IMS
-cnm Ins i u o de
Mic oelec ónica
de Se illa
Ha dwa e Demons a o s o
Oscilla o y Neu al Ne wo ks
A Ve sa ile App oach o B ain-Inspi ed Compu ing,
om Embedded Sys ems o Phase-T ansi ion Ma e ials
Manuel Jiménez T a és
2024
Ph.D. Disse a ion
Ha dwa e Demons a o s o
Oscilla o y Neu al Ne wo ks:
A Ve sa ile App oach o B ain-Inspi ed Compu ing,
om Embedded Sys ems o Phase-T ansi ion Ma e ials
Ph.D. Disse a ion by
Manuel Jiménez T a és
o he deg ee o Doc o o Philosophy by he Uni e sidad de Se illa
wi hin he Doc o a e P og am o Sciences and Physical Technologies
Supe iso s:
D . Ma ía José A edillo de Juan
D . Juan Núñez Ma ínez
Tu o : D . Be nabé Lina es Ba anco
Se illa, 2024
Ins i u o de Mic oelec ónica de Se illa, IMSE-CNM (CSIC, Uni e sidad de Se illa)
C Amé ico Vespucio 28, 41092, Se illa, España
Depa amen o de Elec ónica y Elec omagne ismo
Demos ado es en Ha dwa e de
Redes Neu onales Oscila o ias:
Un En oque Ve sá il a la Compu ación Neu o-Inspi ada,
desde Sis emas Embebidos a Ma e iales de T ansición de Fase
Tesis Doc o al po
Manuel Jiménez T a és
pa a op a al í ulo de Doc o en Filoso ía po la Uni e sidad de Se illa
den o del P og ama de Doc o ado de Ciencias y Tecnologías Físicas
Di ec o es:
D . Ma ía José A edillo de Juan
D . Juan Núñez Ma ínez
Tu o : D . Be nabé Lina es Ba anco
Se illa, 2024
A mi amilia, an o la de sang e como la o jada en el camino.
A mi mad e y, en especial, a la memo ia de mi pad e.
ii
Acknowledgmen s
This Ph.D. Disse a ion has been de eloped a Ins i u o de Mic oelec ónica de Se illa, Cen o
Nacional de Mic oelec ónica (IMSE-CNM, CSIC, Uni e sidad de Se illa). I has been suppo ed
unde he amewo k o he Eu opean H2020 P ojec "Neu ONN: Two-Dimensional Oscilla o y
Neu al Ne wo ks o Ene gy E icien Neu omo phic Compu ing," (G an Numbe : 871501). Also,
i has been pa ially suppo ed by PEROSPIKER (Ho izon Eu ope ERC-2022-ADG - G an ID:
101097688), Conseje ía de Uni e sidad, In es igación e Inno ación de la Jun a de Andalucía
(BioVEO - Jun a de Andalucía PAIDI 2020 - P oyExcel_00060), and by Conseje ía de
T ans o mación Económica, Indus ia, Conocimien o y Uni e sidades de la Jun a de Andalucía,
den o del P og ama Ope a i o FEDER 2014-2020 (P ojec US-1380876). A 3-mon h secondmen ,
unded by Neu ONN and supe ised by D . Sieg ied Ka g, has been ca ied ou a IBM Resea ch
- Zu ich, in collabo a ion wi h he Science o Quan um and In o ma ion Technology depa men .
Fi s o all, I would like o exp ess my immense g a i ude o my hesis supe iso s, Ma ía José
A edillo and Juan Núñez, o hei cons an guidance, pa ience and encou agemen . The e is no
doub ha hey ha e con ibu ed signi ican ly o my g ow h bo h p o essionally, scien i ically and
pe sonally h oughou all hese yea s. Likewise, I ex end my hanks o Be nabé Lina es o his
willingness, suppo and wo k, making my pa icipa ion in Neu ONN and my secondmen in
Swi ze land possible. I would also like o hank he s a o he Technical Uni a IMSE o hei
indispensable wo k, which o en goes unno iced, as well as hei cama ade ie and good spi i .
