Ellipsoidal/radial basis functions neural networks enhanced with the Rvachev function method in application problems

A new type of multi-centered basis function neural networks (MCBFNNs), that are the generalization and extension of ellipsoidal/radial basis functions neural networks (E/RBFNNs), is introduced. This paper aims to further elaborate a method of supervised binary clusters classification and identification using Radial Basis Function Neural Networks (RBFNNs) enhanced with the Rvachev Function Method (RFM) in complex non-convex, disconnected domains. Practical numerical examples are presented only in particular cases of MCBFNNs for E/RBFNNs enhanced with the RFM. R-functions are used to construct complex pattern cluster domains, parameters of which are applied to E/RBFNNs to establish domain boundaries for pattern binary classification or parameters for systems identification. ror functional is a convex quadratic one with respect to weight functions which take weight values on the discrete connectors between neurons. The activation function of neurons of E/RBFNNs is the signum function, and, therefore, the error functional is non-smooth. The feed forward Neural Networks with the delta supervised learning rule during training phase is applied. The sub-gradient of the discretized error function is used rather than its gradient, because it is not smooth. The application of the RFM allows for the creation, implementation, and resolution of large heterogeneous Neural Networks capable of solving diverse sets of binary classification problems with greater accuracy. Numerical explorations in clustering and classification substantiate concepts and assumptions. Applications to human hearing sensitivity and identification of a dynamical system are presented on numerical examples.