Michail Zak

Terminal attractors for addressable memory in neural networks

Abstract A new type of attractors—terminal attractors—for an addressable memory in neural networks operating in continuous time is introduced. These attractors represent singular solutions of the dynamical system. They intersect (or envelope) the families of regular solutions while each regular solution approaches the terminal attractor in a finite time period. It is shown that terminal attractors can be incorporated into neural networks such that any desired set of these attractors with prescribed basins is provided by an appropriate selection of the weight matrix.

Terminal attractors in neural networks

Abstract A new type of attractor—terminal attractors—for content-addressable memory, associative memory, and pattern recognition in artificial neural networks operating in continuous time is introduced. The idea of a terminal attractor is based upon a violation of the Lipschitz condition at a fixed point. As a result, the fixed point becomes a singular solution which envelopes the family of regular solutions, while each regular solution approaches such an attractor in finite time. It will be shown that terminal attractors can be incorporated into neural networks such that any desired set of these attractors with prescribed basins is provided by an appropriate selection of the synaptic weights. The applications of terminal attractors for content-addressable and associative memories, pattern recognition, self-organization, and for dynamical training are illustrated.

Neutral learning of constrained nonlinear transformations

Two issues that are fundamental to developing autonomous intelligent robots, namely rudimentary learning capability and dexterous manipulation, are examined. A powerful neural learning formalism is introduced for addressing a large class of nonlinear mapping problems, including redundant manipulator inverse kinematics, commonly encountered during the design of real-time adaptive control mechanisms. Artificial neural networks with terminal attractor dynamics are used. The rapid network convergence resulting from the infinite local stability of these attractors allows the development of fast neural learning algorithms. Approaches to manipulator inverse kinematics are reviewed, the neurodynamics model is discussed, and the neural learning algorithm is presented.<<ETX>>

Terminal chaos for information processing in neurodynamics

New nonlinear phenomenon — terminal chaos caused by failure of the Lipschitz condition at equilibrium points of dynamical systems is introduced. It is shown that terminal chaos has a well organized probabilistic structure which can be predicted and controlled. This gives an opportunity to exploit this phenomenon for information processing. It appears that chaotic states of neurons activity are associated with higher level of cognitive processes such as generalization and abstraction.

Terminal Model Of Newtonian Dynamics

A new type of dissipation function which does not satisfy the Lipschitz condition at equilibrium states is proposed. Newtonian dynamics supplemented by this dissipation function becomes irreversible and has a well-organized probabilistic structure.

Creative dynamics approach to neural intelligence

The thrust of this paper is to introduce and discuss a substantially new type of dynamical system for modelling biological behavior. The approach was motivated by an attempt to remove one of the most fundamental limitations of artificial neural networks — their rigid behavior compared with even simplest biological systems. This approach exploits a novel paradigm in nonlinear dynamics based upon the concept of terminal attractors and repellers. It was demonstrated that non-Lipschitzian dynamics based upon the failure of Lipschitz condition exhibits a new qualitative effect — a multi-choice response to periodic external excitations. Based upon this property, a substantially new class of dynamical systems — the unpredictable systems — was introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics. Firstly, such systems have zero Jacobian. As a result of that, they have an infinite number of equilibrium points which occupy curves, surfaces or hypersurfaces. Secondly, at all these equilibrium points, the Lipschitz conditions fails, so the equilibrium points become terminal attractors or repellers depending upon the sign of the periodic excitation. Both of these pathological characteristics result in multi-choice response of unpredictable dynamical systems. It has been shown that the unpredictable systems can be controlled by sign strings which uniquely define the system behaviors by specifying the direction of the motions in the critical points. By changing the combinations of signs in the code strings the system can reproduce any prescribed behavior to a prescribed accuracy. That is why the unpredictable systems driven by sign strings are extremely flexible and are highly adaptable to environmental changes. It was also shown that such systems can serve as a powerful tool for temporal pattern memories and complex pattern recognition. It has been demonstrated that new architecture of neural networks based upon non-Lipschitzian dynamics can be utilized for modelling more complex patterns of behavior which can be associated with phenomenological models of creativity and neural intelligence.

Quantum Neural Nets

The capacity of classical neurocomputers islimited by the number of classical degrees of freedom,which is roughly proportional to the size of thecomputer. By contrast, a hypothetical quantumneurocomputer can implement an exponentially larger number ofthe degrees of freedom within the same size. In thispaper an attempt is made to reconcile the linearreversible structure of quantum evolution with nonlinear irreversible dynamics for neuralnets.

Non-Lipschitzian dynamics for neural net modelling

Abstract Failure of the Lipschitz condition in unstable equilibrium points of dynamical systems leads to a multiple-choice response to an initial deterministic input. The evolution of such systems is characterized by a special type of unpredictability measured by unbounded Lyapunov exponents. Possible relation of these systems to future neural networks is discussed.

The problem of irreversibility in Newtonian dynamics

A new type of dissipation function which does not satisfy the Lipschitz condition at equilibrium states is proposed. It is shown that Newtonian dynamics supplemented by this dissipation function becomes irreversible, i.e., it is not invariant with respect to time inversion. Some effects associated with the approaching of equilibria in infinite time are eliminated. New meanings of chaos and turbulence are discussed.

Statics of wrinkling films

This paper presents analytical investigations of some general properties of equilibrium of films in a state of wrinkling. The results are illustrated by examples.