hopfield network solved example

83!0OT$jq,lW,L\d,'-HM@WTT+:5(Z7S5Mj8(flX^N[6^r"'#W]KV@o-b8) endstream endobj 56 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 60 0 obj << /Length 4406 /Filter [/ASCII85Decode /FlateDecode] >> stream 0:E"+A8%gRR'4h=1/;;nOqSHeb#J/GX:/4CC\kn]*IXc+!9-b!,iWLFf2C>20NptR ;,pm8JSCB4eY2u@FaX;Q4LHc)OQ:e6(;%lAUf2)W88k\ne%R\]R^Un)?fF_f@@XO5knZmtXog;[f%X"bB136Y4!BNQKG[n8]RX_plT ]S5JeG,]`1OPnqIen3?D]Pb?l8(. Z^bSNIib6X"s3,f\iIrSJ_VS;`37.1*$3HQ7!I%OpV4b2CllI$KR?q,\;c_XAfC;k d:dND0hA*,S)c,ZeEh;CQht5S$9%#\K1)`HagS&&L_fa fM#g`Bf#j)+i-u1ZK#adWW4I\c6\]@^ad7%!gIbNOY246EV]s$$SC4%Q6A3o[SE#qS9+"a*o'4_,X/Dac/YL+mRuNO9A8jWip\F&eY,QY[*5]oALDf1R+qVT#)82P\]N?`c"+\K\,idr%U0i-lVPn0*_6@?Ga*uS"m[/\>TN%P1:[C)CO-'.Se?2! k!8=1StF1R*eY#lp#f0iNGl_PJ12?mRHpcMIR8Ma5pmJ++UNc6.=\1**`.&nE< ,pBcM'g2,qKd>&E.VW>o$P39 MnG.EkHlqd@Gn[[k$qKPmakJ#>8. >h>rPb7H$?UirXOf)hb`0#s09?ZL5#7?`F&3H-8XTXmS[r?AhZbA(#9YC`&-]8_p; oOXtY*kSCIlhf.feYh>XA#D"nDsQ0TVIH26_bQF/\>qr%3%IMj':c\:(>BRhSK7e+ Neural Network Examples and Demonstrations ... the Hopfield model, for which the weights may easily be determined, and which also settles down to a stable state. -_@=^3@0o:.A^UFZaI)W/jQ_Ak%b@jh+Co=+K-G@B4VdjqI8am,]N!qYd>daesloG cG92/c+E]VFLTScg`"? @;E'GTnDaDS3.^@omY,g+OP>;/TP"qnT/%62oK]Xf>Q]i8H0)6N>E5Y+g4mVXKcXGI[%n6o#.F7^j ?d"EoU5alJnqSOUUGkif9+dY-qS^12W^=!^dnhT-D-;SQX/U0eJ"hI,'nuAmh&'Wc >=ksGjh31C;btF85:@"`+,[te#fn>QHiNIm]NmktiXEJuMQV"Q:aB^_3h96Ke1RPR EhpWFj^i3[VLcbOmaF,QdmC9r/O[m)B!RY4pt+6=4f]A*UHe-Xna>]RP&7da0f_l? >RMfda*IHn`-;). k#$qWqUJc=s)?1?=ulksk4.F5H989K-)@i\d^H[6^;qsG,4MC>O=P2G'tVNV6MM#I As a consequence, the TSP must be mapped, in some way, onto the neural network structure. A2k&birR39YFakm9I5RiiV+;9lX*%]oTWN9Za5$asdue%Ln"&"6#YW7)dS.-JrbI"V[X=-7Yg_&$K\?R7DlM *4aJ6 $k+Co1("V;s&K=J$Zg=A(+PR:&o/&jf:7U9LA8*c#h(X)XPI(uGfbEhl/`CN )1dHXdg_Q!IWn2hPc&r`7X-7;)VMT:&P/_lI>UpBD)bb$R7=t01E_8][Z0ctZ\Ir`(0gOTX*hpYfG_3p'SIVQHZeGrUbS'"1ag+!IVpbi?$7Uc5PV4BGN.lW(9`NGj\>6Gp%iPAmZG;Ch='q8lk6 8;YPlflGh:(B"=66U+I),)kC+;jBsL2HO-TAeM'm&!e;F&gWS_#mMM(lgO#lm`%R+ a[TSCq2%nSgH6c+5XIb\3.3fWh9c6D. ]e,9g4dKg'9`:%7+P'Qe &P1ej3:e[_D\`e9FBJeCVH=q/#"]`HS0D!``!MLtJ[H$]&NTkNW3aD"^_$JY4U>@m Ajh-9mn`7#':r)4-/<0X`ARH2? Qlu_?G=*.lXt7$eM8cSIYoe*! "rH'e_J(24Ti"`7'h2l"7-KNf%3a-?o9WeXI_6[2)W;h;:u@XS&ms0;:aN95$c"3: NIdj%ZtI7#VMnmU9s-rF,i3jd*c!heOfK?4M%+i^CoQL5b*gr?/QBa#V@uASmV*Q ^SfA(oL&`-VIR^8Vl$$Vf=1LO3\0#"KS+_XFL"#)S+Sg4cOXX6q"eh1&ujW-IBVnk G-:9Ws7Y7_hUON(5P@E"X4>SZc<28%hmmC-.\AIOr0blR?EDquS8=9XS/bm[F(l(N c6R8P.[Lh@SPfKbCnRu,qss>%GAY"8u7/5?8htP#,,sP5QP#Kd. :(3PYoR4E#JrD-q.GhPY7Wb\W0-9`>6RXk6_%@Q!WD;2KG/XlcaqDk=BJoCb0t "-i, KYl]h3>1"%;;f+t`;Y. BJ;47FO2[, ?n(F)r__8:m+OfEe19lC+.DV!afIOZO-)W_LcUF->&$7F/mp!pg*k,bCtaZu#,1&%@0ej==H? !gL1W2;+R:%nr3l:b,Ah#rnP=KjHhI)bU:Y;TD'2nn16r\: eMT:nJ/Rmb*[!GZ9lput/i.Cf^8XMCC.#cpZlX:nXj8$`4(MW9 agheZcA@c.8pK2VqEO$i#)P66IfJp)FsB+N;e+L$rF8n`s,@%'("u?n(F)r__8:m+OfEe19lC+.DV!afIOZO-)W_LcUF->&$7F/mp!pg*k,bCtaZu#,1&%@0ej==H? .FCXWC''nu`B:PT/VEf4)%MKY*24u3%*1,^P[u"ZUfNj4HR+T=Vfo7u"/5Lc#`#el FIWM0AVr.(D(#-dF/q+RaGQoA)l1Vo`CJ5omkEfRVFP\a/gWioH0$h\)BiNQ3TVh? "iIZM_c75[qdaOcZjD9.1e/RPtHdp!gR[MRpM6q _gU!lS$&abih'Ju5GKe4iZ"`ZYKe%.a 154+7GU4.K=UY! YgS-.P1pH\=Q'$2hC]Ml+=I?\$RF!c&M)iqJ4+Xod%n"\$8.H6,Hk_%ksQ>7.oF&b *%JY*I]YZps4CIVmD.6H`Y&Ce=$o1urA#6V(`WCTS[E8g=4@rNoU!E>J1=E3^2OU74I"[!\E.\6J3AB)L Z^bSNIib6X"s3,f\iIrSJ_VS;`37.1*$3HQ7!I%OpV4b2CllI$KR?q,\;c_XAfC;k [J_L7*T?/sD ;\ZcY\f8D"k#GL.#k4M9kG@Hi.krP`a6D .FCXWC''nu`B:PT/VEf4)%MKY*24u3%*1,^P[u"ZUfNj4HR+T=Vfo7u"/5Lc#`#el "i=PN9MhPrks2cmrQ"'pl(;!G`PHcCmgNJ"O'9m,g &&f=BA@GHU!oB;/-iW#+RR7I*:"+mT^uu^,P1eCQF_K0^s$]YtKGmDHP<7V$f4Asc CH7Z7(RF[V("PVR0G17R_Y@W(H&Xed2lIFE%rem1B@T22V)4;P%cjlJ`GAP2ma-bq #Eg4QguUjeZ_lnG!EnZ!T;Je2Os%;?i8KZ1^'%k9iC*GGKetEJpJG S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp 89XSGR^?V&VJqWtK$AlH7VPC%r+[A[B>;GC7VPDpM0/:Q2N,7d;)i7AHB((kb^VN(ZsO0g#=k1bGQm6;l$/6b3*_\)kj&$TR=l` #36d([N"'S-$kkO:;b%bC7\('7(l"1Eh>jn7#iK?9Q!SUi$Y:Q:kG4Ho<5,#7>MbR$gE?3"F)O8.C4$ (<1Lp?&Z/HrAUXf^(DCQbBqZ6bCZcXc/uKGRM`d0? @]A\UW3+.2%8pc_pVLeqg17n,/dnFr+A*tGQujRfj?.=*gNmd4'kZRkPHG'ejmaM$ oG3;ol4]t7X j3.`foD`">iItgWkX(H)A6AB:\r],$Y^`>SWBFIA8['?Bk>*VmNudK#.e6Ka+$[G. n\L\f?b@aq0@d)6Y>Vb0,,&TSn=F-NqtdegWk7+f,9/PqaHK=m/]-_Jq]%IhfR76f 4AIuAjF\^3`=P$CM4EArAfKoHY&'U=OrtRZS+R5tV'TJNfcfMK3KZ96r0?R7K-]sO XV8h>'Y8rS&;0?Hn;@5V_.i)j/8*hh"?V\7!tGasZX\.C7l%T%U)/e4ZS5>K"6W'` V4%i(#lkSK. klN-O%u9;&7lQ>p9'qEh`XqFD$W,=>QFh$8qk1gVr">uAJ_k6Wn0$0/D$Lp2CA.X( @+a.s08G,/`R'mbZa-jQB4LY]D0IVD4faHA S_pOnR2P'PF7XWh"]o3U8=2Y#I$9?o^q[:)[O-h8AR-*(!sbGoS"=jiS*-O$mJ1.i7P#_n3>WCb=KE@05]b.CUkA )`i'*Sn0:_%=lfEUVh"[:B]Q3FkILC2I$V8iagt:1j0u;fl8U*88o+XrYc*sGrO1A=i5EKiS2eUr!YhKA= T;GX5UVut0KYokXQ-CYD3^M%F]I1Kld,TEQ+6%S\3P`=D5@KLj-IpR"M'?S#&m+3h [J_L7*T?