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Факултет инжењерских наука Машинске конструкције и механизација Трибологија машинских система Метода оптимизације испитивања триболошких система уз помоћ неуронских мрежа [Семинарски рад] Студент Професор Новаковић Милан 342/2013 др Блажа Стојановић Крагујевац, јун 2014.

Metode Optimizacije Ispitivanja Triboloskih Sistema Uz Pomoc Neuronskih Mreza

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Fakultet inzenjerskih nauka, Tribologija, Masinske konstrukcije i mehanizacija

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  • [ ]

    342/2013

    , 2014.

  • 342/2013

    2

    1. ............................................................................................................................................. 3

    2. .................................................................................... 4

    3. ....................................................................................... 5

    4. ............................................................................................................... 10

    4.1. T ................................................................................................ 12

    5. .......................................... 14

    5.1. ............................................................................................. 17

    6. ................................................................................. 19

    6.1. .......................................................................................... 19

    6.2. ............................................................................................................................ 20

    6.3. (Adaptive Linear Element) .................................................................................... 20

    6.4. J ......................................................................................................... 22

    7. ........................................................................................... 24

    7.1. Hornik stinchombe white-ova teorema (1989) .......................................................................... 24

    7.2. .................................................................................. 25

    7.3. .......................................................................................................... 29

    7.4. ........................ 29

    7.4.1. ................................................................................................. 29

    7.4.2. . (learning constant) .................................................................. 29

    7.4.3. ................................................................................................................... 30

    7.4.4. .......................................................................................................................... 30

    7.4.5. . ............................................................................................................ 31

    7.4.6. . ................................................................................. 32

    7.4.7. ..................................................................................................... 34

    8. ................................. 35

    ..................................................................................................................................... 37

    ............................................................................................................................................. 38

  • 342/2013

    3

    1.

    .

    .

    :

    1. 2. ( )

    .

    .

    ()

    .

    o ,

    .

    .

    . ,

    ,

    (, , , .).

    , ( ,

    ) .

    .

    .

    , (eng. Artificial

    neural network ANN)

    . .

    ():

    ,

    .[0]

  • 342/2013

    4

    2.

    1. , . 2. - , 3. - . 4. .

    . 5. .

    .

    6. . 7. (Very Large Scale Integration) . 8. .

    . -. .

    9. . , , - . [1]

    2.1.

  • 342/2013

    5

    3.

    , ,

    . -

    .

    .

    [2]:

    .

    , ( ) ,

    .

    [3]:

    Bayesian regularisation (BR) Levenberg-Marquardt

    .

    Powell-Beale conjugate gradient algoritm (CGB)

    Polak-Ribiere conjugate gradient algorithm.

    BFGS quasi-Newton method (BFG)

    1,

    ,

    .

    Adaptive learning rate (GDX)

    Levenberg-Marquardt algorithm (LM)

    .

    .

    3.1.

    .

    1

    9 25 1

    1 http://en.wikipedia.org/wiki/Hessian_matrix

  • 342/2013

    6

    3.1.

    3.2.

  • 342/2013

    7

    3.3. [0]

    - [1,3]:

    x 1 2 ...in h outhN N N N N

    inN -

    outN -

    1 2, ,... hN N N -

    [3]:

    1

    13 25 2 - 25 , 13

    2

    3

    9 15 10 5 1 - 3 15, 10, 5 ,

    9 1

    , [2]:

    ( )

    .

    ,

    .

  • 342/2013

    8

    (feed forward single

    layer neural network)

    3.4.

    (feedforward

    multilayer neural network)

    3.5.

  • 342/2013

    9

    ,

    .

    3.6..

    . 1z .

    3.7.

    1z 1z1z

  • 342/2013

    10

    4.

    :

    ijw . , .

    ( ) .

    . [0,1] [-1,1].

    4.1.

    m

    j

    jkjk xwu1

    4.1.

    )( kkk uay

    :

    .,1,)(, 00 kkkkjkjk wxvayxwv

  • 342/2013

    11

    4.2.

    .

    kk bwax 00 ,1 , kb , 4.3.

    4.3..

    .

  • 342/2013

    12

    4.1. T

    ,

    .

    :

    1. (), (supervized learning) 2. (reinforcement learning) 3. ( ), (unsupervised learning)

    4.1.1.

    4.1.2..

  • 342/2013

    13

    4.1.3.

    )()( , ii dX , )(iX , )(id .

    ,

    ,

    {, }. ,

    ,

    ,

    .

    .

  • 342/2013

    14

    5.

    5.1. [4]

    [4]

    .

    ,

    ,

    .

