New Models And Learning Algorithms For Functional Network

Functional network(FN) is a recently introduced generalization of neural network (NN), Every problem that can be solved by NN can be solved by FN, but some that can not be solved by NN can be solved by FN too. Because researches of functional network are relative less in domestic and foreign, there are some basic theories and problems of novel functional network structure for perfecting basic theories, and a novel learning algorithm of functional network is presented. In this paper, we try to use the method of computation mathematics and a new viewpoint is proposed that functional networks can be regarded as a kind of visualization means of some mathematical methods. Then the functional networks representations of mathematical methods or mathematical structure, a novel computation models and methods for traditional numerical computation methods are proposed. Exactly speaking, the functional neuron as the basic units, base on standpoint of map theory, the complex functional networks are combined with several simple functional networks, which is not just a simple assembly, but according to solving problems of transcendent knowledge. Namely, constitute the topology structure of functional networks and design their corresponding learning algorithms of functional network based on problems on hand. Main works are as follows:1. A functional networks approximation method for non-linear functions is proposed, a kind of separable functional networks for using function approximation is designed. Based on functional networks model, a learning algorithm for function approximation is given. It can approximate a given continuous function satisfying given precision. The simulation results demonstrate that the approximation method presented in this paper has convergence and powerful performance.2. A kind of rational fraction functional network model is proposed and the learning algorithm for rational fraction functional network is presented. The parameters of the rational fraction functional network are determined by solving a series of linear equations. The experiment result shows the effectiveness of the rational fraction functional networks in solving approximation problems of the function with a pole, which polynomial functional networks can not solve.3. A new orthogonal functional network model is presented, and its construction method and learning algorithm are given. The orthogonal basis functions that have stronger approximation ability are chosen to approximate the functional neural functions. The experiment result shows that approximation performance of functional networks is greatly strengthened with orthogonal basis functions.4. A functional network model of solving a class of given functional equations is presented. According to the characteristic of the equaction, the computation model and corresponding learning algorithm are desingned. The parameters are carried out by solving linear equations. The simulation results demonstrate that the learning method presented in the paper is more efficient and feasible in finding the roots of functional equations. The method can be also promoted to solve the general functional equations.