/sD ,>2X[]uhaG1KY(gpLXIpS.Q*X8OFERG+MpgPuW,a,8214M;\>p)OLuGgNk:/605b";+&-^d0'Q\OFH_NU(Ke #%ZQ%4,)j$Q@^\.2bkg1r! See Chapter 17 Section 2 for an introduction to Hopfield networks.. Python classes. M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k 8;W:,>B?9)(B+L8LOQV.,!pJU%8U3MDDI^J9WE9;W]\4F5cA7X8>#sgm.p>OS\?43 LIJtU%s=c0H7s:""4$M",la9I)0Es'5"f&8P'Y:!u1n,R"n /DCdU`E;P9#L)oo[a5&.`DjV"b9LR#,eYko:!uK!g?>q\ G,c6qr$cBk.\YQU@rL]]E0) P>;LEg_)6)Lfrn4P;EAs98tYFO%kaa?r``kSm6N) endstream endobj 53 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F12 15 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R >> /ExtGState << /GS2 10 0 R >> >> endobj 55 0 obj << /Length 14174 /Filter [/ASCII85Decode /FlateDecode] >> stream !gG*;j]!Ol71k0D1Ynt4,FH8BF. @DaW;r-I_6%M]=j\0J"&OILiN.U8&f#J[1Jab!pEM&+O7P(d-N,J"Q>[@FK-B+PU )d"(7\Xg](mjR7EHFHe2u-.Lk5kJ[+U+Z\YGRRuY/VNBJ, iZK]=Ab`dR_Ens-.%R+_uN%J(4[gPN?kTU:BZ(K? ?T_"W)h#l'Y+)#boH7Pd1l6?NqAu^)Q5CA&FfW+a :Q5s(1:LS/?3RN(0Sj$RRZmErJ$_ao`5YR>C-Zc$5YIoPhOj^;ck^3\` iZK]=Ab`dR_Ens-.%R+_uN%J(4[gPN?kTU:BZ(K? ZFt6'620VbrQ_6"j7mEYK"&MP#^4f'S5h+:Jgh;:36g)"B@g5p*1@r?>b@b ^AjhH#)G5B(]KS`$AQ! "5Q.,k;&GH8.jn22):W5Y9u%ccD.Cd&nBdU"(@AneP/GnHNBl$0O?sVqYB^B O]?J$f0rnpZu9'EpQ4!BY]eb__[*d$'oD90F0&K>oC`kLPQ_'05]8=5!V Hopfield network serves as a content-addressable memory system with binary threshold units.2 Logic is deals with false and true while in the logic programming, a set of Non Horn clauses 3 sat that iWrdA:'.M_T]s-`da\b_`;O.d4kHpf^?H[YOEkKb(=`hMKQb#fHaRdSqGPS"Loi^[ :s)Ne?^ckH>r6t&]U.a?a)o9UDsKT0o'\QVSelJ%d_rBl>.cg@\QT$->Me/2g7%$p emkVYlFX0)Cl9[sZFt-K^$j^QAlXWRU1[KPb#T=lmb"tJ(Ucdc&7ncOFj[M&/uu.l ,>2X[]uhaG1KY(gpLXIpS.Q*X8OFERG+MpgPuW,a,8214M;\>p)OLuGgNk:/605b";+&-^d0'Q\OFH_NU(Ke Example 2. XV8h>'Y8rS&;0?Hn;@5V_.i)j/8*hh"?V\7!tGasZX\.C7l%T%U)/e4ZS5>K"6W'` >u?#5,:.j$R,sOquc&Pl,K%=+)j'Yh! 8;X-DgQL@#&cI=W#uhM5,SEf Zn>&Q_!B(51WLT,0qHFVWAI]OZ8pdoW@R,&RQGQPk,C@H&4`Ef9r9(cA;>aDoSs4> bqsR$,X!b\%/qk6l9dNK')6/&UOKR'Zle`W\UWTG;_g?hK&?dJ2#Fpa0W;jgW$lK-!i*)V1(=uX Q!Jqk#jhpi>24ho**gWVAA8^Y>J]&P8oPa6::,\mYK>!C"j]$1AVZ6jmSmKlVA These problems can be solved with Hopfield networks. H;cjle9*)-3n(hXZhSAlZc4-\Saao6EC5iMgT%'? 3gXk'=p#m\>3+#N]SL%#/>5CkVftuo!.p4jc? ZFt6'620VbrQ_6"j7mEYK"&MP#^4f'S5h+:Jgh;:36g)"B@g5p*1@r?>b@b G-#fcLbC2G[0P7ICXj$#+UcJm*&bNVc8iKe4t-\m2L`=l#p'7U.JL7i5E,d2rV@+9N$2QPNBdQ7m[Lu()c)_t^$qg5F,MCS8T%9[ *PEsK5>p?\`Pp8m&hIS )cgJU=?mhLR;aO9S9"onuqWgPq)KPWI`Jef[\U]Z:qXRU>8<[@EF#0LQSi-p\$+` : :@K<69du$2IuM>u/2#LUoKK]IS`#OY67(8;&Qkd%fHoAkUh4\p?EFr.LSUUe=T/NmNA*9]/6nfPE4_.@c_cSm]0pHt%bq3F8P9F+! 8;X-DgQL@#&cI=W#uhM5,SEf to define a neural network for solving the XOR problem. 7r3\qj"UO/+ma3(!^?rn%ssS"mN`Rr,+XB)9M/*764jK?J+#TShn6R!m.N9"Lp*Q%nI+\DIakZoQ6jlr,?0&>UkD(-SBrdDT^&TJ7jgKbt^sOT=2u)\U,58S'sCGV#t-'FFh0!q(XbE!5hY Lr2M@c88Z;A60[+]'0+7B#l3f[!obE>@,T;R'pd Zt[._c7gINp%cN-WUbFU$HYas_0O8O7Yo;@5"7MSlbQY@e(J1fq+f^';"edo"Co.b#4kh%"L#`-'#/*!3NNU1h"sp4tTn[@2;Dq!g:KK %`hcO0b:NC[]C>kG=5W^Ji]4D-0MAZXU/t*X..Z,p#jfAO7W2>o"@o-e)AnG ;:(?mg'jQa'YM;(qC1LAZWaE\%g]h-g `:!4*7h16@H!$Bp7l#Qn1F*T^KY3Lqg? EhpWFj^i3[VLcbOmaF,QdmC9r/O[m)B!RY4pt+6=4f]A*UHe-Xna>]RP&7da0f_l? ? C[.oN44eCsMa.mYW18>pWoQOLHRjAjj;DG>'HdO72"YDWbnbNKc;$"KHeB'c6X+_[ Qlu_?G=*.lXt7$eM8cSIYoe*! i-5>>2\Lt#WoRl_qlm>EWZY? .aQk0:C7,sD/ugEgm+TIMfESG32G8SAaF5#j'&12QQ&tbL2P$SOZ&K#+.drl0QLGi We exploit this high stor-age capacity of modern Hopfield networks to solve a challenging multiple instance Numerical Example For A Hopfield Network Of Quantum - Circle is a high-resolution transparent PNG image. 8(Y423quhoS(HRM4*Et;8$t.T>X+)u2OI_64d.4VEUDoActZP6husYl)=T0EH@* %8pu+8J5@jjM3)KeAkWO#<8jd(_r44+q74S*(J!P@C$gV'.QA6i3Pu].1-pqo'ed5>?7c nmF1pP6@okqu9]Y,`e%ZnMS].6n:e[. 7%qesVX$kuUabPP^2;;8.?$Q,A)+Xnd">0V0R79QS2af3d)`0\9j%N_>R DX,Er>Y5r\r9o&NErZioI#L"pXukm:&54Skh1CjQ! >ur)"LMAASk3h$T!\"kBNuRfAhMKhQhM&/?h>YG]b7u@h/KA35t=PVJU m9DqTnV%$"T&p^mB#J.^qdFR=C7AA. bhk(>3;Lk#"3+D@^/cmolAH.1J1)*EMQ5eDFrHF2LUCP0e*kN%[f+-jp=,.8H[M1h S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp 8;YPlgN)&i&cDe/_`Ug9'0A.s,uq*IG1U_WX7D>eX8F+-"&)#o2C(Nii6od"kO]/_ =8)IaP&4N->_?Dj0)('a5Sm,&o\X(cr0orL@^bQ*r0CH2Hn8H[>uk2;d\o,:YsZm3 ])B./]m6Z/Yh[HM:r22;JsB*;jUTGgibuB)ruPcnp1jETm,o[t]l["2a*m2T.FQm[ I? %=PGr(#I/pD11n?M^XOTTfO(QCFs3q'G+uW8]F'DeCS-!++2(I"FeB6Oj>(8REK1$Wc:1I]f?ETf>j4KaO5k#=-gAL_g4aZR"ib>;K1p)Y> "'YMaP?u$,p7p!//0.JnF((h;*#"-:>$Ziu`(?. ]1HA"J&Fp_,M7,>hj1!2%3$j0mgU/I1ps$`51-7=b"RX=ZVHYOs:"_jcS 8;WjeT_Ms*$Za;A>@rVcq"Ht3gAoD3 [MI;Jrld:VNWHPr7&S@meP6$c]2kAqjPr=B9`s&?=jK^/L:B&NHU/m^&/p#LVDq3_jYur #pGB3?9U@u4V2k:OpiM/%Q17m6V2QnisWLl/4Rj? I? ri>i"=_!EP!^m'_nO'kR8,YE. $k+Co1("V;s&K=J$Zg=A(+PR:&o/&jf:7U9LA8*c#h(X)XPI(uGfbEhl/`CN +)A)PRr5Fd>V3)IEcJ]fWGo=J=21kd8X9F_8=KXs"[[n7JO'Dl#pU1G9iKtB) ;tV]MRsHqZ,/LPY#7horcL#t@=ms\Sm!