ENGINEERING COMPUTATION BY ARTIFICIAL NEURAL NETWORKS. Explaining Neural Networks

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Merry Walker
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1 SRK oaz Poltcha Pozaa Ittut Mcha Stooa ul. Potroo 3, Poza EGIEERIG COMPUAIO BY ARIFICIA EURA EWORKS Eplag ural tor ural tor ar copod o pl lt opratg paralll. h lt ar prd b bologcal rvou t. A atur, th tor ucto dtrd largl b th cocto bt lt. W ca tra a ural tor to pror a partcular ucto b adutg th valu o th cocto ght bt lt. Cool ural tor ar adutd, or trad, o that a partcular put lad to a pcc targt output. hr, th tor adutd, bad o a coparo o th output ad th targt, utl th tor output atch th targt. pcall a uch put/targt par ar ud, th uprvd larg, to tra a tor. Batch trag o a tor procd b ag ght ad ba chag bad o a tr t batch o put vctor. Icrtal trag chag th ght ad ba o a tor a dd atr prtato o ach dvdual put vctor. Icrtal trag ot rrrd to a o l or adaptv trag. ural tor hav b trad to pror copl ucto varou ld o applcato cludg pattr rcogto, dtcato, clacato, pch, vo ad cotrol t. oda ural tor ca b trad to olv probl that ar dcult or covtoal coputr or hua bg grg, acal ad othr practcal applcato. Suprvg trag o ural tor

Cool ural tor ar adutd, or trad, o that a partcular put lad to a pcc targt output. hr, th tor adutd, bad o a coparo o th output ad th targt, utl th tor output atch th targt.

2 uro th vctor put ar o uro Multpl lar o uro

3 Spl uro Modl uro AD th uro th lar ha put hr or,...,. [,..., ], tor Archtctur Multpl ar o uro Codrd ultpl-lar ural tor ha: lar th uro,,...,, ach lar dotd a put u [ u,..., u ],,... ; output. AD,,..., ; Bacpropagato Algorth h plt pltato o bacpropagato larg updat th tor ght ad ba th drcto hch th prorac ucto dcra ot rapdl - th gatv o th gradt. hr ar to drt a hch th gradt dct algorth ca b pltd: crtal od ad batch od. I th crtal od, th gradt coputd ad th ght ar updatd atr ach put appld to th tor. I th batch od all o th put ar appld to th tor bor th ght ar updatd.

.., ; Bacpropagato Algorth h plt pltato o bacpropagato larg updat th tor ght ad ba th drcto hch th prorac ucto dcra ot rapdl - th gatv o th

4 t otc that put o uro ollog a Wght vctor o uro AD ar coctd th output o u, AD : hr ot calld ba.,...,..., ] [,...,,,,...., th lar Output o AD ca d a: hr: rar ucto actvato ucto ca b tpcall, or apl:, u > a tp ucto hard-lt ucto: t ;, u a lar ucto: t t ; a god upolar ucto: t u t βt ; t t tg h t t a god bpolar ucto: b β u. t otc that a upolar god ucto drtat-abl ad cotou: du t β u t u t. dt t otc that output ro lat lar uro:,,..., ar output ro tor. W ca copar th th pattr output: d, d,..., d ad gt rror d,,..., W ca orulat global rror a u o rror quar. W ca adut th valu o th cocto ght bt lt ug gradt dct algorth

t u t βt ; t t tg h t t a god bpolar ucto: b β u. t otc that a upolar god ucto drtat-abl ad cotou: du t β u t u t.

5 η hr > η dot valu o corrcto tp. W hav. t o Ad η h thod o drv blog to th lar. For output lat lar hav: d For othr lar hav W ca d rror th lar or th uro,,..., Ad gt:. O o ot o odcato o bacpropagato algorth cot addg tra tr otu tr a α η hr, α.

6 { } X ;.. a { }; { a } a vbrg-marquardt Algorth { l }; l.. { }, ;, a, a Z a a ε a ; Z p ;.., a..,.. p,,,3,... a a a, a,, a, b a a { Z } p b Z, { b } c c Z ε b, { } b b c Z Z Z ; Z Z λ I Z d d S ; X { }; { } d X d X > Bcau S d X d X d ; d S d d X X X X X X S d ; X X ; d X X X

7 Rrc Stała Oo: Sc uroo, Oca Wdacza Poltch Warza, Warzaa,. Stała Oo: Sc uroo ucu algortcz, W, Warzaa, 996. D. Rutoa, M. Pl,. Ruto: Sc uroo, algort gtcz t rozt, PW, Warzaa, 997. Duch W., Korbcz J., Ruto., aducz R.: Bocbrta ra Bodcza, Sc uroo, to 6, Aadca Oca Wdacza Et, Warzaa. Hto G.E. : Coputato b ural tor, atur Arca, vol. 3, p.97. Ptrucha U., Brau R.W. 999: Ug Grog RBF-t Rubbr Idutr Proc Cotrol, ural Coputg & Applcato, 8, Baratt R., Caa B., Fa A., Plo F. : Autoatd Rcurrt ural tor Dg to Modl th Dac o Copl St, ural Coputg & Applcato, 9, 9-. Rao M. Srva J. : oroal Vbrato o Pr-td Blad ug Artcal ural cholog, Egrg th Coputr, 6: -5. abot G. : ural tor rcotructo o lud lo ro tracr-partcl dplact, Eprt Flud, 3:

Baratt R., Caa B., Fa A., Plo F. : Autoatd Rcurrt ural tor Dg to Modl th Dac o Copl St, ural Coputg & Applcato, 9, 9-. Rao M. Srva J.

Finite Dimensional Vector Spaces.

Finite Dimensional Vector Spaces. Lctur 5. Ft Dmsoal Vctor Spacs. To b rad to th musc of th group Spac by D.Maruay DEFINITION OF A LINEAR SPACE Dfto: a vctor spac s a st R togthr wth a oprato calld vctor addto ad aothr oprato calld scalar