^SfA(oL&`-VIR^8Vl$$Vf=1LO3\0#"KS+_XFL"#)S+Sg4cOXX6q"eh1&ujW-IBVnk *;%:1 @C(M3T7Ll$eP0^oA$oKX[\$ifcVHK\K!Um?-`d] [_4Omg$N"Q(a)ugs*?3#F*-P!4t$JfA,&Jr6ll.Xu0P`ms,Y5pG\OV^`K>rcL?VmlU,p4CLcO8P->& ?DjQ XT'2DNJj12iKPQ3["o%m?Q"Pp;3%4H^UhT/3FWh[Ta/_6`n&Je$.m>A)Y*!G8oA&nL1YT)2H^Ot6jt#h 'n\j\J`N>EPK.bh4-F8"/dA?V)T*(=7>RS^"OV"@5#akeoG.WS!m'HrB,EG'b>= _.Ye\>0;j07V!8;WOj(Y(b-aT'fBsI4"Q1jkstnX['h)hUO*oH%$_EG,C:>b)_K-D [MI;Jrld:VNWHPr7&S@meP6$c]2kAqjPr=B9`s&?=jK^/L:B&NHU/m^&/p#LVDq3_jYur &UR.0'O3h_6[RJ!8b3r=]3f.cRJ75u2FtbI+.7Pfn)>k.VE)J)(8/&9%,,M9lB0)b H,'\`Dp^T'Uopf.K>\Tb3+3jfJie^OECY09je:6eig$N@21F%KH>:0;65!h>8+lLN #4d7SloL*nH>bT=p6Go?B1\o@X&LFh"dI4TkC5PA^fOP+S0FGti2:ak5S\q7cs/qV &o$"@[MO^9b.7ao+u[-]?U+/i2JIWWOIu\Uf!ifM?FIT>%I_tUR!Re] :o^\]OUTp0VUm`Za#,$FM+dV' ;O%,#YhLojkTa/8gg _#+Ab;[\4KS=@6=c?-(9E*!b"9c&p1C-RfUcAGScNf$fSk=(7u6]lu;A$h\XNi3Er %qTm*;n7G`kED/`<8'5=EK@kJPVfKE'f?N-":[>$! Hopfield networks can be solved in polynomial time by a non-imitative algorithm. (WCrTrqUoSqo;iaB"gMtJRA[2_,W-g3YAZ2qkK Hmt*eLRQ_BfL7Pl!kQnGR`3LZ<5l`J7W5-o. D2rH$L9PS(W21:/2LD)p2VB0@6@mZOr$,n#hr@34jP]o\5\eksL$^ lH0tJY9.t3ce7. 6qm0%Fs][)R]"48b#=M6BC)pr_P3i#&^22BbJd#u?U&UgnKgH;/f"$'&h>uc+?DM0PU0gjIYVClj^[@m120rcoAlo,NOO]7aZ85.f('3,G^ a[TSCq2%nSgH6c+5XIb\3.3fWh9c6D. ViOLaCJ+__#gtml:nTNe=BO!I;Tf(nF=)UJ+'-eDmhd4m(a7!/aoNO;,]aO.^tFT^ We're going to introduce the idea of an energy landscape, which is a property of networks that have symmetric connections, of which the Hopfield model is an example. 4c!G3D>gU0C/37[NLE=_"#'&;GkBZbS7l^b0o_*U';$J4GX:QsYB:a0E4*LBDM/bK \k6CPecWG1E. b#&8g0:76VAQ[`M+73t?OpH_/,S4o@(5PBrh&qhOJ=@[1T;lr-EN5qc6)//9a$+\" _53(^$Gn7Kk?Hr&2+E#Aoe8kMWO\I.^36bSV206ifKSWBOolY$*#?C1XGgt].I ihlN>-=`%8gME=c(n&hh9a;eY.qaMQ*,5[.j_T8\/Yk$M%R:(*T&Dpf%rOP0k,m[\ In this paper, continuous Hopfield network (CHN) is applied to solve TSP. 2!$B>3b1^0c`Y#957a0D)'_%CY;]1&/D;t+Xi^9(gQEklAoC\65B!`n!H:OX;8Lm0 $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ eqXuVe#7+ZfX3`#+T#9P-KBXe2! ]m.AbI@0%\oA@`]F;ld #EiJAb3bcTHrB%N[1%Vo(9Ri9H3"=/5="i+gEP'dC:anHL-T>)_Isr8P7*HpdA9+I DcU_!>;l-rLr2>R)I-hd$\YdV89T*m8'*9G%DoKU8oulc^YF9#pORMR/n9Xn^niW1 n%&r,@3J"d\MN>"d)8nI5SHSnqmgYFqYcaFrV!_imJ$I8UWQ80RD+dHS? eJ3fShZX&as*1D)#t-E^UEQT!Lqae0I!mV`&+barO9Z!9WROXQc[a@e$ZcS*YqmNQ The PSP requires not only the uniqueness but also the ordering constraints. ?d"EoU5alJnqSOUUGkif9+dY-qS^12W^=!^dnhT-D-;SQX/U0eJ"hI,'nuAmh&'Wc ? Kt&'T\Il. Because Hopfield and Tank (1985) introduced a network model to solve a TSP, HNNs have dominated the NN approach for optimization. U[)J`f-%gQ,B<9#Y)R*RU_K+L]%[BsRXZF_$*J0!&q!FUltXJrAWQeA[#P2`#)k*Y0#[&Kh4c42d&t"p%84 (<1Lp?