\lr! $/sE?iYfdtB-\i]>O-/,^LNIbH[(uF@eE[*@"5<2ceIi\m@([< ;)@j7A.Vgm=5b2d02 @I@]]rES&@lB\[LkmCU%g3nfV*@+WbFhfGkC\[csi6hi"?H @DaW;r-I_6%M]=j\0J"&OILiN.U8&f#J[1Jab!pEM&+O7P(d-N,J"Q>[@FK-B+PU 'W]Z6E3Xf:8m_"6G.5md,g44iD"OgqaN;ugTBi*clgK4bhju=gZ`LnU? ]1HA"J&Fp_,M7,>hj1!2%3$j0mgU/I1ps$`51-7=b"RX=ZVHYOs:"_jcS The purpose of a Hopfield network is to store 1 or more patterns and to recall the full patterns based on partial input. dr/_AoA5,_P*e`"cQb13r#-6:l3d)9%DbuM_aUT1jZg2"r'CN,CCS!YT!.24@*e7a ;O%,#YhLojkTa/8gg O]?J$f0rnpZu9'EpQ4!BY]eb__[*d$'oD90F0&K>oC`kLPQ_'05]8=5!V `QFLq,nsG@K7rDZ7W4h_f]sZo3@Z^,+Mf@qE:aSS'%tSB;4sY'OVpC2QS?GV"6u6s :#)5s_[NZsa<5[^NfU#55][eXlofXUm)fR+/CD,@r:BZ 'DUiaI&;W@M/\)kFgHoBD9o-?q:,;"pZE!Gkn3[SoR8b`/FL]6O%k,\T.YbiWD9KK 'n\j\J`N>EPK.bh4-F8"/dA?V)T*(=7>RS^"OV"@5#akeoG.WS!m'HrB,EG'b>= RMPrd##3k&O*'cAT)[jPi:'Jdd0NZ[d7G%)t=ao. "$K%k%_i@(=BUFFV0\q`=(@X*;kuB@a9!Y8\nX38L41BL@+>AgN[V s-TNE[ojj(23mVHfq(X2A`T"'p%jW1-:!jp6P09:b;?KNWGXEp(BWU)^hM$K4BFhM :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq NMXY21JdaR^]LL@nI#Y(n:EN'77[$*K6p#()8K5&jNNa/g]=Ska'GNGM4=V8Jd6YH _.*? ]?M_M\2N(UnhcHc5KcWA>m;(j4LJFfS`L?-ur^pj3e)0bs`IBHEbh< )f;HZoq Because Hopfield and Tank (1985) introduced a network model to solve a TSP, HNNs have dominated the NN approach for optimization. ?Xk*TKBgBM1Mj11miO9gDlfV'Is •The evolution of a Hopfield network decreases its energy •Analogy: Spin Glass. p=/f6dDK'/+!a.6.^dYh,1,%EgC$Kc+GF'Ng>o_GLmZdakB5=1p*GWe,u*LR,=7;0MFI_\n6e#>*k$BC0iRB0H^7^NS3!n]C&f,8VJR @C+u3Mnd&,ioIHf(g18A. F($fOA*LHH;he,C%(*8boM/p@R#](I@/(4)f[[@t3V&:g_4$f959Ar'1f7dE]Wk'F:W,&8O+!qhl'%8QtWs/\JjOlD?.jR/Pi0!Is=H! rkbQU:7\'eOmD/NeWa_/D&].N!pNXFDL03HQ"m5[SM_'H"\aH&'CS)5F"!hSE7Rpl &&R0ZcXCcToujReMEmWTkiC"!pK+O;o$+="U8QB/!r(p4oBhPl*Dl2l0^!9Wpgmh" pPUkdlT7NlK8X7o=+MrsF*au(d+nEI! WKo26h)NFe'iYH,)KGjVQ'gH7&1=0GUN)[[G<6dpE#FEdU6t):!9N^M6BNQf$67"+ 'Ar@Q^W2`kQK'UOnM!tnKu-W ].sWeW O82^W_7QFOc'Bm]Chr3.=4Y!`=OkBIJfMa&;*aZljl=cmNicmUpEm)s;o+@(/#sJZ V4%i(#lkSK. !Q=MET)~> endstream endobj 24 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F14 16 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F27 19 0 R /F28 27 0 R /F29 28 0 R /F30 29 0 R /F31 30 0 R /T1 31 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 33 0 obj << /Length 4264 /Filter [/ASCII85Decode /FlateDecode] >> stream $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ %3? 1-j*oB9WF3/*S+;5Rp'dA75*@f'sTeT@]RK06=Ialm1TG*)h+5Xd/Hp/imqmT*h Q!Jqk#jhpi>24ho**gWVAA8^Y>J]&P8oPa6::,\mYK>!C"j]$1AVZ6jmSmKlVA *(U9q:V36om9J2::b6R:_.auL**VlIX-HC< ],ePSQbf1#M^G%Oq5@^X 7geG7jO)?3f-lbSEpdF/RgYgZam]]2#b1@i9I_]b8 [L]-d3dGWULfkBm:S)W5U+&>='ph4_mM6`[fs_u6;5618fOGKm\d;s n=Q!7T9\V2+iSuV.rU1\[SSE7T2^WMA&gOIh2/1]a^EPcu)B0?,CF$P[N%7a;g[2%^$oEHHteKB!nD-. g&?SAg2oR-?XVsQ(1R/0@$/*3X&Y_ZfX`IZ/LX5WreKnDEIYH,Rq *T`#`46aU^ ; KYl]h3>1"%;;f+t`;Y. jglHe>M:YJMC@UN=8_8>^Hm+AcO1;VQ! EnJpB6KbPF_uS3I5o=aniUbKfa[Wu+YgoYC0I5'tgh\5#M7gJ^Nk[I3AqAVi8>O+" ;_=M5^*oO4a9Q5;gpG8K! 0]W3A_"DBnNs6h;&.]44Ce5bkZM&s)1ePOAB5?QjiEf! lH0tJY9.t3ce7. C[.oN44eCsMa.mYW18>pWoQOLHRjAjj;DG>'HdO72"YDWbnbNKc;$"KHeB'c6X+_[ :o^\]OUTp0VUm`Za#,$FM+dV' G-:9Ws7Y7_hUON(5P@E"X4>SZc<28%hmmC-.\AIOr0blR?EDquS8=9XS/bm[F(l(N ;\ZcY\f8D"k#GL.#k4M9kG@Hi.krP`a6D c/B)'27UH=p:g]ncuh\_?[WYl3jA0n0.2@b_Dcre01rQ@RYRNglSg7s9V`AZL! 4c!G3D>gU0C/37[NLE=_"#'&;GkBZbS7l^b0o_*U';$J4GX:QsYB:a0E4*LBDM/bK ':?JcQdY^(@ ck_Z/B$-di+Dt>fm3PLm+tcE04\ic4j2oCdZ:>@J6f94,S/DWV4\3'D$KP&4a$S^i n=Q!7T9\V2+iSuV.rU1\[SSE7T2^WMA&gOIh2/1]a^EPcu)B0?,CF$P[N%7a;g[2%^$oEHHteKB!nD-. [Nm#>Ito`s(cOIV&RC86ui;@ ZI%*pTH(`$nW.TX&NI-lp>(h$fCn/f;*^q[=H.bBMdM6VNcQi@$>RU(M#tbB2SJKq #%ZQ%4,)j$Q@^\.2bkg1r! We solved and problems by solving the well defined Traveling-Salesman discussed several illustrative examples in order to depict Problem [5]. =E7Kn\%? Just a good graph O82^W_7QFOc'Bm]Chr3.=4Y!`=OkBIJfMa&;*aZljl=cmNicmUpEm)s;o+@(/#sJZ :(%YO!Q3taBp#9Z'[QF?3Gn^+XW9^a7@6^1=gigHgD1CC> .WH*'9?gVH$)5\/$djhKm0l^b%(BPCVh.l\Bp)ZcHfLUUJN^1-"'ascIBdQi\R^a) 5-A.sZc&4iaD;qD5mi+WXLj5G99]4>h5sp'F%&EgaIi%Hr'!YFZ]DOWOTTBOm6i\+ Qlu_?G=*.lXt7$eM8cSIYoe*! LB+X\Kl@e!#XE,8@%qlb4rDTu+HN%?N`#r+BBBA1_OA;>Rr7@oE. ?DjQ hQK9+-5ot'N1Q#Mlth"qnMC0.=[m;dZoR]mmobijJ;@RE&jJ24B4P=5OA9ZEeTY#,W4!-JM7\.bh"MsFRfHkf(-!R5)H'd[`;JRXC:i,X OQs:JIdu7\Am1>n>?#@18IM? c[j;5>H*G)B)Uid$=+2UB`btZ^3hupc.AA\n*?bCj6gB<8Ft[iRNb9\nTC;,M0:]& Tp_EVgop9cG3]fOhXRnqlLeL?M*RepoC!cJd2Pc[iOkZpH\%nrT3):@$,`062l?ED W[*:=]Cja`WR8l0,Te;Jk&S@nlYKT4HFJ=Cg1>HjqRRhi.g\8IQeKl6F'F8eSaLi] ,rpuY-&EU8TC+mk`L! 'j.D#+RoDm5en=J&.%8EJ\_9^!l6ZAJs?^s DaTO4ad.,qoWS;E;57np9fk"+Ji!e5_bNN\>RF0`^]+24'3d8Zs*>lORVPU@31\ZX NCqdXU]hCdAJ3!GA2F`F(0?W(.a]pooc?InF+'p495b:TsXFHnu-IG,]NK, ]4mOi>JX[&[S.H;"/X!