Th oughou he de elopmen o Neu ONN, I had he p i ilege o lea ning om highly p o essional
colleagues who c ea ed a e y s imula ing en i onmen . I would like o hank all he membe s o
he conso ium du ing hese yea s, in pa icula Aida Tod i o he excellen wo k as coo dina o , as
well as my ellow PhD s uden s. I am especially g a e ul o Sigi Ka g, Oli ie Mahe , and Be nd
Go smann o hei wa m welcome du ing my s ay a IBM Resea ch. They, oge he wi h he o he
g oup membe s and iends li ing in Swi ze land, made his expe ience uly unique and en iching.
I am g a e ul o he lea ning, kindness and suppo I ecei ed om my colleagues and iends bo h in
di icul imes and momen s o laugh e , whe he inside o ou side IMSE. The e a e oo many
un o ge able people o name in hese lines. I is impossible no o conside amily o hose who ha e
been by my side o so many yea s. I dedica e a special men ion o my mecha onics iends (including
Ma io), o Juan No mando, and o Jesús Pa a, o being aluable sou ces o guidance and suppo . A
special hanks o Ra ael Vallejo Ju ado o his con ibu ion o he co e .
I would like o end by dedica ing hese lines o my amily, especially my g andpa en s and my
mo he , Juani, as well as my b o he om ano he mo he , Manuel. I will also ondly ca y wi h
me all ha I ha e lea ned om my uncle Pepe and my aun Cha i, and hei dedica ion.
Thank you all o being he e.
x
Figu e 3.4: Noisy A-J pa e ns om he noise es se wi h (a) 10, (b) 20, (c) 30 and (d) 40 co up ed
pixels. .......................................................................................................................................... 43
Figu e 3.5: Quan iza ion impac on accu acy o 60-neu on ne wo ks ained o s o e 5 digi pa e ns
(‘0’-‘4’). ....................................................................................................................................... 45
Figu e 3.6: P oposed IRPUSH lea ning ule. .............................................................................. 48
Figu e 3.7: Re ie al accu acy e sus noise le el o ne wo ks ained wi h IRPUSH wi h di e en T
alues. .......................................................................................................................................... 50
Figu e 3.8: Accu acy e sus PF o IRPUSH wi h T=95. ........................................................... 51
Figu e 3.9: (a) Re ie al accu acy e sus noise le el using IRPUSH wi h pa ial ac o s o 100%
and 33% and T=150. (b) Maximum noise le el ole a ed keeping e ie al accu acy abo e 95%
using IH and IRPUSH wi h pa ial ac o s o 100% and 33%. Conside ed T alue was he one
p o iding he bes esul o each case. ......................................................................................... 53
Figu e 3.10: Re ie al accu acy compa ison wi h o he lea ning ules. ...................................... 54
Figu e 4.1: (a) Schema ic o a di e en ial oscilla o . (b) Schema ic o he emula o ci cui o he
VO2 de ice. .................................................................................................................................. 59
Figu e 4.2: Physical layou o he di e en ial oscilla o included in he ASIC. ......................... 59
Figu e 4.3: (a) Schema ic o oscilla o calib a ion ol age selec ion. (b) Scheme o a di e en ial
oscilla o . ...................................................................................................................................... 60
Figu e 4.4: (a) Schema ic o he synapsis. (b) DONN implemen a ion om [124]. ................... 60
Figu e 4.5: Physical layou o he synap ic ci cui . ..................................................................... 60
Figu e 4.6: Layou o he ab ica ed ci cui . ................................................................................ 61
Figu e 4.7: Foo p in o he QFN64 package and names o he signals. ..................................... 63
Figu e 4.8: Schema ic o he designed Schmi -T igge bu e . .................................................. 64
Figu e 4.9: (a) Block diag am and (b) pho og aph o he expe imen al se up ............................ 66
Figu e 4.10: (a) Diag am o ins umen s in ol ed in he se -up. (b) F on panel o he GUI-app
o con olling he ONN expe imen s. ......................................................................................... 67
Figu e 4.11: Expe imen al wa e o ms o a di e en ial oscilla o wi h analog ou pu s. ............ 68
Figu e 4.12: Expe imen al wa e o ms o di e en ial oscilla o ou pu s a e digi aliza ion. .... 68
Figu e 4.13: Impac o he con igu a ion o he ou pu bu e s age on he du y cycle o he ou pu
ol age. ........................................................................................................................................ 69
Figu e 4.14: Expe imen al wa e o ms co esponding o he ou pu s o one o he oscilla o s a e
digi aliza ion applying wo calib a ion ol ages: (a) 1.2V and (b) 0.9V. .................................... 69
Figu e 4.15: Ou pu s o wo uncoupled oscilla o s wi h SHIL: (a) ou -o -phase, (b) in-phase. (c)
Wi hou SHIL. .............................................................................................................................. 70
Figu e 4.16: Expe imen al wa e o ms o posi i e ou pu o oscilla o s 1, 4 and 9. .................. 71
Figu e 4.17: Two coupled oscilla o s in which he synapse is p og ammed wi h (a) Nega i e
coupling: VP=0V and VN=0.95V and (b) Posi i e coupling: VP=0.95V and VN=0V. .................. 73
Figu e 4.18: (a) Th ee oscilla o s wi h all- o all connec ions. Ou pu s (b) wi hou SHIL and (c) wi h
SHIL. ........................................................................................................................................... 74
Figu e 4.19: (a) The lea ned pa e ns and (b) he es pa e ns selec ed o he 3x3 DONN AM.