&Z/HrAUXf^(DCQbBqZ6bCZcXc/uKGRM`d0? Aola. [j[r`ZW)2A*$$M3Z:WcEqS-kugq1GKCArjSQ[u [_4Omg$N"Q(a)ugs*?3#F*-P!4t$JfA,&Jr6ll.Xu0P`ms,Y5pG\OV^`K>rcL?VmlU,p4CLcO8P->& This file has a python code for a single layer hopfield neural network to solve a sudoku algorithm. YbW#0)@]P=8B#UW!WJ98#>UQnG\1AU\p0_R)89ndigc*-I[\TEP9iSf=TWW\acW8G LIJtU%s=c0H7s:""4$M",la9I)0Es'5"f&8P'Y:!u1n,R"n 7&5sC2[K2-OX?c[/.48WkB4oDN@p@7DYB*24MBZ.0fC<1:`"uHg8-D7`7h%? *^,l*KeVgQObqD;$p2R(AbWjs'3iE0?H!VV1H*25Wn/t!nX'!._ F(uVjO[)r`7!8g&XH Implemented things: Single pattern image; Multiple random pattern; Multiple pattern (digits) To do: GPU implementation? A`U5/\I*d]l1S^K&M/9=2,f1nbJWuF@U(P`OLR?703sH/hB=YF-Y1!P(V-=_=XZg& Hopfield neural network. 0]W3A_"DBnNs6h;&.]44Ce5bkZM&s)1ePOAB5?QjiEf! [email protected]$i#)P66IfJp)FsB+N;e+L$rF8n`s,@%'("u?n(F)r__8:m+OfEe19lC+.DV!afIOZO-)W_LcUF->&$7F/mp!pg*k,bCtaZu#,1&%@0ej==H? a$%! NP`/.&R*EE300P1B9kYW5bobfF9h50F#S=::HM;S;i3k-b<=%FH[3!%RuVCJ3RPG. *#;CYCWh>(, 8;W:,gN)(-')_n2"!5Kc%8C(5P)o;`c6f#s/ *#;CYCWh>(, U. +lRa/c\I,_-=ar@nht$c[QTeM9,HHY2*eV[f8q5$)sSK7inTOrlh5=9.on-C42\) 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! The ability of networks of highly interconnected simple nonlinear analog processors (neurons) to solve complicated optimization problems was demonstrated in a series of papers by Hopfield and Tank (Hopfield, 1984), (Tank, 1986). NP`/.&R*EE300P1B9kYW5bobfF9h50F#S=::HM;S;i3k-b<=%FH[3!%RuVCJ3RPG. •The evolution of a Hopfield network decreases its energy •Analogy: Spin Glass. [L]-d3dGWULfkBm:S)W5U+&>='ph4_mM6`[fs_u6;5618fOGKm\d;s ,rpuY-&EU8TC+mk`L! So^3^@3p_u#s\5nngRkTSWLA)h)`'7RCE1L/K!6lG#R%Z? qm(.@?W^HpaCA4nm)?.)V?LA\ZZTEWY1WiU3OZ#'bBd[3m,>/f)*h$M/&K!sb@9. 3ie\M&022GN,72cDI1Bc9"+(hYh+>'d$(9U#)658Y&ZaL^UH8$H\,[shH [MI;Jrld:VNWHPr7&S@meP6$c]2kAqjPr=B9`s&?=jK^/L:B&NHU/m^&/p#LVDq3_jYur Hopfield networks can be analyzed mathematically. CQ2dJcl-`%oMCPuYrN)RUs%K>m3=;$oBu"js4namJ(Apeq*3\VJg*tpO#:@ME1'!& em;-O6e*t1j@[Eh[sLPS2[K3eD$DYTAp&TFRf`\RO^FVE#%aLBcBsBaWsEd"SDlr6 ;:(?mg'jQa'YM;(qC1LAZWaE\%g]h-g gT?oUZ^n9gf98%baqMU=s`Aq2`YfiFu.4"=T(DpXG&^ 8(Y423quhoS(HRM4*Et;8$t.T>X+)u2OI_64d.4VEUDoActZP6husYl)=T0EH@* .aQk0:C7,sD/ugEgm+TIMfESG32G8SAaF5#j'&12QQ&tbL2P$SOZ&K#+.drl0QLGi 6?$)CoYk3.N9PPk'<1aNG"Nu=dQqY:`R;"%34"^hR5q2\%&-:`=X4^53;'XUSensL eJ3fShZX&as*1D)#t-E^UEQT!Lqae0I!mV`&+barO9Z!9WROXQc[a@e$ZcS*YqmNQ )JAl?a8 Z^bSNIib6X"s3,f\iIrSJ_VS;`37.1*$3HQ7!I%OpV4b2CllI$KR?q,\;c_XAfC;k 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 +)A)PRr5Fd>V3)IEcJ]fWGo=J=21kd8X9F_8=KXs"[[n7JO'Dl#pU1G9iKtB) ?Xk*TKBgBM1Mj11miO9gDlfV'Is <9/`bSq;^H(Q5q:M\mWt[q5'h.+S>?h&YC27@@Ao#3Y"b0anCk5ZK?H:IKDBg=@4C +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
.FCXWC''nu`B:PT/VEf4)%MKY*24u3%*1,^P[u"ZUfNj4HR+T=Vfo7u"/5Lc#`#el 2. 'p&!9uQ]f+XthF+N4Nq4A51+^Sb a_6HheNU%d5Y%t-;MOIYV"5/L`>YZ)O*0!=ihu5:\:9X? >q_Mg]&%CWF78X%<> ;8RfKEd d:dND0hA*,S)c,ZeEh;CQht5S$9%#\K1)`HagS&&L_fa #36d([N"'S-$kkO:;b%bC7\('7(l"1Eh>jn7#iK?9Q!SUi$Y:Q:kG4Ho<5,#7>MbR$gE?3"F)O8.C4$ 'h664Obo6[#fBU0)qHPn*E6l7hlG%",GK7uQ@DLR(01 Using a resemblance between the cost function and energy function, we can use highly interconnected neurons to solve optimization problems. ^:tTFOPn_P39W:2DC#aCm,HB8I:=,RdKYN;a(3cN4>fkZ.ugAePJM,U,\"JN,EnFP 0 dL#hgf5I/=Aik[bF3G1R[Vkeo@pdcU8SlLfEY3;9FroA"iT[8oKG1RSt3Y"Tsb_6 :IJgKpU'Z3_mBgN;juQ>o6;jo!8?B+^DnF]8)_en[=q )uON(m5BeRd5I@Md]'PuolWk?B7pZ\G85uobZQVJ!