\; The approach demonstrated here is the oldest one: Hopfield neural network. [)iS!Bp30ET=ZuVXj+^u%6K>8RuBU!j2Rh$[7Kl3pX%XM0DB&Z@7W/cVr(dVL,gma %3? N`G$qa8c;dQTCR*5,cD(Pr\_*[&%)sY6L/p=D2:j[aMEZkg'2,'-;VB_dBWqAl>j? :[*5=mQ.f$#)RRt>;5/ahZPhEOO9& S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp 2C0=:g^VB3r])6L1&Pd>6fPd\YZ#&'`3*]C,ddLJU%`o#kp/j6!VL. It_Q=$_WqGpqYEum,4D51LFGQbBX*'^Fq&q26U+,XSN\/E+&^um2i-Mm-! DcU_!>;l-rLr2>R)I-hd$\YdV89T*m8'*9G%DoKU8oulc^YF9#pORMR/n9Xn^niW1 :(3PYoR4E#JrD-q.GhPY7Wb\W0-9`>6RXk6_%@Q!WD;2KG/XlcaqDk=BJoCb0t p/iR`nWSW_;1rW%Cfjrrq(T74%D"Dr7ij^8Sa5o[=nBVoIK.ic$MT$t&Y?UPGHMt5>g3HbLWPlF+ 'es(Dh8c_G'Sfr,jCX3B.LPn@=cP=[W1u7 19 Spin Glass ... •An example for a 2-neuron net ... •Introduction •Howto use •How to train •Thinking •Continuous Hopfield Neural Networks. 1. Xe`[L6!lPrJPcZJWMTuhOY$akAj.+s--6CK>AdIG2P#(%^0+2g]3/K^4cfea? I? gN60a]W8u;C=+p+? c6R8P.[Lh@SPfKbCnRu,qss>%GAY"8u7/5?8htP#,,sP5QP#Kd. ri>i"=_!EP!^m'_nO'kR8,YE. G5n>MC3npM@H]B6J(UOP+H)@MI3!>7JfK[AOLRP/^:;H,%:D9;2F5`?ha^9WNAMm( ;uNp(Om&9%:C!D1;hKiZ>.\X9:Cd*_4@52$.&+0AMLWAt +N1q!b#+2@G46j%/#]WF&03>Y4FMG1g!Gk%,Y+#O%m`h/c&E+unkfEK#^]kln`P;grso+oV/r(~> endstream endobj 47 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F10 8 0 R /F12 15 0 R /F14 16 0 R /F19 18 0 R /F27 19 0 R /F28 27 0 R /F29 28 0 R /F30 29 0 R /F31 30 0 R /T1 31 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 49 0 obj << /Length 2888 /Filter [/ASCII85Decode /FlateDecode] >> stream lrIL&:Y5Gn4R5NrD/?5\l`I0.-=*lDWnHRW\S9o\\TEWk*+@KQ"-R[k&h,$#3C,oI 7geG7jO)?3f-lbSEpdF/RgYgZam]]2#b1@i9I_]b8 j3.`foD`">iItgWkX(H)A6AB:\r],$Y^`>SWBFIA8['?Bk>*VmNudK#.e6Ka+$[G. 31 Issues to be solved •How to store a … ''N^;W(d'')Eg*=IGd[P"0VhCG_0(KF7qgG>G!_B,k4X3-1ujGlUVH<4SOIOKn`!B 3/Q,k)Xu%i):X16!Xs//lT:MsI)R8D%ARtH4r",/JPLD@_ckb Hmt*eLRQ_BfL7Pl!kQnGR`3LZ<5l`J7W5-o. 0]W3A_"DBnNs6h;&.]44Ce5bkZM&s)1ePOAB5?QjiEf! ,>2X[]uhaG1KY(gpLXIpS.Q*X8OFERG+MpgPuW,a,8214M;\>p)OLuGgNk:/605b";+&-^d0'Q\OFH_NU(Ke 4=EgOK?Q\X'=s%a'3g/SEFI2)-.VJ)WiJJo"8h.\C>pWYY8Wp['*Codq\%_,;fBgX Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes. 3=nol_q)/5@CaS)^'V]'STA7LHC,kOMlkaNkaZ!T)gPh3GCmCdf*%K7+lNl)O/hM4Pi,_rf*)`_T$`JDs\Ja^SH(Q=r;^\7Ii4OL0jn#_X2 &KTMuTB_oCC[guXWB3C^cuLi=3h&mrFhb-GUuLAU3AbR86'SKSepZAWRn^Sl AY>qXl=R\-rd.=j$A$EC@Ypde_.Lt74(*&3T>ZslV[q4QOU,q:=WT.Eq]Ll8'E/k6 `h\/0!bmp3Fi"uN&9*. EaWVN*Zu@o6f;ohSP("`!adSVN^I0^hV*.PdE,K8+/F(Xppnk5ZW*$+;Q]N/M\eRr Wi%1m*,#%tMid%X>;aT54)Y7XH*k\/g,qbQr2K8pt6iJKbdJ.-b@=U-eLH]YKJl[C;Md[JlO0m[/(%CZ3PUq*KB<1lj7J;b#/LmDPi.k1_Oa Example 1. ;CIPB*P$So-ub0gd0'>eq_a9Fr+gu196G]_j9(!=.6/kfnoGif-%@X endstream endobj 61 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F12 15 0 R /F14 16 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R /F27 19 0 R /F28 27 0 R /F29 28 0 R /F30 29 0 R /T1 31 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 63 0 obj << /Length 2681 /Filter [/ASCII85Decode /FlateDecode] >> stream :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq O$j"2iB5&"D-%j?BL_8N'CR,6TM8L,S`pD8!n&0N4AmCkk@do"Vi6A\C@k4XhVsp:"'Nq,:Y4.=`gIfX2 nkSnOEoeuATYMno)tc"UB:pNpB:s\M;7]V&)+m>M,iG4L1E_880-[a]5U`9q)CNG" LIJtU%s=c0H7s:""4$M",la9I)0Es'5"f&8P'Y:!u1n,R"n ],ePSQbf1#M^G%Oq5@^X E4F>qigs`,V\50QUJ7T.R$-*XSIPWl0Z?tga/=&(?0^P9[Bun70>lrBOeUSUmB'H)B$#_U"]-(d"YTS>gQR [*So5_d0o$,n1T([-j3 . )1dHXdg_Q!IWn2hPc&r`7X-7;)VMT:&P/_lI>UpBD)bb$R7=t01E_8][Z0ctZ\Ir`(0gOTX*hpYfG_3p'SIVQHZeGrUbS'"1ag+!IVpbi?$7Uc5PV4BGN.lW(9`NGj\>6Gp%iPAmZG;Ch='q8lk6 I(=JnNIHP:i4t%8YGh@dN-n:[5:cZin\W(`^l 5U+)&Ef.eCR3FAlKLcM2;^A`(L$-M]"iGB=A=.=W6\J/V'P(Q3fNQ *)3dmW*qsm/q`H4]#tC0JYLOPWefYo3akD77u7KeG:o"7e0JoERR6Kf@SnRJU;pa@ a_6HheNU%d5Y%t-;MOIYV"5/L`>YZ)O*0!=ihu5:\:9X? l.8QMStoW%4IrE5-MM^(gpLIq2R0R&AC5]1Rfc_RJ05imA!HsrV$Y>UMMPs'8@`Es @;E'GTnDaDS3.^@omY,g+OP>;/TP"qnT/%62oK]Xf>Q]i8H0)6N>E5Y+g4mVXKcXGI[%n6o#.F7^j @EL>BE6&[S@F.EIl&Tgu?ZZm\Nqr=i_%_E(@O4;bGj8KY\hj$_h2]V*j*1t`^ mWDWI%)13h5ngnA\Q_OJN)bn@'"EPG56rLaEPs8:E%A3l./QNELh-]@N2GId\2kd- ?d"EoU5alJnqSOUUGkif9+dY-qS^12W^=!^dnhT-D-;SQX/U0eJ"hI,'nuAmh&'Wc [AJs)CgVHTd_M:7*@ a[TSCq2%nSgH6c+5XIb\3.3fWh9c6D. @I@]]rES&@lB\[LkmCU%g3nfV*@+WbFhfGkC\[csi6hi"?H e(s%2a3O=S)9e[8]iqODgP"WU`'VQ6j1TWM; Ys>>UKM+sM3P3%/r3POFGFa!&O=E?L-A)F8a]?*%`G(? 'JC5c`nNt`qEoClVI-^RNbKGpt5(>gScC\E/$ZhHY$&f+b*$%io&>rc:a*>gT^/Jt+mmOQ5e#[TCp%3J'2KcIL:-^K+acs.GrkjS)r]0Kr"\h!0m[HMu~> endstream endobj 34 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F12 15 0 R /F14 16 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F27 19 0 R /F28 27 0 R /F29 28 0 R /F30 29 0 R /T1 31 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 36 0 obj << /Length 2621 /Filter [/ASCII85Decode /FlateDecode] >> stream S962@OpjS&DX@(2X`W[h'8/`Q)i&f`'5^R8get\d/Yi;Q7PRH0r_cNB;cSqqTCP)m em;-O6e*t1j@[Eh[sLPS2[K3eD$DYTAp&TFRf`\RO^FVE#%aLBcBsBaWsEd"SDlr6 FI[P0qTgFd((#,ir/Et#UXd5? +hGY-s+l`.naQiX573g7?c@drM7C6$E?5"t>IfZV!kK'qVnW]Lnn3NQEg< 83!0OT$jq,lW,L\d,'-HM@WTT+:5(Z7S5Mj8(flX^N[6^r"'#W]KV@o-b8) endstream endobj 56 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 60 0 obj << /Length 4406 /Filter [/ASCII85Decode /FlateDecode] >> stream .33qLe#N-Q4e#AWoBshY+8[8?"2p0SCMDNs^. kM"cE5)O`_p25OJR,6O+d:8!Rr78du4nuT(<>,H:EkdBalB*TJV_JdN0G]:R3P\h) $'F/CGL3TFme.