..................................................................................................................................................... 76
Figu e 4.20: T aining pa e ns o he h ee-pa e ns AM e alua ion. ........................................ 78
Figu e 4.21: G aphs wi h 9 nodes p oposed o he Max-Cu ha d-p oblem: (a) G9A, (b) G9B, (c)
G9C. ............................................................................................................................................ 79
Figu e 4.22: Tempo al e olu ion o he numbe o ials ha achie ed he op imum cu alue o
each g aph: (a) G9A, (b) G9B, (c) G9C. ..................................................................................... 80
Figu e 4.23: Numbe o measu ed op imum s a es upon each ial. (a) G9A, (b) G9B, (c) G9C.
Inse ables depic he pe cen age o ials wi h he same numbe o measu ed op imum. ......... 81
Figu e 5.1: Illus a ion o ab ica ed die s uc u e. ...................................................................... 84
Figu e 5.2: Pic u es o (a) he IBM labo a o y se up and (b) he in e ace PCB. ........................ 85
x i
Figu e 5.3: (a) Scanning elec on mic oscope image o c ossba VO2 de ice. (b) Schema ic o he
nano-oscilla o implemen ed wi h he VO2 de ice. (c) Elec ical measu emen on he c ossba
VO2 de ice. (d) Oscilla ion ol age measu emen s o a single VO2 c ossba oscilla o . Ma e ial
adap ed wi h pe mission om [62]. ............................................................................................ 86
Figu e 5.4: (a) Schema ic o he VO2-based oscilla o wi h SHIL injec ed h ough capaci ance a
he ou pu node. (b) Schema ic o ou coupled oscilla o s wi h squa e con igu a ion. A e age
phase and measu ed wa e o ms o : (c) case wi hou SHIL, and (d) wi h SHIL applied. .......... 88
Figu e 5.5: Expe imen al esul s o coupled elaxa ion VO2 oscilla o s, in ol ing 4 o 9 nodes o
sol e he Max-Cu p oblem: a) Inpu p oblem. b) Solu ion ound, and measu ed. c) Phase ela ionships
o he ONN ou pu s. d) Numbe o cycles equi ed o ge o he s able s a e. Ma e ial ep oduced wi h
pe mission om [62]. ................................................................................................................... 90
Figu e 5.6: Dis ibu ion o he cu s a ained o g aphs in ol ing 6 o 9 coupled VO2 oscilla o s
o sol e he Max-cu p oblem. The op imal solu ion is ound in mos ins ances o g aphs C, E,
and G. Ma e ial ep oduced wi h pe mission om [62]. ............................................................. 91
Figu e 5.7: Dis ibu ion o he numbe o sa is ying assignmen s 𝐾 a ained o g aphs in ol ing
6 o 9 coupled VO2 oscilla o s o sol e he Max-3SAT p oblem. Ma e ial om [62]. ............... 93
Figu e 5.8: Expe imen al esul s in ol ing 3 o 6 nodes o sol e he G aph colo ing p oblem: a)
Inpu p oblem, b) Oscilla o g aph, measu ed c) Wa e o ms and d) Phase ela ionships o he
ONN ou pu s. e) Numbe o cycles equi ed o ge o he s able s a e. ....................................... 95
Figu e 5.9: Schema ic and measu ed wa e o ms o DONN. (a) Posi i e coupling. (b) Nega i e
coupling. ...................................................................................................................................... 97
Figu e 5.10: Wa e o ms o he 4-neu on DONN showing a s able solu ion. S o ed pa e ns
encoded in he nega i e weigh ma ix (Equa ion 5.7). ............................................................... 98
Figu e 5.11: Schema ic o he VO2-based mino i y ga e wi h he capaci i ely-coupled inpu s and
SHIL. ........................................................................................................................................... 99
Figu e 5.12: Mino i y ga e ope a ion. Vol age wa e o ms o he SHIL and he inpu s signals.