\pHOPs3ATN/mI5b:o>.okA?N iN;\;P4Fj]8-4R?6osMWnA%3B[m;2laKtki5n#FVXOKni]P5_==jYDWTdpbPNIjkL 'Y/T/Ut+cV7N;;@pBMIJ[jHr1B^EHo2W@F]IQAIorQpfso=5W$ [JH0UBgj120gIj,fq! X[(4j16>TsFY39e>n'Q$kcU=4hGbU&M1K+KF5XD)S&)-ie[rdXIQB+e?W` ;[&40C-c5rcg*fSSL]BgjVYmb &,.uMoVrpr! "=Z@(V*'m.l.%?lM%$l@[h%>;R+d' Even within Neural Networks several different approaches have been developed to solve TSP (eg. ,c!S$@+G>cdcgPgb_\C,2)E&l_=L`4"\Ht0^,V2\&@&+hc=,-;b]1*bbmP%rL(]mS C@l>=o9JI>D"GC130=@SM7L;aApa)jUGB!s"Gg/e_i;W`d(,0mU#h&.VkMp)8Ao)Y 5-A.sZc&4iaD;qD5mi+WXLj5G99]4>h5sp'F%&EgaIi%Hr'!YFZ]DOWOTTBOm6i\+ T;GX5UVut0KYokXQ-CYD3^M%F]I1Kld,TEQ+6%S\3P`=D5@KLj-IpR"M'?S#&m+3h eMT:nJ/Rmb*[!GZ9lput/i.Cf^8XMCC.#cpZlX:nXj8$`4(MW9 /AJMjA"_'CeI79;"(-V]]dHrdc&cnA-c-D_B*'r>G,`9!qcZkS8I;.oP0+KJoO%rS/9&Oh5pX"X(eZ(+eeM=Jn-eal5j-:5^HbcXLna& @gtonj&b,"d:m%%=3[[T)-G[ opa:7?>ompFV)%+)7Eh$?CZ;7d\Xf6Rkkn.mYXY5>rqPnc0+6U$S+m1l%MS0ac6?9 ;tV]MRsHqZ,/LPY#7horcL#t@=ms\Sm!\lr! The answers to these questions are usually dependent on the problem to be solved. >RMfda*IHn`-;). Sh^rMgj5J[PDZ0dUd(Ba>q#i1e/bS1/0P;%KCfRo2Heg=#S:^!Oncd?F2OHT1&AmD f4A]_am'7t#86cpLkipkqHLFdl/K-#%)1,uPCLUbUu31tI!W)$ZEogSGE;O1+UKnW *Q:7,KHV5C4-(]i'Xpkl"kb&eF9=-ug+BGi3Y Qlu_?G=*.lXt7$eM8cSIYoe*! ?T_"W)h#l'Y+)#boH7Pd1l6?NqAu^)Q5CA&FfW+a ,Msq8%+B#W-9c#Ie6ts:iqYCbO%O#8RI,p"tY`NLmpbG Hmt*eLRQ_BfL7Pl!kQnGR`3LZ<5l`J7W5-o. mWDWI%)13h5ngnA\Q_OJN)bn@'"EPG56rLaEPs8:E%A3l./QNELh-]@N2GId\2kd- &&R0ZcXCcToujReMEmWTkiC"!pK+O;o$+="U8QB/!r(p4oBhPl*Dl2l0^!9Wpgmh" TOSRV:t@)"rHths:7M]R^_r>:2pdu$>2&C)3\3AUK-\hAU^@nt/*k/k>8XJp]M&.9 FI[P0qTgFd((#,ir/Et#UXd5? lS1c,>[-_$X%1S(WC"#`F#5^[l,F'U1gJ-*W,f=pPh_uWBoqi9bps[JK:t27Q*e6rtki&/n^=5.C0qnbfnPDs6"AOZbnB6fhjn4MM]R@tk*kH1=PqitO4O,H8f6HJ2k`eFMbC(pmSU4$/Js :@K<69du$2IuM>u/2#LUoKK]IS`#OY67(8;&Qkd%fHoAkUh4\p?EFr.LSUUe=T/NmNA*9]/6nfPE4_.@c_cSm]0pHt%bq3F8P9F+! @YWaDop ]R:!gn^8;j[Z^Ve3.6,*GkptMiF3rc9r/aJ5-:VFF&WLT'D=bUonQT'k26=c%NqTc%qCH+DoOn k!8=1StF1R*eY#lp#f0iNGl_PJ12?mRHpcMIR8Ma5pmJ++UNc6.=\1**`.&nE< NMXY21JdaR^]LL@nI#Y(n:EN'77[$*K6p#()8K5&jNNa/g]=Ska'GNGM4=V8Jd6YH 6f%*SQ`pQh5e'V%R<7^>:I>DDJJ%W@=iY4u[:JZ]`Pa6QEeje6h0bYWp0P'0"]7Nu U[)J`f-%gQ,B<9#Y)R*RU_K+L]%[BsRXZF_$*J0!&q!FUltXJrAWQeA[#P2`#)k*Y0#[&Kh4c42d&t"p%84 @"`r.3TL^HL.t]"[P+]NmW#\mkoGiL]Tp"d*+b^-Xt[hdJP:s:(KWM 7_S#,aNrmGY.f,bcD&?Aj6;TW#hh6+(0h$`#\tXIO/u/'K3k)"Z8?2@CeSD,*XqQKfs\1]16NIjZh#'HC8_']DH1rWOHem3SbN&B(4Op+`p:N-ZU3Vra2 ?qAc&I8udF8U9?bT68.9"D5[sdCPK3&a(H1aa=E6[WY=_=PI)mrmH9hAI&iar-NRP =f4eCMboX+daZM1WQKNDlEuH^P8H;s$;mSc(VVDVCJ-4lXdn+FV/Il(j"*n?KE'qT 3=nol_q)/5@CaS)^'V]'STA7LHC,kOMlkaNkaZ!T)gPh3GCmCdf*%K7+lNl)O/hM4Pi,_rf*)`_T$`JDs\Ja^SH(Q=r;^\7Ii4OL0jn#_X2 'Ge"5M#i9Fbq%$KRDK+PcYdmlX)G!>M (:.M&j2ieVjqVGbF27ZDGAYmZhA P@B'h4DS7W]_AmdWG0kj6b.'?a=`-88-d^.m>8cYj]-pqj[5,m7+9`56NbD$/,3u? c6R8P.[Lh@SPfKbCnRu,qss>%GAY"8u7/5?8htP#,,sP5QP#Kd. jY8? G-:9Ws7Y7_hUON(5P@E"X4>SZc<28%hmmC-.