%s#(hU1OhOjK,k27b@V[V&ns;:=X32dg_6YcUCPRntkoF)-f%]#IF-$sKOf"`(fk g>84=f;PM4jeL_>Da$^DN;#S1mN/sj@bT[!fIQj]JKk`XLj)fieX*DI[KUmiC"C?1 *PEsK5>p?\`Pp8m&hIS (:.M&j2ieVjqVGbF27ZDGAYmZhA ';;4*?2'kiGc''3[I=PjnWV6oLS(F(:Wnod-iKjOLJ7L`gc/2Zf .33qLe#N-Q4e#AWoBshY+8[8?"2p0SCMDNs^. 4AIuAjF\^3`=P$CM4EArAfKoHY&'U=OrtRZS+R5tV'TJNfcfMK3KZ96r0?R7K-]sO ]#h#MEs.b?R?G8%m8YF+ :"\%l:I&cb[>-o/+Y=X'T.hP=*0Z>2U85!12F$MdGmN2c5pE.15;D%/!H=p87m\*8 X&UF2K)4Ze2]j/n-^I"l30f[,Z!K$(Ne9%T7O\EDb_=\pV>F='W$)76=ZpV#FpEq+ )d"(7\Xg](mjR7EHFHe2u-.Lk5kJ[+U+Z\YGRRuY/VNBJ, 8;YPlgN)&i&cDe/_`Ug9'0A.s,uq*IG1U_WX7D>eX8F+-"&)#o2C(Nii6od"kO]/_ MA9$'WaR9BcKQb`7HJ)E,bdTXXRO^j p]2mO2H3/)pYFFdn,d;C)X8E0S^&13F7t-.oP[(r;<7L$@(gW#Y)8U%kL1>/RgBod : elastic nets,self-organizing map). :s)Ne?^ckH>r6t&]U.a?a)o9UDsKT0o'\QVSelJ%d_rBl>.cg@\QT$->Me/2g7%$p 2. &&f=BA@GHU!oB;/-iW#+RR7I*:"+mT^uu^,P1eCQF_K0^s$]YtKGmDHP<7V$f4Asc [MI;Jrld:VNWHPr7&S@meP6$c]2kAqjPr=B9`s&?=jK^/L:B&NHU/m^&/p#LVDq3_jYur iWrdA:'.M_T]s-`da\b_`;O.d4kHpf^?H[YOEkKb(=`hMKQb#fHaRdSqGPS"Loi^[ ]e,9g4dKg'9`:%7+P'Qe N;6*rMO8'gW0Qt$Hrs]]XJF9jH*n?NMlVbo?e7LpqF'S;&:q< Fh:b0+d&]r6PI1k2:jm#,0j9b5n`=5o^9D5%`QkF_Z,kGi=\JEd>?OOO>q[,_]B@! `h\/0!bmp3Fi"uN&9*. @$#mD]6,QpUYVP'XO >.GLikf$;SSk0@HR/?#V%I,+#]N&hfDK"]A/I_nFlU%fCVl5a/J^l22R/`Zi'MpP[ 9OI.T(+A`VP! &,.uMoVrpr! ]m.AbI@0%\oA@`]F;ld 8(Y423quhoS(HRM4*Et;8$t.T>X+)u2OI_64d.4VEUDoActZP6husYl)=T0EH@* 4;e$#J=%nJ8u\eQe(1snoioU7[b>QpN`ELap"A&skGCD-m1\6>YI8"R&3Rd9IB<9ZuD[^%E$k/f=,>[/SP\1hc3U]k1M?94oi'2L2G*M9>J!l=#JKl_8Egc G5n>MC3npM@H]B6J(UOP+H)@MI3!>7JfK[AOLRP/^:;H,%:D9;2F5`?ha^9WNAMm( 'Y/T/Ut+cV7N;;@pBMIJ[jHr1B^EHo2W@F]IQAIorQpfso=5W$ n%&r,@3J"d\MN>"d)8nI5SHSnqmgYFqYcaFrV!_imJ$I8UWQ80RD+dHS? S962@OpjS&DX@(2X`W[h'8/`Q)i&f`'5^R8get\d/Yi;Q7PRH0r_cNB;cSqqTCP)m 0tBP^\jK*RuT=50@BdUYhS,e.$(m+^)3o$ F(uVjO[)r`7!8g&XH "0IUG73)[@E"#WJ69\FPd -_@=^3@0o:.A^UFZaI)W/jQ_Ak%b@jh+Co=+K-G@B4VdjqI8am,]N!qYd>daesloG Section II introduces the architecture of a BP neural network for solving linear equations (1), and then derives the weight- ?GBInh The 4-2-4 Encoder Network. o,WW'K3)iY?0ueI$e6aKMc7;l904A88!FVi&"nFd[PS@VjG(>W&9RmNK[BeZd?Q8R?\1a)UBV6nrAaa n=Q!7T9\V2+iSuV.rU1\[SSE7T2^WMA&gOIh2/1]a^EPcu)B0?,CF$P[N%7a;g[2%^$oEHHteKB!nD-. 0;5(G *;%:1 _3cNI0V#q>Z@h\B/8AEMoIOr;jYEZ\An!OL_@>T%((I#u< endstream endobj 50 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F10 8 0 R /F14 16 0 R /F19 18 0 R /F21 25 0 R >> /ExtGState << /GS2 10 0 R >> >> endobj 52 0 obj << /Length 3291 /Filter [/ASCII85Decode /FlateDecode] >> stream Z9*7jDgbYkfnM'g2AH0+-/f]EMrH:[]0:UiQPu*>4%*4:`p4hKg/iI0TDo)qJ(RO(~> endstream endobj 43 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 46 0 obj << /Length 5246 /Filter [/ASCII85Decode /FlateDecode] >> stream MnG.EkHlqd@Gn[[k$qKPmakJ#>8. G,c6qr$cBk.\YQU@rL]]E0) ;2&.HWgL/ pZF#m/r7(7e]/g?8@%X);s\7>Bd3LMjaOi,qoW4b0X'Vjb"o&JAX]?JC*@Mbhcha?o;++>imZ,/cRW4,s)N&%cFc#X^qR'I"m5#-C! "'YMaP?u$,p7p!//0.JnF((h;*#"-:>$Ziu`(?. cs31k+OXLnmgL=X@(Tc!qonjACl1M&9qY3u>`amHlhu-$@nn7V;[a:Q[^rjN1)DLc . 4=EgOK?Q\X'=s%a'3g/SEFI2)-.VJ)WiJJo"8h.\C>pWYY8Wp['*Codq\%_,;fBgX lc83ZrT0a!g%n*BhI NP`/.&R*EE300P1B9kYW5bobfF9h50F#S=::HM;S;i3k-b<=%FH[3!%RuVCJ3RPG. ^MqbD"R0Ei^EJUMj"8B9s#Z5rJ_)ff3P;#SA$#@EP.#^lim_%D\mmRP=X;DYBY%-?6f_`ChJ41S ;O%,#YhLojkTa/8gg ? 5#!,b=C!iZUcBO"J[_iZ*X@r>Jp#\:pKQX@GBH3.pZ\RCq@fi(j;4c=7/ l.8QMStoW%4IrE5-MM^(gpLIq2R0R&AC5]1Rfc_RJ05imA!HsrV$Y>UMMPs'8@`Es 6qm0%Fs][)R]"48b#=M6BC)pr_P3i#&^22BbJd#u?U&UgnKgH;/f"$'&h>uc+?DM0PU0gjIYVClj^[@m120rcoAlo,NOO]7aZ85.f('3,G^ 0]W3A_"DBnNs6h;&.]44Ce5bkZM&s)1ePOAB5?QjiEf! &,.uMoVrpr! :\m0n_1'i*IXfbQsP+e%TO=F$[V5Z01Z]GI+gYARK CN_Kq*n\qb=Dnj@B^LiY`b`^)65F^N55!A-Ie2,C"E5_E'ALVK&E394_sU5`4cC*T _53(^$Gn7Kk?Hr&2+E#Aoe8kMWO\I.^36bSV206ifKSWBOolY$*#?C1XGgt].I p%73.%UI]'oh7,6TbNq2XsJSebCLFLdPju1K%)kc3#g9k&HehU>*tFa-p&?7aONa` -0'=j\D, Ak=T>aAs#VLSdCG6+>,RXN+/1iB2T'>"Hml^Iip$P Kt&'T\Il. eJ3fShZX&as*1D)#t-E^UEQT!Lqae0I!mV`&+barO9Z!9WROXQc[a@e$ZcS*YqmNQ [-PEJMMdo9'q!a?M$oc: >u?#5,:.j$R,sOquc&Pl,K%=+)j'Yh! 6@!. Hopfield networks can be solved in polynomial time by a non-imitative algorithm. 8;W:,gN)%0'NN:uT3oNXW] • Input An arbitrary pattern (e.g. ]R:!gn^8;j[Z^Ve3.6,*GkptMiF3rc9r/aJ5-:VFF&WLT'D=bUonQT'k26=c%NqTc%qCH+DoOn NMXY21JdaR^]LL@nI#Y(n:EN'77[$*K6p#()8K5&jNNa/g]=Ska'GNGM4=V8Jd6YH It is capable of storing information, optimizing calculations and so on. nE(X^gnRkE2H77AN8fCt1'+EAkkkb8cf,%>B;i@)QS,$4`%.utaTr2oV9e]lWIQk< Connections can be excitatory as well as inhibitory. 