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its.

Artificial neural networks and support vector machines

(SVMs) are capable of building general models, by

“learning” from the data in the so-called “training proc-

ess”. In the paper [17] the complete evolution of the con-

sidered discharges is followed from the beginning to the

end to determine whether a disruption if fortcoming or

not. We will try to describe the using of artificial neural

networks in investigation of behavior of nonlinear Vlasov-

Poisson-Fokker-Planck systems that describe the turbu-

lence.

In the past decades, the fuzzy logic theory has been

shown to be an effective approach to dealing with the

analysis and synthesis problems of nonlinear systems.

Among various models available to fuzzy systems, the

Takagi-Sugeno (T-S) fuzzy model was a notible one

which was a linear system constructed to approximate a

nonlinear plant. Roughly speaking, the T-S fuzzy model

is a system described by fuzzy IF-THEN rules which can

give local linear representation of the nonlinear system

by decomposing the whole input space into several par-

tial fuzzy spaces and representing each output space with

a linear equation.

We consider a generalization of the traditional Ta-

kagi-Sugeno fuzzy system that includes both the pa-

rameter uncertainities and the stochastic disturbances.

The corresponding ith fuzzy rule is formulated in the

following form:

Plant rule i:

If is

1t

1i

and...

pt

ip

is

then

d

,,

ii i

ft

d ,

i

A

A fttftft

t t

1, 2,,i r

I

(6)

q

itR

are uncorrelated zero mean

Gaussian white noise processes with covariances .

q

There been performed studies on the correlation be-

tween electrostatic and magnetic fluctuations. The con-

nection between turbulent density fluctuations and low-

dimensional chaotic dynamics has been quantitatively

verified by using suitable numerical algorithms like the

correlation dimension and the statistical distribution of

recurrence times.

One of the most difficult challenges in computational

statistical physics is sampling of low-temperature equi-

librium states of the systems. Standard Markov Chains

Monte Carlo (MCMC) methods become stuck in local

minima and are unable to correctly sample the equilib-

rium distribution. In parallel tempering, many replicas of

the system are simultaneously simulated using a standard

MCMC method with each replica at a different tempera-

ture. Replica exchanges are allowed that permit replicas

to move between low and high temperatures. We must

use the adaptive Monte Carlo.

Competitive learning neural networks are regarded as

a powerful tool for online data clustering to represent a

non-stationary probability distribution with a fixed nu-

mer of weight vectors. One difficulty in applications of

competitive learning neural networks to online data clus-

tering is that most of them require heuristically-prede-

termined threshold parameters of balancing a trade-off

between accuracy, i.e. error minimization performance,

and speed of adaptation to the changes in source statistics.

Both quick adaptation and error minimization are simul-

teneously accomplished without any carefully predefined

parameters.

When modeling transport processes in tokamak plas-

mas the following assumption prevails: widely separated

regions of the plasma do not significantly interact with

each other. This paradigm states that any particle or heat

flux can accurately be described using a set of local

transport coefficients.

Transport events are random and accurately described

by a classical Gauss-Markov process. Nor is taken into

account the fact that a large localized event may have a

significant aftereffect in the same region it occured in.

Any transport model should endeavor to encompass the

possibility for a nonlocal, nondiffusive action at a dis-

tance. The heat flux Q(r, t) has been temporally averaged

over a collision time; we thus only addressed the spatial

question of the nonlocal, nondiffusive (non-Gauss-Markov)

behavior of Q. An equally question is now to address its

Markov (temporal) counterpart. A strong nonlocal, non-

diffusive spatial behavior is found, stremming from the

selforganized transport.

At least, the available data does not allow to conclude

on the possibility of non-Markovian (memory) effects.

With the introducing of artificial neural networks for the

non-Markovian process is transfered into Markovian

process [18].

Copyright © 2012 SciRes. JMP

D. RASTOVIC 1865

6. Drift Kinetic Equations and Advanced

Optimal Control

The analysis of electrostatic fluctuations is of great in-

terest in understanding the degradation in confinement of

a magnetically confined plasma. Diagnostic methods are

mostly based on digital correlation of data applying dif-

ferent models for their analysis. Some diagnostics are

based on a statistical processing of signals.

We must identify systematically and properly the kind

of instability responsible for the turbulence and the

anomalous transport. The anomalous particle transport,

as well as the correlation between the existence of an

instability and its contribution to the transport, is one of

the most important issues in understanding the confine-

ment of a magnetized plasma.