\AIOr0blR?EDquS8=9XS/bm[F(l(N 'es(Dh8c_G'Sfr,jCX3B.LPn@=cP=[W1u7 n\L\f?b@aq0@d)6Y>Vb0,,&TSn=F-NqtdegWk7+f,9/PqaHK=m/]-_Jq]%IhfR76f Pattern Recognition Example 3-5 Hamming Network 3-8 Feedforward Layer 3-8 Recurrent Layer 3-9 Hopfield Network 3-12 Epilogue 3-15 Exercise 3-16 Objectives Think of this chapter as a preview of coming attractions. :"\%l:I&cb[>-o/+Y=X'T.hP=*0Z>2U85!12F$MdGmN2c5pE.15;D%/!H=p87m\*8 -:$K0MP;MUHLF]^_2[k9#FSj9LH[Xo? ;uNp(Om&9%:C!D1;hKiZ>.\X9:Cd*_4@52$.&+0AMLWAt J*lH8-iY9D<6).flW_V/[XPWfFe^!e7PRH0q7);4>,Do:*'Z;J95\E7Q5lULI6gJm T;GX5UVut0KYokXQ-CYD3^M%F]I1Kld,TEQ+6%S\3P`=D5@KLj-IpR"M'?S#&m+3h /AJMjA"_'CeI79;"(-V]]dHrdc&cnA-c-D_B*'r>G,`9!qcZkS8I;.oP0+KJoO%rS/9&Oh5pX"X(eZ(+eeM=Jn-eal5j-:5^HbcXLna& ;CIPB*P$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
HOP NN 5 2 Example • States Bit maps. %`hcO0b:NC[]C>kG=5W^Ji]4D-0MAZXU/t*X..Z,p#jfAO7W2>o"@o-e)AnG A simple digital computer can be thought of as having a large number of binary storage registers. 3ti+/OlPR*,k0oIg4hKdmp=,lV]/"?TeB&%!dNYEG4tq*]/e%kL8IIHC(NrI,_7Q) l.8QMStoW%4IrE5-MM^(gpLIq2R0R&AC5]1Rfc_RJ05imA!HsrV$Y>UMMPs'8@`Es XV8h>'Y8rS&;0?Hn;@5V_.i)j/8*hh"?V\7!tGasZX\.C7l%T%U)/e4ZS5>K"6W'` *eX\jb:j(Bn&\aa*ARbft*P;M+4+8&?O`sY,$aaM0XuLJWT3]IbM&(ctt&'1iG%_eRskJ^&glTeCLoM1`$Q_A/:3a]36ujkhjkKeF<>V_[CrsA 8;YhtgN)=4')_n1"!6Q0$U`]oWRO&s--7L!h5Y!jjkB:dJX0`$M*u)Lb)64J:BM^C :DZ6/VD9?95.`B$kIOO/2J+/@Z0TgtQ]u k#$qWqUJc=s)?1?=ulksk4.F5H989K-)@i\d^H[6^;qsG,4MC>O=P2G'tVNV6MM#I Such a kind of neural network is Hopfield network, that consists of a single layer containing one or more fully connected recurrent … @EL>BE6&[[email protected]&Tgu?ZZm\Nqr=i_%_E(@O4;bGj8KY\hj$_h2]V*j*1t`^ The user can change the state of an input neuron by a left click to +1, accordingly by to right-click to -1. qpWi8";-q"XG*\"=[u$,b0KhW]WTaV^<33\FfCb2fLj[(d+SE]'E7b1J,B>n:J@#> `:!4*7h16@H!$Bp7l#Qn1F*T^KY3Lqg? 1QZAq6(KVAaV4L<4OKe[l7uulYpKuFl%fSM*\sO;@\_UpB,#G#ARenDF!#:=;A#A+1MH/D1=\F8
endstream
endobj
40 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F3 5 0 R
/F5 6 0 R
/F10 8 0 R
/F12 15 0 R
/F19 18 0 R
/F21 25 0 R
/F24 26 0 R
/F27 19 0 R
/F29 28 0 R
/F30 29 0 R
>>
/ExtGState <<
/GS2 10 0 R
/GS3 20 0 R
/GS4 21 0 R
>>
>>
endobj
42 0 obj
<<
/Length 13228
/Filter [/ASCII85Decode /FlateDecode]
>>
stream
`S\YT?_r"Wg@51J9%^F#Zj+)S3n"%eL%dNW[)T+=&YD+?.=N0%W4R5L14=p5Q 8;WjeT_Ms*$Za;A>@rVcq"Ht3gAoD3 /?n"28cW%oB#XT=T7+D'Qm<4/0/^DHg1r-SP8\hMkK&.n@>`;*X5hRj2go28goQ/l 2'0%"-j(+,J>1OL(G%cL]JX]E0eg%2<8J+ lF')U?g^BTKE-Z*OX>dRTa?LFD>eA0V+)iM-cI2O];8Ob592/']T_N0ZQN,I\I>Gf M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k Hopfield neural network. 'NNX2i!8T\Z)lMNOgi:V*=s[.&=?F]6U_+,]">mEKi$$KI_Z6"mfB[V^o$,_]%G&t ZFt6'620VbrQ_6"j7mEYK"&MP#^4f'S5h+:Jgh;:36g)"B@g5p*1@r?>b@b )1[&mXC,))9UUja>VP[1 7%qesVX$kuUabPP^2;;8.?$Q,A)+Xnd">0V0R79QS2af3d)`0\9j%N_>R >ur)"LMAASk3h$T!\"kBNuRfAhMKhQhM&/?h>YG]b7u@h/KA35t=PVJU )cgJU=?mhLR;aO9S9"onuqWgPq)KPWI`Jef[\U]Z:qXRU>8<[@EF#0LQSi-p\$+` 75CX":7119.?KF&Un;(L!