    ,

    0 1.

    () . [5]

    :

    .

    .

    .

  • 342/2013

    15

    iX ,

    . jY

    . , .

    1

    j act ij i

    i

    Y f W X

    ijW - iX jY

    jY .

    ( kO )

    .

    1

    k act jk j

    i

    O f W Y

    jkW - jY

    (eng. Logistic Sigmoid function) :

    1

    1act x

    f xe

    .

    , ,

    [3,4,5].

    .

    . ,

    .

    ( ):

    1

    1 n i i

    i i

    d O

    n d

    E - ,

    id -

    iO -

    n -

  • 342/2013

    16

    [2], :

    k k ke n d n y n

    21

    1

    2

    N

    av k

    n k C

    e nN

    ke -

    av -

    N -

    C -

    [6] :

    2

    1 1

    1 Q N

    n n

    m n

    E d m y mN Q

    E -

    Q -

    N -

    d - ( )

    m -

    :

    , ,

    .

    , ,

    . ,

    ,

    .

    :

    jiji

    EW

    W

    jiW -

  • 342/2013

    17

    5.1.

    . [0,1] [-1,1]. :

    0,0

    0,1)(

    v

    vva

    5.1.1. .

    (1943)

    2/1,0

    2/12/1,2/1

    2/1,1

    )(

    v

    vv

    v

    va

    5.1.2.

    ()

    1

    1 bva v

    e

    5.1.3. () .

    b .

    a(v)

    1

    0 v

    a(v)

    1

    -1/2 1/2

    v

    a(v)

    v

    1 1b

    2b

    21 bb

  • 342/2013

    18

    [-1,1],

    0,1

    0,0

    0,1

    )(

    v

    v

    v

    va

    5.1.4. (sgn(v))

    1

    2 1

    v

    v

    v ea v tg

    e

    5.1.5.

    ( )

    a(v)

    1

    0 v

    -1

    a(v)

    v

    1 1b

    2b

    21 bb

    -1

  • 342/2013

    19

    6.

    6.1.

    6.1.2. i-

    niwwww Timiii ,...,2,1,),...,,( 21 - i-

    )()( txrtwi ,

    - .

    ,

    ),,( iir dxwfr ,

    ,)())(),(),(()()1( txtdtxtwftwtw iirii -

    ,

    .

  • 342/2013

    20

    6.2.

    .

    :

    B,

    , ,

    .

    ., xywyr iii

    ,

    . ,

    mjnixyw jiij ,...,2,1,,...,2,1,

    - ji xy , ijw (

    ), . , , , .

    6.3. (Adaptive Linear Element)

    .

    ).(),...,,( )()()1()1( pp dxdx .

    iw ,

    pkdxwm

    j

    kk

    jj ,...,2,1,1

    )()(

    .)(2

    1)(

    2

    1)(

    2

    1)(

    1

    2

    1

    )()(

    1

    2)()(

    1

    2)()(

    p

    k

    m

    j

    k

    jj

    kp

    k

    kTkp

    k

    kk xwdxWdydwE

    .

    )(wEw w ,

    mjxxWdw

    Ew kj

    p

    k

    kTk

    j

    j ,...,2,1,)()(

    1

    )()(

  • 342/2013

    21

    )(kx ,

    ,)( )()()( kjkTk

    j xxWdw

    Vidrov-Hofovo . LMS

    ( , Least Mean Square).

    Vidrov-Hofovo

    ,

    xWdydr T .

    (w) w,

    (),

    , .

    6.3.1. -

    w.

    minE

    0w )(nw)1( nw

    w

    eEw

    )(

    w

    wE

    )(

    )(wE

    w

  • 342/2013

    22

    6.4. J

    ().

    .

    6.4.1.

    nnmmm www ...,,, 2211

    m

    n

    p

    ..,2,1,,,2,1,)()(

    1

    )()( pknidxwaxWay kik

    j

    m

    j

    ij

    kT

    i

    k

    i

    : TimiiT

    i wwwW ,,, 21 , .

    ,

    ,

    p

    k

    p

    k

    n

    i

    k

    j

    m

    j

    ij

    k

    i

    n

    i

    kT

    i

    k

    i

    p

    k

    n

    i

    k

    i

    k

    i xwadxwadydwE1 1 1

    2

    )(

    1

    )(

    1

    2)()(

    1 1

    2)( .2

    1)(

    2

    1

    2

    1)(

    ,1

    )()(')()(

    p

    k

    k

    j

    k

    i

    k

    i

    k

    i

    ij

    xnetanetadw

    E

    )()( kT

    i

    k

    i xWnet - i- k- .

