Functional-link Neural Networks and Visualization Means of Some Mathematical Methods

This chapter focuses on functional-link neural networks. Beginning with the XOR problem, we discuss the mathematical essence and the structures of functional-link neural networks. Extending this idea, we give the visualization means of mathematical methods. We also give neural network representations of linear programming and fuzzy linear programming. A single-layer neural network, first studied by Minsky and Papert, was named perceptron in 1969 [l]. It is well known that a single-layer perceptron network cannot solve a nonlinear problem. A typical problem is the Exclusive-OR (XOR) problem. Generally, there are two approaches to solve this nonlinear problem by modifying the architecture of this single-layer perceptron. The first one is to increase number of the hidden layers, and the second one is to add higher order input terms. There are numerous applications using either of these approaches [2-41. Here we will illustrate that these two approaches, in fact, are essentially mathematical equivalence. is the same neuron with a higher order term, 2 1 .x2 shows a simple neuron with two inputs.