................................................................................................................................................... 100
Figu e 5.13: Mino i y ga e ope a ion. (a) Wa e o ms o ‘000’ inpu , e alua ed wo imes in he
same expe imen . (b) inse A and (c) inse B espec i ely show he imes whe e inpu s a e
ac i a ed, he p e ious and pos e io ou pu logic- alue encoded in he phase di e ence wi h
espec o a e e ence signal. ..................................................................................................... 101
Figu e 5.14: Mino i y ga e ope a ion. (a) Wa e o ms o ‘100’ inpu , e alua ed wo imes in he
same expe imen . (b) Inse A and (c) inse B espec i ely show he imes whe e inpu s a e
ac i a ed, he p e ious and pos e io ou pu logic- alue encoded in he phase di e ence wi h
espec o a e e ence signal. ..................................................................................................... 102
Figu e 5.15: Schema ic and measu ed wa e o ms o ou uncoupled oscilla o s wi h SHIL signal
added o VDD. .............................................................................................................................. 103
Figu e 5.16: Measu ed wa e o ms o ou oscilla o s con igu ed as he squa e g aph injec ing FHIL
signal added o VDD. Di e en FHIL ampli udes a e shown: (a) 125mV, (b) 250mV, (c) 0.5V, and (d)
1.0V. .......................................................................................................................................... 104
Figu e 5.17: Measu ed wa e o ms o oscilla o s con igu ed as (a) he ou -node squa e g aph and (b)
he six-node g aph, injec ing a SHIL signal wi h 250mV ampli ude added o VDD. ....................... 104
Figu e 5.18: Measu ed wa e o ms o i e isola ed oscilla o s (a) wi hou SHIL injec ion and (b) wi h
SHIL injec ion gene a ed wi h an addi ional double- equency oscilla o . .................................... 105
Figu e 5.19: A 4-node Max-Cu p oblem. (a) Op imal solu ion (4 cu s). (b) Ene gy landscape. ....... 107
Figu e 5.20: E olu ion o he numbe o cu s wi h ime o (a) wo di e en noise le els and (b)
wo di e en SHIL ampli udes. ................................................................................................. 108
Figu e 5.21: Wa e o ms o wo SHIL schedules, a) cons an ampli ude and b) inc easing
ampli ude. .................................................................................................................................. 109
Figu e 5.22: The p oposed SHIL schedule and he pa ame e s ha desc ibe i s con igu a ion. 109
Figu e 5.23: 6-node Max-Cu p oblem wi h wo op imum solu ions wi h 8 cu s. .................... 110
x ii
Figu e 5.24: SHIL schedules used o he e alua ion o he oscilla o y IM. ............................. 110
Figu e 5.25: Success a io e sus HD be ween he ini ial s a e and he closes op imum solu ion
o he h ee SHIL schedules. Simula ion pa ame e s: VMAX = 0.4V, σNOISE = 1mV, TCOMP3. ... 112
Figu e 5.26: A e age and s anda d de ia ion o he success a e o he h ee SHIL schemes
e alua ed a i e di e en compu a ion imes. .......................................................................... 112
Figu e 5.27: a) 8-node Max-Cu p oblem epo ed in [86] wi h 12 cu s (b) His og am ep esen ing
he p obabili y o success o ob aining a leas 10, 11 o 12 cu s. ............................................. 113
x iii
Lis o Tables
Table 2.1: Dis ances, nea es pa e n o e ie e and ONN esul s o 5x3 expe imen . ............. 22
Table 2.2: Dis ances, nea es pa e n o e ie e and ONN esul s o 10x6 expe imen . ........... 24
Table 2.3: Resou ces and maximum sys em clock equency on Zybo-Z20 FPGA o ONN
designs. ........................................................................................................................................ 25
