Well reservoir model identification using translation and scale invariant higher order networks

This paper describes the use of higher order neural networks to identify well reservoir response models from test data. Well reservoir response models are characterised by a family of parametrically related curves. Neural networks can in principle offer an interesting approach to the identification problem as data are often uncertain and incomplete. However, it turns out that the well reservoir model, viewed as a curve in two dimensions, is invariant with respect to translation and changes of scale of the axes. This poses severe problems for a standard backpropagation network using the two-dimensional plot as an input retina. This difficulty can be overcome by using a higher order network in which the output is forced to be invariant with respect to the required transformations of the retina. In this way, the potentially huge number of weights is significantly reduced using the invariance condition as a constraint which acts so as to divide the weights into equivalence classes within which they are equal. The resulting network can then be trained using standard techniques. We contrast this network approach with classical methods of model identification.