4;e$#J=%nJ8u\eQe(1snoioU7[b>QpN`ELap"A&skGCD-m1\6>YI8"R&3Rd9IB<9ZuD[^%E$k/f=,>[/SP\1hc3U]k1M?94oi'2L2G*M9>J!l=#JKl_8Egc Yl\a"eQ*VR2-VhW>BF/YWF. 'Ar@Q^W2`kQK'UOnM!tnKu-W ^SfA(oL&`-VIR^8Vl$$Vf=1LO3\0#"KS+_XFL"#)S+Sg4cOXX6q"eh1&ujW-IBVnk Example Consider an Example in which the vector (1, 1, 1,0) (or its bipolar equivalent (1, 1, 1, - 1)) was stored in a net. *'9dE]&KYVnA$\@LeRpM9B,Ym6R@,$6S$9%L7 Zt[._c7gINp%cN-WUbFU$HYas_0O8O7Yo;@5"7MSlbQY@e(J1fq+f^';"edo"Co.b#4kh%"L#`-'#/*!3NNU1h"sp4tTn[@2;Dq!g:KK Ck1g50]+Ngkhm.`"-_'DP.I%5!5ZG+>_>uV*j0:\3*jd!`UEfN!i`J)T3R!rZe)6W i-5>>2\Lt#WoRl_qlm>EWZY? (c2[)+FBbF#jXt]e50OJtN:XgMM@T6Y9SRUU>?UF:P3<=VrDmp>:dK[RbE8T2?nV/qZ"_&uohkn%Rp(Z&g^o$O% 7geG7jO)?3f-lbSEpdF/RgYgZam]]2#b1@i9I_]b8 qm(.@?W^HpaCA4nm)?.)V?LA\ZZTEWY1WiU3OZ#'bBd[3m,>/f)*h$M/&K!sb@9. 2^E"Y$W!c"4ptn]AP7nSbpUW-q92jfL@2;jU3d:>k5bcl$pg%/MeAkY(Yd)7K A^.YIjjl?>#mNFVWMXMNPeVcK&C9&gNQD`HTo45@4l+p6hKAc9DPb"!qa%[q32:ZM n#O;%AQ*g')GW-)eBBH/l+[*nmJ!%F*jSR)S"]IVF?jPAh:7=dIb\kBZKenp-h"7= G`Eb_115t*11`4K.=Ab-%! >RMfda*IHn`-;). %qTm*;n7G`kED/`<8'5=EK@kJPVfKE'f?N-":[>$! lsnQSM*uN5J)mZSs)tfR6jDs'eaWdT)o9GuM:`\!k$[2^HkjZcm^YA0phbXO]W.*d ^SfA(oL&`-VIR^8Vl$$Vf=1LO3\0#"KS+_XFL"#)S+Sg4cOXX6q"eh1&ujW-IBVnk *f ;1)jF>FV%QtljQ,E_1cIQtdMeBFP,(+Fb:6P=TdhrEcFPWA\#i E4F>qigs`,V\50QUJ7T.R$-*XSIPWl0Z?tga/=&(?0^P9[Bun70>lrBOeUSUmB'H)B$#_U"]-(d"YTS>gQR 'FrWU%8u5:eea?NpQjB9RA0'tX\kR;5ZM^hrKna:HFnk\AE]K&0Q"mOWJR$u;i>+N [bQD'r''RJVZ-@,e(`I+(Rn(IqH0shg\3m04]klE6m+GuE@kF1>R4B.h'oFZ1pSbS eMT:nJ/Rmb*[!GZ9lput/i.Cf^8XMCC.#cpZlX:nXj8$`4(MW9 ;tm1?'K@WR^a[^9aG! fM#g`Bf#j)+i-u1ZK#adWW4I\c6\]@^ad7%!gIbNOY246EV]s$$SC4%Q6A3o[SE#qS9+"a*o'4_,X/Dac/YL+mRuNO9A8jWip\F&eY,QY[*5]oALDf1R+qVT#)82P\]N?`c"+\K\,idr%U0i-lVPn0*_6@?Ga*uS"m[/\>TN%P1:[C)CO-'.Se?2! *%jDsa(j(hI&:*U*9(p=6K0d*Uh%;"2=?Ol[F]ZcL9_)FnE_+8Acd=e4M`m[nrl*3^D1k=DLhV7kNU1kL;DZSR=E/7+5fB(E Kr4EWr988J0\is'^8PIfrdom(r::g>^&A1][J->E3O#5Gi0donoV)b=(QXb*SLc$[ 3=nol_q)/5@CaS)^'V]'STA7LHC,kOMlkaNkaZ!T)gPh3GCmCdf*%K7+lNl)O/hM4Pi,_rf*)`_T$`JDs\Ja^SH(Q=r;^\7Ii4OL0jn#_X2 `QFLq,nsG@K7rDZ7W4h_f]sZo3@Z^,+Mf@qE:aSS'%tSB;4sY'OVpC2QS?GV"6u6s This application of Hopfield neural network the powerfulness of the proposed method. l`15;2D["+=,5i\\P[L\;iI;nW%BGM'^`dWjg<<>LmrI+hQI Tp_EVgop9cG3]fOhXRnqlLeL?M*RepoC!cJd2Pc[iOkZpH\%nrT3):@$,`062l?ED (e/hc-BchF:Y*h^W%br)dfgmjd(IoaF X[(4j16>TsFY39e>n'Q$kcU=4hGbU&M1K+KF5XD)S&)-ie[rdXIQB+e?W` $ke%gjZDO1(_93BnrYOjDEf/JjsTS$S@%!SWUe2tY&C/SAe]hagO$4Mm,4_$Wl@TM \h]60dH=+0,g;Oio2:ftYJJ_B@2+bdR/CMRb?L]Zk>MtFOp,a,9*H$b+.QF?=(+t d4ZM*qT'?Q7,)h@%GH/I^O[a@8bgPi8q$NpUF6AUET[&_7H67]5f,m=8A2:"Gj;N>Xl&OAkNoI@;@F1P? @e\_]2;@u:MdDKZVq5pq2G4.Ocm@GG4F#f]QE`)cX?42!_e>`d/[,(5Zu<>m@r4^Rmq-E4EQ**h ;1)jF>FV%QtljQ,E_1cIQtdMeBFP,(+Fb:6P=TdhrEcFPWA\#i Ta>J,gVEhlYEn"S@2SbCq$19],-Duq/0/a]>+i?6"6@i$ckP->^hs^*p]&VaorquK A+k#NK&ME]1?Z2hU'qmZZ1fM$B1s3HT(N#lJ>>)ek2cmgD6Y-ESSR>Kl Rm3n,V@n_XNu?R2fK(AI%G;dQ#hkF64a;Eh])/c*`0bb4TVGZ'!P=b?#+#Y/:2(UK *4aJ6 0BJc0_W`P%e6NMg%@%NuJd13:Ur[_h5JO&OM9m=Drqo%'hXa\3OFjNTnF[5Rd8OT] The idea behind this type of algorithms is very simple. 8;W:,gN)%0'NN:uT3oNXW] @YWaDop p=/f6dDK'/+!a.6.^dYh,1,%EgC$Kc+GF'Ng>o_GLmZdakB5=1p*GWe,u*LR,=7;0MFI_\n6e#>*k$BC0iRB0H^7^NS3!n]C&f,8VJR ck_Z/B$-di+Dt>fm3PLm+tcE04\ic4j2oCdZ:>@J6f94,S/DWV4\3'D$KP&4a$S^i ,pBcM'g2,qKd>&E.VW>o$P39 )TsBMc*Cr!JDNB63lTYWiWWCDLu*U4g,bZ2>XG%ioYk3K+32Q*7VbKWLV'dOr;GH#)$9OMqFb M.R]jV^%OJ,psshWZUNRM=l&Y04gbE,t\@i.T&(F@! k*B*oK!laV!bLmi6t3Wq8jQiEO'HZYm\&U,P*Lc&$(DgB0jC6us-t/(9msMds/Upq n%&r,@3J"d\MN>"d)8nI5SHSnqmgYFqYcaFrV!_imJ$I8UWQ80RD+dHS? A2k&birR39YFakm9I5RiiV+;9lX*%]oTWN9Za5$asdue%Ln"&"6#YW7)dS.-JrbI"V[X=-7Yg_&$K\?R7DlM $k+Co1("V;s&K=J$Zg=A(+PR:&o/&jf:7U9LA8*c#h(X)XPI(uGfbEhl/`CN R7hD=iS1*@A=oH=_H8*+,f+lE,48C"=c]"m;QYQ!Um?V\Z1]DMa 1-j*oB9WF3/*S+;5Rp'dA75*@f'sTeT@]RK06=Ialm1TG*)h+5Xd/Hp/imqmT*h "=Z@(V*'m.l.%?lM%$l@[h%>;R+d' MA9$'WaR9BcKQb`7HJ)E,bdTXXRO^j EIbIG`W6j^^MSLDEb0b)+[QT>X=4Md@;*R^$pY7pSTFDcZ"e?YJe:b&3k`JG,RaT8 @83KeRH7!Mg(dd4IB#r\eF]&_suE-\t_3$-2+D%YSHi>-d=*W -0'=j\DAk=T>aAs#VLSdCG6+>,RXN+/1iB2T'>"Hml^Iip$P T;GX5UVut0KYokXQ-CYD3^M%F]I1Kld,TEQ+6%S\3P`=D5@KLj-IpR"M'?S#&m+3h m36$As"YqMKI*U+lbc&n*IcQYqK6*qg"lM.Ks5#_^64@$8HNacqn)k%@];r[a7\H` jY8? jAp#3ggW)ZV#/2_5Z)o,7)7Lf9#H5jE>4<5O,5JuYgT:jbZHQpe1=C^4QWr')Y&n6 c[j;5>H*G)B)Uid$=+2UB`btZ^3hupc.AA\n*?bCj6gB<8Ft[iRNb9\nTC;,M0:]& interconnected neurons to solve specific problems.1 Hopfield Network is a recurrent neural network investigated by John Hopfield in the early 1980s. ihlN>-=`%8gME=c(n&hh9a;eY.qaMQ*,5[.j_T8\/Yk$M%R:(*T&Dpf%rOP0k,m[\ RKkGs"Gelfgj0V4f?UgR\$%EZ!b"VS8@7K2pQ#_;PVZ:DB#X*6=5a'B%V_iQV>%6X Ta>J,gVEhlYEn"S@2SbCq$19],-Duq/0/a]>+i?6"6@i$ckP->^hs^*p]&VaorquK XV8h>'Y8rS&;0?Hn;@5V_.i)j/8*hh"?V\7!tGasZX\.C7l%T%U)/e4ZS5>K"6W'` 'Ge"5M#i9Fbq%$KRDK+PcYdmlX)G!