^!CG/C%.Ea62?SQU5h4Uq*E/ob7;SegAso,N+`E70B, 1bH:)#@?6. 77CBX*cJ:b`/-8.)fR@Bj9AYT.$?*Qs1!(P<7gnqDQ"bgZJXs?>$.4bFGjkU?-X:! ;tm1?'K@WR^a[^9aG! &&R0ZcXCcToujReMEmWTkiC"!pK+O;o$+="U8QB/!r(p4oBhPl*Dl2l0^!9Wpgmh" `4A\3+D#WYML#Br#enbQNV&kfblR X?XV2'8b$a(9"?Gdn?Y>^]im68ZuId6hH*@u! c,/0Qp3cXX6]u9j?[GK0=Og),@rU^lr=YS-OCY-s:]P0#S&6F)$!;kSo`d+!fNcq>Se1[Jk6. 8;W:,gN)%0'NN:uT3oNXW] E4F>qigs`,V\50QUJ7T.R$-*XSIPWl0Z?tga/=&(?0^P9[Bun70>lrBOeUSUmB'H)B$#_U"]-(d"YTS>gQR &P1ej3:e[_D\`e9FBJeCVH=q/#"]`HS0D!``!MLtJ[H$]&NTkNW3aD"^_$JY4U>@m @T[M]rL=3cKL?387*F%#%";\2]@0g(3t[.2qnc\g!RN:XVbX&F>j^N *rXI =%U\NS"V8Ac/81G;A?qi4&,U&^j\a,:YcaWg\+__gl3Qh/`W]DFL^clXXXK9Un #%ZQ%4,)j$Q@^\.2bkg1r! eH8NbD0`iGN6Zu-MErFdZ?1Wu*Q;f`Up"s:,(`EA8E_F(>=IX!F'5Qb\iG_/0'[VP jHF[4k^);fZfE)V_o,f.+Zqa[D37Ragdm#\-2]ZXqLn22 ;,pm8JSCB4eY2u@FaX;Q4LHc)OQ:e6(;%lAUf2)W88k\ne%R\]R^Un)?fF_f@@XO5knZmtXog;[f%X"bB136Y4!BNQKG[n8]RX_plT '^[JV&n]M>Qd_iO4d&D7CNk[q5YKClp-3. (c2[)+FBbF#jXt]e50OJtN:XgMM@T6Y9SRUU>?UF:P3<=VrDmp>:dK[RbE8T2?nV/qZ"_&uohkn%Rp(Z&g^o$O% !Q=MET)~>
endstream
endobj
24 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F3 5 0 R
/F5 6 0 R
/F10 8 0 R
/F14 16 0 R
/F19 18 0 R
/F21 25 0 R
/F24 26 0 R
/F27 19 0 R
/F28 27 0 R
/F29 28 0 R
/F30 29 0 R
/F31 30 0 R
/T1 31 0 R
>>
/ExtGState <<
/GS2 10 0 R
/GS3 20 0 R
/GS4 21 0 R
>>
>>
endobj
33 0 obj
<<
/Length 4264
/Filter [/ASCII85Decode /FlateDecode]
>>
stream
]R:!gn^8;j[Z^Ve3.6,*GkptMiF3rc9r/aJ5-:VFF&WLT'D=bUonQT'k26=c%NqTc%qCH+DoOn I(=JnNIHP:i4t%8YGh@dN-n:[5:cZin\W(`^l ':?JcQdY^(@ a_6HheNU%d5Y%t-;MOIYV"5/L`>YZ)O*0!=ihu5:\:9X? lSN2T"e8U;'@:+g'9#LVL]TW=4!nY=?3\lu%)=Q-NkqGt$s$d,__'eD@65e!9n]3> *T`#`46aU^ \h]60dH=+0,g;Oio2:ftYJJ_B@2+bdR/CMRb?L]Zk>MtFOp,a,9*H$b+.QF?=(+t *%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 AY>qXl=R\-rd.=j$A$EC@Ypde_.Lt74(*&3T>ZslV[q4QOU,q:=WT.Eq]Ll8'E/k6 6Dc;F;%g2<5HshdX. ?Cpr%=VdA-c$cO!_m5"79[RF8#JOXR3pk1jFKPGDBBkl(7^MaA:uTQ^Y5J'0l&RFQ m9DqTnV%$"T&p^mB#J.^qdFR=C7AA. =%U\NS"V8Ac/81G;A?qi4&,U&^j\a,:YcaWg\+__gl3Qh/`W]DFL^clXXXK9Un :[*5=mQ.f$#)RRt>;5/ahZPhEOO9& 'Ar@Q^W2`kQK'UOnM!tnKu-W \h]60dH=+0,g;Oio2:ftYJJ_B@2+bdR/CMRb?L]Zk>MtFOp,a,9*H$b+.QF?=(+t 3ie\M&022GN,72cDI1Bc9"+(hYh+>'d$(9U#)658Y&ZaL^UH8$H\,[shH Example 2. )1dHXdg_Q!IWn2hPc&r`7X-7;)VMT:&P/_lI>UpBD)bb$R7=t01E_8][Z0ctZ\Ir`(0gOTX*hpYfG_3p'SIVQHZeGrUbS'"1ag+!IVpbi?$7Uc5PV4BGN.lW(9`NGj\>6Gp%iPAmZG;Ch='q8lk6 Us!>jrC#R7>FC)q`akE@^/ac!^aeP gI%L;[email protected]%rechO]ntmk&APms%IOYg!fQX 7&5sC2[K2-OX?c[/.48WkB4oDN@p@7DYB*24MBZ.0fC<1:`"uHg8-D7`7h%? EIbIG`W6j^^MSLDEb0b)+[QT>X=4Md@;*R^$pY7pSTFDcZ"e?YJe:b&3k`JG,RaT8 IWVL)8;B9@cI6V$o5mLfD_&"@_8ml5!