  • 342/2013

    23

    .)(

    )(

    )('

    k

    i

    k

    ik

    inet

    netaneta

    ijw -

    ,)( )()(')()( kjkikikiij

    ij xnetanetadw

    Ew

    (delta learning rule),

    )()( ' xwaxwadr TiTii .

    .

    ,

    , ,

    a.

  • 342/2013

    24

    7.

    (feed forward artificial neural networks FFANN),

    ,

    ,

    (backpropagation algorithm).

    7.1. Hornik stinchombe white-ova teorema (1989)

    HSW

    1. 1)(lim

    a

    2. )1(0)(lim

    a

    )(a

    , , .

    (

    ).

    FFANN . FFANN

    , ,

    , ,

    FFANN.

    , HSW

    , .

    ,

    .

  • 342/2013

    25

    7.2.

    :

    1. )(kx , )(ky .

    2. ,

    ijw .

    (BP )

    m l n .

    1x mxjx

    1y iy ny

    qqw1

    iqw

    nqw

    1qv qjv qmv

    7.2.1.

    7.2.1. (x,d)

    . :

    qnet - q ,

    ,1

    j

    m

    j

    qjq xvnet

    qz - q

  • 342/2013

    26

    m

    j

    jqjqq xvanetaz1

    i-

    l

    q

    l

    q

    m

    j

    jqjiqqiqi xvawzwnet1 1 1

    .

    .1 11

    l

    q

    m

    j

    jqjiqq

    l

    q

    iqii xvawazwanetay

    , .

    n

    i

    n

    k

    n

    i

    l

    q

    qiqiiiii zwadnetadydwE1 1 1

    2

    1

    22.

    2

    1

    2

    1

    2

    1)(

    ,

    iq

    iqw

    Ew

    ,

    iqwE / ,

    ,0 qiqiiiiq

    i

    i

    i

    i

    iq zznetaydw

    net

    net

    y

    y

    Ew

    i0

    ,0 iiii

    i

    ii

    i netaydnet

    y

    y

    E

    net

    E

    inet ,

    i

    i

    inet

    netaneta

    .

    qz .

  • 342/2013

    27

    q

    n

    i

    jqiqiii

    qj

    q

    q

    q

    qqj

    q

    qqj

    qj xnetawnetaydv

    net

    net

    z

    z

    E

    v

    net

    net

    E

    v

    Ev

    1

    .

    i0 ,

    ,1

    0

    n

    i

    jhqjqiqiqj xxnetawv

    hq q

    ,1

    0 iq

    n

    i

    iq

    q

    q

    qq

    hq wnetanet

    z

    z

    E

    net

    E

    qnet q.

    hq q

    i0

    . , ,

    () .

    , .

    , ,

    jinputioutputjiij xxw ,

    output-i input-j i.

    , .

    Q , q=1,2,...,Q

    i

    q net - i- q-

    i

    q y - q- .

    . ijq w

    j

    q y1 iq y .

  • 342/2013

    28

    : pkdx kk ,...,2,1,, )()(

    0: () 0 maxE (

    ). . =0,

    k=1.

    1: k- (q=1):

    )(1 k

    iii

    q xyy , .

    2: ( ).

    qiywanetayj

    j

    q

    ij

    q

    i

    q

    i

    q ,1

    iQ y .

    3: ( iQ )

    EydEn

    i

    i

    Qk

    i

    2

    1

    )(

    2

    1 ,

    iQiQkiiQ netaydy )( .

    4: ( ).

    iq 1 :

    ,,1 ijqolld

    ij

    qnew

    ij

    q

    j

    q

    i

    q

    ij

    q wwwyw

    j

    j

    q

    ji

    q

    i

    q

    i

    q QQqzawneta .2,...,1,,11

    5:

    . je k

  • 342/2013

    29

    . , .

    .

    . (batch mod training) ,

    .

    7.3.

    (error surface )

    .

    . :

    , .

    . , . .

    7.4.

    7.4.1.

    .

    .

    .

    ii kk

    3,

    3,

    ik i.

    7.4.2. . (learning constant)

    ,

    , .

    ,

    0.001 10. , .

    , 0,

    , , 0, 0

    0,

    a E

    a bb

    , .

    E .

  • 342/2013

    30

    7.4.3.

    .

    i0 , .

    pL

    ,1,1

    pydp

    Ei

    p

    ii

    iii

    ydL sup .

    7.4.4.

    ,

    . .

    .

    , .

    1,0,,)1()()( twtEtw ,

    ( 0.9).

    7.4.4.1.

    .