Table 2.4: Time measu emen s om pos -implemen a ion simula ions o ONN designs. ........ 26
Table 2.5: Resou ces on Zybo-Z20 o ONN designs wi h he p og ammable synapses. ............. 26
Table 2.6. Success ul in e ences and a e age ime o ONN designs and he HNN model. ....... 27
Table 2.7: Al e na i e 40 neu on ONN aining con igu a ion e alua ion.................................. 32
Table 2.8: 8-ONN aining con igu a ion e alua ion. ................................................................. 33
Table 3.1: Hamming Dis ances – 10x6 Digi s. ........................................................................... 40
Table 3.2: Capaci y esul s on digi se 10x6. ............................................................................. 41
Table 3.3: Hamming Dis ances – 12x8 Cha ac e s. .................................................................... 41
Table 3.4: Capaci y esul s on cha ac e se 12x8. ...................................................................... 42
Table 3.5: Accu acy esul s o digi se 10x6............................................................................. 44
Table 3.6: Accu acy esul s o cha ac e se 12x8. .................................................................... 44
Table 3.7: Quan iza ion impac on digi se 10x6 – 5 pa e ns. ................................................... 45
Table 3.8: Quan iza ion impac on cha ac e se 12x8 – 4 pa e ns............................................. 46
Table 3.9: Quan iza ion impac on cha ac e se 12x8 – 10 pa e ns. .......................................... 46
Table 3.10: Capaci y wi h di e en T alues and weigh p ecision. ........................................... 47
Table 3.11: Minimum numbe o bi s and numbe o lea ning i e a ions wi h di e en T alues
o IH and IRPUSH wi hou pa ial upda e o he 26 pa e ns example. ................................... 49
Table 3.12: Minimum numbe o bi s and numbe o lea ning i e a ions wi h di e en T alues
o IRPUSH using a pa ial ac o o 33% and 100% (no pa ial upda e). .................................. 49
Table 3.13: Re ie al accu acy wi h di e en T alues and weigh p ecision. ........................... 50
Table 3.14: Accu acy o IRPUSH s T wi h pa ial upda e. ....................................................... 52
Table 3.15: Compa ison o minimum numbe o bi s equi ed o s o e he 26 pa e ns. ............ 55
Table 4.1: Singled-ended oscilla o s design pa ame e s. ............................................................ 59
Table 4.2: ASIC ex e nal signals. ................................................................................................ 62
Table 4.3: Con igu a ion o he DONN ou pu mul iplexe s acco ding o he con ol bi s. ........ 64
Table 4.4: Con igu a ion o he ou pu mul iplexe o he se o simple ci cui s acco ding o
con ol bi s. .................................................................................................................................. 65
Table 4.5: F equency cha ac e iza ion o uncoupled oscilla o s. ................................................ 71
Table 4.6: Oscilla o calib a ion esul s. ..................................................................................... 72
Table 4.7: F equency cha ac e iza ion o uncoupled oscilla o s applying calib a ion. ............... 72
Table 4.8: DONN expe imen esul s co esponding o he AM................................................. 76
Table 4.9: Summa y o expe imen al esul s co esponding o he AM. .................................... 77
Table 4.10: ASIC esul s o sol ing Max-Cu p oblem associa ed o he 9-node g aphs. ......... 79
Table 5.1: Values o he ci cui pa ame e s used o he Max-Cu p oblems in Figu e 5.5. Ma e ial
om [62]...................................................................................................................................... 89
Table 5.2. Ci cui pa ame e s alues used o sol e he Max-3SAT p oblems in Figu e 5.7. ...... 93