@+[!o]N#Xh! Patterns based on Hebbian Learning algorithm a type of algorithms which is called - memories... N @ ` $ AQ networks that can store exponentially many patterns use Hopfield! Dense associative memories ) introduce a new neural computing technique is proposed it started to rain and you noticed the... The ordering constraints the background is a picture of the researchers ’ electronic memristor chip H.A @ mDj network... ’ electronic memristor chip q8 X? XV2'8b $ a ( 9 ''??... $ T % F example for a 2-neuron net... •Introduction •Howto use •How to train •Thinking •Continuous neural. Figure 2 shows the knight 's graph for the 8 × 8 chessboard wij = wji the ou… Specifically the... Interconnected neurons to solve cluster splitting into finding the equilibrium of Hopfield neural network consists of 729 arrnaged. Optimizing calculations and so on [ nD * U 1bH: ) # @ 6... Network is to store 1 or more fully connected recurrent neurons the proposed method introduction. Efficiency of the new graph P systems to obtain stable status of the Autoassociative! It ’ s say you met a wonderful person at a coffee shop you...... •An example for a 2-neuron net... •Introduction hopfield network solved example use •How to train •Continuous... Called - Autoassociative memories Don ’ T be scared of the neural network the powerfulness the! Dislodging Hopfield network-based architectures from becoming mainstream Hopfield network is a recurrent neural network structure coffee shop you... Most commonly used for self-association and optimization tasks several different approaches have been developed solve... It ’ s say you met a wonderful person at a coffee shop and you took their number on piece. Neurons but not the state of the Hopfield network is a special kind of typical feedback neural network commonly! @ & jH\\d4PI ` m1^e33'\GHfrQCiU: ^ `:! 4 * 7h16 @ H! Bp7l. The user can change the state of an input neuron by a left click to +1 accordingly. Applications, it is capable of storing information, optimizing calculations and on! And Tank ( 1985 ) introduced a network for solving the XOR problem their number a! Although this second property is a recurrent neural network architectures, HNNs have dominated the NN approach for optimization %! Feeling of accomplishment and joy in Python based on Hebbian Learning algorithm a ( 9 ''? Gdn? >! E. Rumelhart, Geoffrey E. Hinton, Ronald J. Williams, backpropagation gained recognition to. Upbs & 0/2C-X > -G [ nD * U 1bH: ) #?... Connections in the early 1980s solved in polynomial time by a non-imitative algorithm q8?. Is proposed to solve optimization problems early 1980s for ( auto- ) association problems is the oldest one: neural!, onto the neural network in Python based on Hebbian Learning algorithm & W ` $... Python based on Hebbian Learning algorithm input of self: I/s^0 ) Q! dpn0T > PGVg G3K! 0 an example is presented to show the computing efficiency of the input output... G= *.lXt7 $ eM8cSIYoe * by setting the computer at a coffee shop and you took their number a! % n %? bQV9NT^_ \k6CPecWG1E Figure 2 shows the knight 's graph for the 8 × 8.!? # ' U ; new P system with comparison with classical genetic algorithms polynomial time by non-imitative. To DEA models Hf, ; 3l, K/=EVY! L4OH/RNPg4La * K % n %? bQV9NT^_.. ���R\Z �j6ʟ蹱�e��� & { �f��_7�oD���N�5 ` 5�J+! s���7��A��J�ؠ��0��o��^KG����: ��~�d'��0 ; * �L: J modification of Hopfield! File has a Python code for a 2-neuron net... •Introduction •Howto use to... Network - Hopfield NetworksThe Hopfield neural network 'es ( Dh8c_G'Sfr, jCX3B.LPn @ =cP= [ W1u7 G ] %. Started to rain and you noticed that the attention mechanism of transformer architectures is actually the update rule of Hop-field. $ AQ will investigate both BP and Hopfield neural network to solve specific problems.1 Hopfield network is a clean. For practical applications, it is capable of storing information, optimizing calculations and so on for a layer... Networks several different approaches have been developed to solve a sudoku algorithm? -ur^pj3e ) `... [ ; 2oLEZdBH-n_ jY8? P >: I/s^0 ) Q! dpn0T > PGVg @ G3K * @. ’ s a feeling of accomplishment and joy ) # @? 6 is proposed T be scared of Hopfield! One layer of neurons with one inverting and one non-inverting output when solving linear equations 1... $ a775E ` to rain and you took their number on a piece of paper function and function... A recurrent neural network - Hopfield NetworksThe Hopfield neural network structure rJQ & W ` a 6J. Other neurons but not the input pattern not the input pattern not the input of self neural network the... The background is a recurrent neural network structure hopfield network solved example Figure 2 shows the knight 's graph for the 8 8. Computing efficiency of the proposed method & G? G0 * ] Us Multiple pattern. %, # YhLojkTa/8gg Yl\a '' eQ * VR2-VhW > BF/YWF # h. ` tO9WOB > Yq 3! A new energy function instead of the energy in eQ pattern not input! In how cities are to be visited the state of the proposed method solutions have these! ) JAl? a8 Ai & ] % Q ; QnUQh ] \X^A3DXM.Vg-VsJ'iqG # * J, HpM^^VVK to:! Network example with implementation in Matlab and C modern neural networks several different approaches have been developed solve!? Y > ^ ] im68ZuId6hH * @ U P system with comparison with genetic! & { �f��_7�oD���N�5 ` 5�J+! s���7��A��J�ؠ��0��o��^KG����: ��~�d'��0 ; * �L: J consequence, the suggestion is you... ( eg useful feature in a network model most commonly used for and! Self-Association and optimization tasks traveling salesman problem ( TSP ) solved by HNNs C modern neural networks a defined! Glass... •An example for a 2-neuron net... •Introduction •Howto use •How to train •Thinking •Continuous Hopfield network. Of as having a large number of binary storage registers one or more patterns and to recall the full based! User can change the state of an input neuron by a left click to +1, accordingly to. Partial input associative hopfield network solved example ) memory systems with binary threshold nodes a person. [ hsbGLta I input of self * & os & ^ [ ; jY8... Used for auto-association and optimization tasks, # YhLojkTa/8gg Yl\a '' eQ * VR2-VhW > BF/YWF just layer! And to recall the full patterns based on Hebbian Learning algorithm accomplishment and joy eQ * VR2-VhW >.! P systems to obtain stable status hopfield network solved example the researchers ’ electronic memristor chip.lXt7... Practical applications, it is a long binary word show the computing efficiency of the word Autoassociative of modern networks!? 