounding environment by adjusting the weights to minimize the error.

Disadvantages: We use LM algorithm to adjusting weights. The mean square error surface of a multiple network may get stuck in the local minima instead of converging into the global minimum.

Multiple network system: 

An actual multiple network system could consist of a mixture of ensemble and modular combinations at different levels. The architecture of the networks for predicting porosity and water saturation are ensemble combination while the architecture of the networks for predicting permeability and lithofacies are modular and ensemble combination.

Ensemble combination:

1.    The bias of the ensemble averaged-function  pertaining to the CM is same as that of the function  pertaining to a single neural network.

2.    The variance of  is less than that of .

3.     The individual expert should be purposely over-trained to reduce the bias at the cost of the variance.

Training parameters: the initial weights, the training data, the topology of the networks and the training algorithm.

The problem with this method is that it requires large amounts of data.

1.     The training sets are adaptively resampled.

2.     Picking n samples from a training dataset of N samples

3.     Virtual samples

Providing weights to each network by two approaches:

Unconstrained approach:

Combined output: , where  is output from the individual network,  are the weights.

Approximation error is: .

Constrained approach:

There is an additional constraint, that is .

The weights can be calculated from the training dataset, or partially on the training and test dataset. This method is called the optimal linear combination (OLC) method.

Modular combination:

The single network has a large number of adjustable parameters and hence the risk of overfitting the training dataset increases.

The training time for such a large network is likely to be longer than for all the experts trained in parallel.

1.     Avoid of overfitting, save time

2.     Reduce model complexity, making the overall system easier to understand, modify, and extend

Class decomposition, automatic decomposition

There are four different models of combining component networks: cooperative, competitive, sequential and supervisory.

Ensembles are always involved in cooperative combination.

The input-output relationship is linear in MLR whereas the relationship is nonlinear in neural network. The neural network method does not force predicted values to lie near the mean values and thus preserves the natural variability in the data.

Tomorrow, I will read more of the dissertation.