    (error surface) . AA

    .

    . BB

    ().

    .

    .

  • 342/2013

    31

    7.4.4.1.

    7.4.5. .

    .

    .

    . wE

    ))(()(2

    1)()()()( 00000

    TTT wwwHwwwEwwwEwE

    .,)()(2

    2

    ji

    ijww

    EHwEwH

    (w), 0)( wE ,

    .0))(()()( 000 wwwHwEwE

    ,

    ,)()( 01

    0 wEwHww

    ,)()( )()(1)()1( kkkk wEwHww

  • 342/2013

    32

    ,

    , .

    :

    .

    .

    .

    7.4.6. .

    .

    ,

    .

    ,

    . overfitting.

    ,

    , (

    ).

    ,

    .

    . .

    ,

    .

    ,

    . ,

    , , , 1,... , 1,...i i ix d d i m j p

    ,2

    122

    2

    2

    1

    n

    fff

    bx

    E

    x

    E

    x

    EE

    fE .

    bE , ,

    fE , .

  • 342/2013

    33

    ..

    .

    EE~

    ,

    , . ,

    E~

    ,

    overfittinga.

    i

    iw2

    2

    1,

    weight decay ,

    .

    .

    . ,

    ,

    , ,

    ,

    overfittinga .

    . (early stopping).

    ( )

    .

    ( )

    , .

    overfittinga, ,

    .

    .

  • 342/2013

    34

    7.4.6.1.

    7.4.7.

    .

    ,

    ,

    . :

    .

    , , ,

    . .

    m-

    M ,

    . mN ,

    .,0,10

    m

    j

    m

    mm

    m jNzaj

    Njegde

    j

    NMN

    mN

    m- , maxM

    .,

    !

    11

    !2

    11

    0

    max mNzam

    jNNNNNN

    j

    NM m

    mmmmmm

    m

    j

    m

    mNm , ,2maxmNM

    max2log MNm .

    () .

  • 342/2013

    35

    8.

    , ,

    .

    ,

    , ,

    .

    ,

    ,

    .

    Pruning.

    8.1. Pruning[0]

    , [0],

    ( ,

    ).

    ,

    , Pruning.

    B [3,7].

    2

    1

    2

    1

    1

    i

    Mi

    pi

    Mi

    i

    O O

    B

    O O

    (1.1)

    ipO - i-

    i

    O - i-

    M -

  • 342/2013

    36

    2R . , ,

    , .

    B 0 1B

    , 0,9 B .

    ,

    .

    ( ) (

    ), 2

    .

    :

    .1. .

    . .

    obukaE validE .

    .2.

    ,

    ,

    .

    .3. lil bajas .

    .4.

    . validE . validvalid EE ,

    validvalid EE 2., .

    ,

    , .

    .

    . (brain demage),

    . ,

    ,

    .

    2 pruning -

  • 342/2013

    37

    [0] L. Gyurova, P. Minino-Justel, A. Schlarb. Modeling the sliding wear and friction

    properties of polyphenylene sulfide composites using artificial neural networks.

    [1] . , ,

    ,

    , 2005.

    [2] R. Koker, N. Altinkok, A. Demir. Neural network based prediction of mechanical

    properties of particulate reinforced metal matrix composites using various training

    algorithms.

    [3] Z. Zhang, K.Friedrich, K. Velten. Prediction on tribological properties of short fibre

    composites using artificial neural networks.

    [4] K. Genel, S. C. Kurnaz, M. Durman. Modeling of tribological properties of alumina

    fiber reinforced zinc-aluminum composites using artificial neural network.

    [5] M. T. Hayajneh, A.M. Hassan, A.T. Mayyas. Artificial neural network modeling of

    the drilling process of self-lubricated aluminum/alumina/graphite hybrid composites

    synthesized by powder metallurgy technique.

    [6] A. A. Tofigh, M. R. Rahimipour, M. O. Shabani, M. Alizadeh, F. Heydari, A.

    Mazahery, M. Razavi. Optimized processing power and trainability of neural network

    in numerical modeling of Al Matrix nano composites.

    [7] Z. Zhang, K. Friedrich. Artificial neural networks applied to polymer composites: a

    review.

    [8] C.M. Bishop, Neural Networks for Pattern Recognition, Oxford university press,

    2000. [9] C.T. Lin, C.S.George Lee, Neural Fuzzy Systems, Prentice Hall, 1996.

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    38

    [4]

    [0]

    C. S. Ramesh, R. Suesh Kumar. Mathematical and neural network models for prediction of

    wear of mild steel coated with Inconel 718- A comparative study