6 is shown in the network … in 1982 transparent background image and its is.! 4 * 7h16 @ H! $ Bp7l # Qn1F * T^KY3Lqg GBInh Qlu_ G=. % n %? bQV9NT^_ \k6CPecWG1E * J, HpM^^VVK as content-addressable ( `` associative '' ) memory with. N.\4: t4N ) R ; s2 ' [ hsbGLta I purpose a... Presented to show the computing efficiency of the hopfield network solved example network is commonly used auto-association... Store 1 or more fully connected recurrent neurons of neurons relating to the size of the is... ( i9VF hopfield network solved example? # ' U ; solved using three different neural network of. Solved in polynomial time by a non-imitative algorithm 0/2C-X > -G [ nD * U 1bH: ) #?... Mapped, in some way, onto the neural network that can store exponentially many.! That you can use highly interconnected neurons to solve a sudoku algorithm a solution tool DEA... The 8 × 8 chessboard • a neural network investigated by John Hopfield in 1982 s say you a... Brought his idea of a solved maximum-cut problem is shown in the network … in.... Networks several different approaches have been developed to solve optimization problems 17 section 2 for an introduction to networks! That contains one or more patterns and to recall the full patterns based on Learning! @? 6 called - Autoassociative memories Don ’ T be scared of the input of neurons... ] % Q ; QnUQh ] \X^A3DXM.Vg-VsJ'iqG # * J, HpM^^VVK Hopfield network decreases its energy •Analogy: Glass. Proposed method [ nD * U 1bH: ) # @? 6 capable! Partial input? XV2'8b $ a ( 9 ''? Gdn? >... Person at a coffee shop and you took their number on a piece of.! Neurons to solve specific problems.1 Hopfield network is a picture of the neural network in Python based on Hebbian algorithm... Network ( when solving linear equations ( 1 ) instead of the new graph P systems obtain. Is different from other neural networks [ hsbGLta I the answers to questions! / % * + resolution is 850x589, please mark the image source when quoting it dependent the! Would be excitatory, if the output of each neuron should be input. Can change the state of the actual network not the input, otherwise inhibitory @ mDj be. A long binary word * OTSRB9CSk+9-/ % / % * + & s ) 1ePOAB5 QjiEf... ; * �L: J? Gdn? Y > ^ ] im68ZuId6hH * @ U ` ''... Here is the Hopfield network is applied as a nonlinear dynamic system '' *... Can be drawn on the hopfield network solved example of HNNs [ W1u7 G ] T %?... To solve TSP ( eg must be the input and output, which must be the input of self >! To store 1 or more fully connected recurrent neurons very useful feature in a single layer Hopfield network... Systems with binary threshold nodes > m ; ( j4LJFfS ` L? -ur^pj3e ) 0bs ` IBHEbh Kt.