The most notable one is that toroidal rotation reverses

its direction after a low-mode ( L-mode) to H-mode tran-

sition. To explain this phenomenon, one has to deal with

the toroidal momentum equation directly. To this end, we

adopt the neoclassical quasilinear theory to calculate tor-

oidal stress induced by electrostatic fluctuations and ex-

amine its qualitative behavior. Of course, this neoclassi-

cal theory must take into account the fact that the pres-

ence of magnetic island breaks the toroidal symmetry of

a tokamak.

The linear drift kinetic equation with plasma flows as

the driving terms is

2

2

32

1

2123

25

E

t

2

M

f

vVnV fC

t

vvvv

VBBLq

f

B Bpf

v

(7)

where f is the perturbed particle distribution function,

is particle speed paralel to B, v is the particle speed,

E

V

EB

t

is the drift, E is the electric field, C is the test

particle part of the Coulomb collision operator, V is the

plasma flow velocity, q is the heat flow, is the ther-

mal speed of the particle,

v

32 2

152Lx is a Laquerre

polinomial, t

x

vv and

M

f

is the equilibrium shifted

Maxwellian distribution function [19].

In Ohmically heated tokamak plasmas or in RF-heated

tokamak plasmas where wave momentum is zero, there is

no obvious toroidal momentum source. The steady-state

toroidal rotation velocity profile is determined approxi-

mately by balancing the diffusion term and the convec-

tive term. There are many partial results of the applica-

tion of drift kinetic equation for different regimes of to-

kamak plasma confinements.

Many physical dynamical systems are spatially ex-

tended and exhibit cooperative wave patterns as well as

high-dimensional chaotic attractors. Example for this

kind of complex dynamics is interacting spiral waves in

excitable media.

Only some model parameters have to be determined

from experimental data. Often, however, even the struc-

ture of the models is not or only partially known and has

to be identified using observed data. Structure identifica-

tion and parameter estimation, several methods have

been suggested based on non-linear regression, state

space reconstruction, perturbation or synchronization.

More difficult to identify and to model are structural

inhomogeneties which occur due to long-range connec-

tions between different(remote) locations in the spatially

extended system introducing additional coupling. Correct

identification of these long-range links is of crucial im-

portance for correct modeling. A network is balanced if

the in-degree of any nodes is equal to its out-degree.

We shall suggest a system identification method based

on steady-state stabilization. To stabilize a steady state,

we add to control signal i such that we have con-

trolled network. Then we exploit the dependence of the

equilibrium on control parameters. In this way we obtain

all relevant information to identify the underlying dy-

namics as well as coupling structure [20].

u

We shall consider in the following time-spatially ex-

tended systems that can approximately be described by

large sets of partial differential equations. We first stabi-

lize a stationary state (at least in some region) and then

exploit the dependence of the equilibrium on control pa-

rameters. In this way we obtain all relevant information

to identify the underlying dynamics as well as coupling

structure.

Statistical behavior described by power-laws is ap-

pealing to attribute to the existence of “criticality” in the

underlying dynamics. In the statistical mechanics sense,

this means the proximity of a phase transition, and one of

the perminent ideas in this respect is “self-organized

criticality” (SOC). This is usually taken to mean that no

fine-tunning to a critical point in the classical sense is

needed and the system at hand is driven to such a critical

state.

The measured signal can be considered after e.g. spa-

tial or temporal thresholding. Example is given by the

local particle flux in the perimeter of a fusion device. In

particular for classical SOC systems, with uncorrelated

driving, global waiting time statistics are described most

often by a Poisson process.

Understanding the field lines chaotic structures in sta-

tionary flows can be considered as the first step towards

understanding transport in these flows. This chaos of

field lines has been investigated for a long time in plasma

physics, especially when concerned with the conception

of magnetically confining devices such as tokamaks.

Potential interest in computing field lines for a flux

free field is the possibility to compute Poincare sections

of the field lines. We should also be able to discriminate

Copyright © 2012 SciRes. JMP

D. RASTOVIC

1866

z

is the charge number of the particle species the physical importance of islands of regular motion

within a stochastic sea in the Poincare section. Let v be a

three-dimensional vector field. Field lines of v are curves

which are tangent to the field at any point.

,ie

v

, E is the poloidal electric field, b is direction

of magnetic field, ~~
~~Simulation and Control of Turbulence at Tokamaks with Artificial Intelligence Methods
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