On functional analysis with neural networks
Hans‐Georg Herrmann · Smart Materials and Structures · 1998
A new method for data set preprocessing and designing feed-forward neural networks as an approximation tool for a given physical problem is introduced. By using further available knowledge about the problem the presented method can improve considerably the network's ability to generalize and can enhance its overall performance properties. The first new element of the proposed method, dimensional analysis, is based on the mathematically proven -theorem and can be applied to the whole group of physical and engineering phenomena which can be expressed in dimensionally homogeneous equations. Therefore a neural network has to satisfy the fundamental mathematical prerequisite of dimensional homogeneity when employed as an approximation tool for a given dimensionally homogeneous relation. The second element of the proposed method needs the assumption of a specific function which describes adequately the character of the given phenomena. A problem-specific topology of the network can be easily developed by determining the number of hidden units, the way of connecting neurons and the suitable transfer functions. To overcome the inherent disadvantages of a simple mapping between input and output and a limited ability to generalize, this method yields an improved generalization of a neural network. Furthermore, a considerable improvement in the learning speed and accuracy of this network type can be observed. Provided that the physical relation is homogeneous in dimension, it has been shown that this developed neural network is able to learn a presupposed function exactly and therefore to generalize even outside the range of the learning data. Further available knowledge about the problem can be easily implemented in the design of the network and may reduce the net topology in size.