Neural Networks with Anticipation: Problems and Prospects

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AACIMP 2010 Summer School lecture by Alexander Makarenko. "Applied Mathematics" stream. "General Tasks and Problems of Modelling of Social Systems. Problems and Models in Sustainable Development" course. Part 6.
More info at http://summerschool.ssa.org.ua

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Neural Networks with Anticipation: Problems and Prospects

  1. 1. NEURAL NETWORKS WITH ANTICIPATION: PROBLEMS AND PROSPECTS Alexander MAKARENKO Institute for Applied System Analysis at National Technical University of Ukraine (KPI)
  2. 2. INTRODUCTION Nonlinear networks science (problems and effects): stability bifurcations chaos sinchronisation turbulence chimera states
  3. 3. MODELS AND SYSTEMS (RECENTLY): coupled oscillators coupled maps neural networks cellular automata o.d.e. system ……… MAINLY of NEUTRAL or WITH DELAY
  4. 4. 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 - …)
  5. 5. 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].
  6. 6. 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): si (t 1) Gi ({si (t )},...,{si (t g (i))}, R),   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.
  7. 7. “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.
  8. 8. HOPFIELD TYPE NETWORK WITH ANTICIPATION
  9. 9. SOME EXAMPLES OF MODELS x j (n 1) f w ji xi (n) w ji xi (n 1) N N x j (n 1) f (1 ) w ji xi (n) w ji xi (n 1) i1 i1
  10. 10. EXAMPLE OF ACTIVATION FUNCTION 0, якщо x 0 f ( x) x, якщо x [0,1) 1, якщо x 1
  11. 11. Network with 2 coupled neurons Single-valued periodicity
  12. 12. Neuronert with 2 neuroons 1,2 Multi-valued ciclicity 1 0,8 2 нейрон 0,6 0,4 0,2 0 -0,2 0 0,2 0,4 0,6 0,8 1 1,2 -0,2 1 нейрон
  13. 13. Netework with 6 neurons. Ciclicity
  14. 14. Network with 8 neurons
  15. 15. The influence on anticipation parameter
  16. 16. PROBLEMS AND PROSPECTS
  17. 17. RESEARCH DIRECTIONS  I. General investigations of abstract mathematical objects:  Definitions of regimes:  Periodicity;  Chaos;  Solitons;  Chimera states;  Bifurcations;  Attractors;  Etc.
  18. 18. 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.
  19. 19. RESEARCH DIRECTIONS  III. Interpretations and applications  Traffic modeling  Crowds movement  Socio- economical systems  Control applications  Neuroscience  Conscious problem  Physics  IT
  20. 20. 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.

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