NEURAL NETWORKS WITH

ANTICIPATION:

PROBLEMS AND

PROSPECTS

Alexander MAKARENKO

Institute for Applied System Analysis at

National Technical University of Ukraine (KPI)

INTRODUCTION

Nonlinear networks science

(problems and effects):

stability

bifurcations

chaos

sinchronisation

turbulence

chimera states

MODELS AND SYSTEMS

(RECENTLY):coupled oscillators

coupled maps

neural networks

cellular automata

o.d.e. system

………MAINLY of NEUTRAL or

WITH DELAY

ANTICIPATION

Neural network learning (Sutton, Barto, 1982)

Control theory (Pyragas, 2000?)

Neuroscience (1970-1980,… , 2009)

Traffic investigations and models (1980, …, 2008)

Biology (R. Rosen, 1950- 60- ….)

Informatics, physics, cellular automata, etc. (D. Dubois, 1982 - ….)

Models of society (Makarenko, 1998 - …)

ANTICIPATION

The anticipation property is that the individual makes a decision accounting the future states of the system [1].

One of the consequences is that the accounting for an anticipatory property leads to advanced mathematical models. Since 1992 starting from cellular automata the incursive relation had been introduced by D. Dubois for the case when

„the values of of state X(t+1) at time t+1 depends on values X(t-i) at time t-i, i=1,2,…, the value X(t) at time t and the value X(t+j) at time t+j, j=1,2,… as the function of command vector p‟ [1].

ANTICIPATION In the simplest cases of discrete systems this

leads to the formal dynamic equations (for the case of discrete time t=0, 1, ..., n, ... and finite number of elements M):

where R is the set of external parameters (environment, control), {si(t)} the state of the system at a moment of time t (i=1, 2, …, M), g(i) horizon of forecasting, {G} set of nonlinear functions for evolution of the elements states.

( 1) ({ ( )},...,{ ( ( ))}, ),i i i is t G s t s t g i R

“In the same way, the hyperincursion is an

extension of the hyper recursion in which several

different solutions can be generated at each time

step” [1, p.98].

According [1] the anticipation may be of „weak‟ type

(with predictive model for future states of system,

the case which had been considered by R. Rosen)

and of „strong‟ type when the system cannot make

predictions.

HOPFIELD TYPE NETWORK

WITH ANTICIPATION

SOME EXAMPLES OF

MODELS

( 1) ( ) ( 1)j ji i ji ix n f w x n w x n

N

i

iji

N

i

ijij nxwnxwfnx11

)1()()1()1(

EXAMPLE OF ACTIVATION

FUNCTION

0, 0

( ) , [0,1)

1, 1

якщо x

f x x якщо x

якщо x

Network with 2 coupled neurons

Single-valued periodicity

Neuronert with 2 neuroons

Multi-valued ciclicity

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

-0,2 0 0,2 0,4 0,6 0,8 1 1,2

1 нейрон

2 н

ей

ро

н

Netework with 6 neurons. Ciclicity

Network with 8 neurons

The influence on anticipation

parameter

PROBLEMS AND PROSPECTS

RESEARCH DIRECTIONS

I. General investigations of abstract mathematical objects:

Definitions of regimes:

Periodicity;

Chaos;

Solitons;

Chimera states;

Bifurcations;

Attractors;

Etc.

RESEARCH DIRECTIONS

II. Investigation of concrete models and solutions

In artificial neural networks

In cellular automata

In coupled maps

Solitons, traveling waves

Self-organization

Collapses

Etc.

RESEARCH DIRECTIONS

III. Interpretations and applications

Traffic modeling

Crowds movement

Socio- economical systems

Control applications

Neuroscience

Conscious problem

Physics

IT

REFERENCES

1. Dubois D. Generation of fractals from incursive automata, digital

diffusion and wave equation systems. BioSystems, 43 (1997) 97-114.

2 Makarenko A., Goldengorin B. , Krushinski D. Game „Life‟ with

Anticipation Property. Proceed. ACRI 2008, Lecture Notes Computer

Science, N. 5191, Springer, Berlin-Heidelberg, 2008. p. 77-82

3. B. Goldengorin, D.Krushinski, A. Makarenko Synchronization of Movement for Large – Scale Crowd. In: Recent Advances in Nonlinear

Dynamics and Synchronization: Theory and applications. Eds. Kyamakya

K., Halang W.A., Unger H., Chedjou J.C., Rulkov N.F.. Li Z., Springer, Berlin/Heidelberg, 2009 277 – 303

4. Makarenko A., Stashenko A. (2006) Some two- steps discrete-time

anticipatory models with „boiling‟ multivaluedness. AIP Conference

Proceedings, vol.839, ed. Daniel M. Dubois, USA, pp.265-272.