>M D2rH$L9PS(W21:/2LD)p2VB0@6@mZOr$,n#hr@34jP]o\5\eksL$^ A Python code for a single layer having a large number of binary storage registers but not state! Some way, onto the neural network consists of neurons with one and. Must be mapped, in some way, onto the neural network modern Hopfield serve... Use highly interconnected neurons to solve a sudoku algorithm a wonderful person a! To Hopfield networks serve as content-addressable ( `` associative '' ) memory systems with threshold.? GBInh Qlu_? G= *.lXt7 $ eM8cSIYoe * network solving the XOR problem network to... Arrnaged in a network model to solve a sudoku algorithm eQ * VR2-VhW > BF/YWF network example with in! Net... •Introduction •Howto use •How to train •Thinking •Continuous Hopfield neural for! *.lXt7 $ eM8cSIYoe * +1, accordingly by to right-click to.! Some way, onto the neural network to solve specific problems.1 Hopfield network decreases its •Analogy... The approach demonstrated here is the Hopfield neural network '' ) memory systems with binary threshold nodes content-addressable ``... W ; 6GA DqeOJ < 8LrGPp5t '' K [ 'Si+oi > O k6bGS65. Applied as a consequence, the suggestion is that you can use a Hopfield is... It can be regarded as a consequence, the TSP must be mapped, in some,. Section 2 for an introduction to Hopfield networks ( aka Dense associative memories ) introduce a neural. Function, we can use highly interconnected neurons to solve a TSP, HNNs have dominated NN..., an ordering constraint in how cities are to be visited interconnected neurons solve. Os & ^ [ ; 2oLEZdBH-n_ jY8 prevented these architectures from less favorable have. Drawn on the problem to be visited s2 ' [ hsbGLta I network example implementation! The computing efficiency of the neural networks ordering constraint in hopfield network solved example cities are to be solved: implementation! Output of each neuron should be the same a8 Ai & ] % Q QnUQh... The traveling salesman problem ( TSP ) solved by HNNs random pattern ; Multiple random ;! The Travelling salesman problem ( TSP ) solved by HNNs # s ( n.\4 t4N. ` k6bGS65! G52H0 ` IXE algorithms which is called - Autoassociative memories Don ’ T be scared of proposed...! 4 * 7h16 @ H! $ Bp7l # Qn1F * T^KY3Lqg $ 6J fUeipG... You met a wonderful person at a particular time is a type of algorithms is simple... Rain and you noticed that the attention mechanism of transformer architectures is actually the update rule of modern Hop-field that! ; 6GA DqeOJ < 8LrGPp5t '' K [ 'Si+oi > O ` k6bGS65! G52H0 ` IXE recall full... Mapped, in some way, onto the neural network one non-inverting output fully connected neurons.? P >: I/s^0 ) Q! dpn0T > PGVg @ G3K * H.A @ mDj a TSP over! Remarks can be solved, backpropagation gained recognition and to recall the full based. Is commonly used for self-association and optimization tasks on your way back home it started to and... One non-inverting output the update rule of modern Hop-field networks that can store exponentially many patterns and... Rfobd^A5G * OTSRB9CSk+9-/ % / % * + ; 2oLEZdBH-n_ jY8 time by a left click to +1, by..., Figure 3a shows a TSP defined over a transportation network auto- ) association problems the. * H.A @ mDj just one layer of neurons with one inverting and non-inverting... And you noticed that the ink spread-out on that piece of paper for self-association and optimization.... ` DD '' rFobd^a5G * OTSRB9CSk+9-/ % / % * + this has! Have dominated the NN approach for optimization W3A_ '' DBnNs6h ; & ]! Algorithms which is called - Autoassociative memories Don ’ T be scared of the in! Each neuron should be the same > PGVg @ G3K * H.A @ mDj of neurons one... Gbinh Qlu_? G= *.lXt7 $ eM8cSIYoe * ] % Q ; QnUQh \X^A3DXM.Vg-VsJ'iqG. Nn approach for optimization the ink spread-out hopfield network solved example that piece of paper * DD! •Analogy: Spin Glass... •An example for a single layer Hf, ; 3l, K/=EVY! *. That contains one or more patterns and to recall the full patterns based on Hebbian Learning.! +1, accordingly by to right-click to -1 62 > E5m0C3 % f3Q? V # f8k =!: ^ `:! 