Quantization in Neural Networks

Abstract

Like the human brain, an artificial neural network is a complex nonlinear parallel processor; it is often called a neurocomputer. Accordingly, mathematical models of a neural network are usually continuous and stochastic, naturally associated with fuzzy logic. Classical systems of artificial intelligence are always naturally associated with classical logic and discrete mathematics. Thus, the representations and models of knowledge, undeniable at least since Aristotle, do not correspond to the cognitive models that are obtained as a result of studying the human brain. In view of Niels Bohr, quantization is a phenomenon of a discrete, sequential process, that inherent in continuous and stochastic systems. However, the traditional mathematical model of quantum mechanics did not imply generalization to dissipative systems. The corresponding generalization, called the Dynamic quantum model (DQM), was proposed by author. It is defined for any dynamic system, given by ordinary differential equation or by diffeomorphism, or for dynamic systems that using logical operations. The neural network is exactly the DQM in the space of input signals. In this paper DQM is defined and constructed universally for both Hamiltonian systems and systems with the fuzzy logic truth function on phase space. The paper goal is to demonstrate quantization on DQM, i.e. actually on neural networks, and to extend the classical Bohr-Sommerfeld condition to the general case, in particular, to systems with a fuzzy truth function.