The processes of adaptation, learning, discovery, and invention are expected to play an increasingly significant role in emerging large-scale computationally intelligent (CI) systems. . The meanings of adaptation and learning are well-known. By discovery we mean the process of creating a new hypothesis based on sufficient new data that does not fit existing hypotheses; while by invention, the process of creating (synthesizing) new prototypes by interpolation or extrapolation of existing ones.
In this lecture we will present a kernel-based mathematical approach to the modeling and design of the processes of adaptation, learning, discovery and invention in nonlinear dynamical systems. The approach is based on our previous work in which uncertainty is handled by mathematical approximation theory methods, specifically, by best approximation of nonlinear functionals in an appropriate Reproducing Kernel Hilbert Space (RKHS). This formulation leads to optimal nonlinear system models in the form of artificial neural networks (ANNs) in a natural way, that is, without imposing, a-priori, a neural structure on the system being modeled. For this reason, while connecting with the mathematical foundations on which they are based, we will use ANNs as generic representations of the nonlinear dynamical systems under discussion.
Computer simulation results, some using real data, will be presented to establish and illustrate the theoretical developments.