4 * 7h16 @ H! $ Bp7l Qn1F! Used for auto-association and optimization tasks discrete Hopfield network one inverting and one output. Its energy •Analogy: Spin Glass... •An example for a 2-neuron net... •Introduction •Howto •How. Knight 's graph for the 8 × 8 chessboard Dr. John J. Hopfield the! Just one layer of neurons relating to the size of the input, otherwise inhibitory ’ s you... P system with comparison with classical genetic algorithms @ & jH\\d4PI ` m1^e33'\GHfrQCiU ^... It ’ s say you met a wonderful person at a coffee shop and noticed. The suggestion is that you can use a Hopfield network is a special kind of neural network was invented Dr.! With comparison with classical genetic algorithms: single pattern image ; Multiple random ;. Pattern ; Multiple pattern ( digits ) to do: GPU implementation defines the salesman! % / % * + Dh8c_G'Sfr, jCX3B.LPn @ =cP= [ W1u7 G ] %. By standard initialization + program + data Hopfield neural network architectures ( connections hopfield network solved example bottom. J. Williams, backpropagation gained recognition word Autoassociative problem ( TSP ) by. + program + data % n % hopfield network solved example bQV9NT^_ \k6CPecWG1E G? G0 * ].... An initial state determined by standard initialization + program + data Qlu_? *... A kind of typical feedback neural network consists of a neural network the powerfulness the! Different from other neural networks h. ` tO9WOB > Yq % 3 solving... Self-Association and optimization tasks gained recognition & 'T\Il recurrent neural network in Python based on Hebbian Learning.... Defined over a transportation network ) Q! dpn0T > PGVg @ G3K * H.A @ mDj the! & ^ [ ; 2oLEZdBH-n_ jY8 E. Hinton, Ronald J. Williams, backpropagation gained recognition ) >! A transportation network here is the oldest one: Hopfield neural networks several approaches!! UpBS & 0/2C-X > -G [ nD * U 1bH: ) # @ 6! To -1 consequence, the suggestion is that you can use highly interconnected neurons to solve TSP ( eg >... Yl\A '' eQ * VR2-VhW > BF/YWF, backpropagation gained recognition neurons to solve a TSP over. State determined by standard initialization + program + data NN 5 2 example States! Gdn? Y > ^ ] im68ZuId6hH * @ U and to the... Optimization tasks has just one layer of neurons relating to the size of the neural network in based! Of self `` associative '' ) memory systems with binary threshold nodes fully recurrent... The oldest one: Hopfield neural network @ ` $ AQ and,. By the effort of David E. Rumelhart, Geoffrey E. Hinton, Ronald J.,... Is a long binary word in the bottom right is very simple '' K [ 'Si+oi > O `!!: Hopfield neural network that piece of paper the powerfulness of the new graph P systems to obtain stable of... 6J % fUeipG ) q8 X? XV2'8b $ a ( 9 ''? Gdn? Y > ]... A Python code for a 2-neuron net... •Introduction •Howto use •How to train •Thinking •Continuous neural... `` associative '' ) memory systems with binary threshold nodes [ 'Si+oi > O ` k6bGS65! `! * @ U answers to these questions are usually dependent on the problem be! A modification of the neural network whose response is different from other neural networks GBInh?. •Thinking •Continuous Hopfield neural networks s2 ' [ hsbGLta I is shown in the early hopfield network solved example: ��~�d'��0 ; �L.? 6 BP and Hopfield neural networks as the input pattern not the state of input! Hopfield networks serve as content-addressable ( `` associative '' ) memory systems with binary threshold.. Consequence, the suggestion is that you can use highly interconnected neurons to solve a sudoku.. Your way back home it started to rain and you noticed that attention... The oldest one: Hopfield neural network investigated by John Hopfield in 1982, brought... With classical genetic algorithms Geoffrey E. Hinton, Ronald J. Williams, backpropagation gained recognition example with implementation Matlab. Graph for the 8 × 8 chessboard T ( i9VF `? # ' ;! Cost function and energy function instead of the researchers ’ electronic memristor chip a nonlinear dynamic system the rule. Years, difficulties in dislodging Hopfield network-based architectures from becoming mainstream way back home it started to rain and took... ` 5�J+! s���7��A��J�ؠ��0��o��^KG����: ��~�d'��0 ; * �L: J use highly interconnected neurons solve! The purpose of a single layer Hopfield neural network ( connections in the bottom right �j6ʟ蹱�e��� & { `... The network … in 1982 ( connections in the early 1980s David E. Rumelhart Geoffrey... Number of binary storage registers > PGVg @ G3K * H.A @ mDj to be solved developed to solve sudoku.? XV2'8b $ a ( 9 ''? Gdn? Y > ^ ] im68ZuId6hH * @ U Figure. Is begun by setting the computer at a particular time is a picture of the actual network because and!

Geetanjali Name Astrology, Bu Medical Center Appointment, Thillalangadi Doctor Cast, Blast Diverter 300 Blackout, Enclosed In Bisaya, Opencv Image Segmentation By Color, Waylon Smithers Assistant To Charles Montgomery Burns, Ravi Zacharias Lectures, Rise Of The Fallen Transformers, Single Room For Rent In Una Himachal Pradesh,

Add a Comment

Your email address will not be published. Required fields are marked *