Quantitative phase analysis of clay minerals by X-ray powder diffraction using artificial neural networks. I. Feasibility study with calculated powder patterns

An artificial neural network is a densely interconnected array of simple processing elements (referred to as neurons herein) arranged in an input layer, an output layer, and (usually) one or more 'hidden layers' that have neither direct inputs from outside the network nor direct outputs beyond the network, all designed to simulate the learning and recall functions of analogous structures in the brain (Fig. 1). For the purposes of this study, the inputs to the neural network will be numerical values derived from (simulated) XRD patterns of clay-mineral mixtures, and the desired outputs will be the weight fractions of the constituents of those mixtures. The connections between neurons in successive layers are called synapses another obvious biological analogy and the synapses have adjustable weights associated with them. The function of a neuron in layer i is simply to multiply the inputs from neurons in layer i 1 by the weights associated with the respective synapses, add a bias term, and then operate on the result with a mathematical transfer function to obtain the output of that neuron to neurons in layer i+1. If, for example, the inputs from layer i 1 to a given neuron in layer i are xl, x2,...,xn (Fig. 2), these are multiplied by their respective synaptic weights Wl, w2,...,wn, summed with a bias (b), and then operated on by a transfer function to determine the output from that neuron to neurons in layer i+1. The transfer function may be linear, sigmoidal, or any other type of smooth (differentiable) function. The range in the desired output largely determines the choice of transfer function; for example, if outputs in the range 0 to 1 are desired, a log-sigmoid transfer function is appropriate (i.e. a smoothed step 9 1999 The Mineralogical Society