Table 5.3: Ci cui pa ame e s alues used o he G aph Colo ing p oblems in Figu e 5.8. ....... 94
Table 5.4: T u h able o 3-inpu majo i y and mino i y ga es. ................................................... 98
Table 5.5: Ini ial s a es ha con e ged o an op imum solu ion wi h SHIL1, SHIL2, and SHIL3,
espec i ely. ............................................................................................................................... 111
xix
Lis o Ac onyms
AM: Associa i e Memo y
ASIC: Applica ion-Speci ic In eg a ed Ci cui
CMOS: Complemen a y Me al-Oxide-Semiconduc o
COP: Combina o ial Op imiza ion P oblem
CSV: Comma-Sepa a ed Value
DEB: Descen Exponen ial Ba ie
DONN: Di e en ial Oscilla o y Neu al Ne wo k
D&O: Diede ich and Oppe
FPGA: Field P og ammable Ga e A ay
FSM: Fini e-S a e Machine
FP: Floa ing Poin
GKM: Ga dne -K au h-Meza d
HD: Hamming Dis ance
HNN: Hop ield Neu al Ne wo k
IH: I e a i e Hebbian
IM: Ising Machine
IMT / MIT: PTM’s s a e Insula ing o Me allic / Me allic o Insula ing T ansi ion
INS / MET: PTM’s s a e: Insula ing / Me allic
IRPUSH: I e a i e Random Pa ial Upda e Symme ic Hebbian
MC: Mon e Ca lo
MSB: Mos Signi ican Bi
MOSFET: Me al-Oxide-Semiconduc o Field-E ec T ansis o
OBC: Oscilla o y-Based Compu ing
ONN: Oscilla o y Neu al Ne wo k
PCB: P in ed-Ci cui Boa d
PTM: Phase-T ansi ion Ma e ial
QFN: Quad-Fla No-Leads
SHIL: Second-Ha monic Injec ion Locking
SMA: SubMinia u e e sion A
Chap e 1
1.In oduc ion
Mo i a ion
Today's compu ing pla o ms ace he dual challenge o handling as amoun s o in o ma ion and
execu ing complex ope a ions. This signi ican ly impac s powe consump ion, and as en i onmen al
sus ainabili y becomes an inc easing conce n, minimizing he ca bon oo p in o compu ing
p ocesses has become e en mo e c ucial. Addi ionally, hese sys ems deal wi h he 'bo leneck' limi
associa ed wi h on Neumann- ype a chi ec u es. Meanwhile, con en ional complemen a y me al-
oxide-semiconduc o (CMOS) echnology is physically cons ained in e ms o ene gy e iciency,
as i s con inued minia u iza ion leads o inc eased ene gy losses om leakage cu en s [1], [2]. The
ise o Edge Compu ing and he In e ne o Things u he unde sco es he need o mo e e icien
and decen alized compu ing a chi ec u es [3], [4], [5]. As he demand o eal- ime p ocessing,
educed la ency, and low powe consump ion in ensi ies, adi ional CMOS-based a chi ec u es
app oach hei inhe en limi s, necessi a ing a pa adigm shi owa ds inno a i e solu ions.
Di e se pe spec i es, spanning he physical, ci cui , a chi ec u al, and sys em le els, a e cu en ly
being in es iga ed o ackle hese challenges. Inno a i e app oaches, aimed a p o iding medium-
e m solu ions, explo e al e na i e de ices o ei he eplace o complemen he MOS ansis o ,
as well as al e na i e compu ing pa adigms. One such non- on Neumann app oach eso s o
exploi ing dynamical sys ems [6], [7], [8], [9]. The undamen al concep in ol es mapping he
solu ion o a p oblem o he ene gy unc ion o an app op ia ely designed dynamical sys em [9],
[10], [11], [12]. As he sys em e ol es o minimize i s ene gy, i inhe en ly compu es he solu ion
o he p oblem. This app oach (“le ’s physics compu e”) d aws inspi a ion om he na u al wo ld,
whe e eal- ime in o ma ion p ocessing can be obse ed in dynamical sys ems such as he
ac i a ion pa e ns o neu al ci cui s [13], [14], cellula signaling mechanisms [15], and
in o ma ion low in social ne wo ks [16]. The physical sys em pe o ms ‘compu a ion’ h ough
i s inhe en dynamics in a collec i e, pa allel manne , esul ing in compe i i e compu a ion imes
and educed ene gy consump ion [7], [17], [18].
Oscilla o y-based compu ing (OBC) [17], [19] belongs o his ca ego y o no el compu ing
pa adigms. I le e ages he ich, non-linea dynamics o a sys em o coupled oscilla o s o
implemen ma hema ical unc ions ha link up a sys em ou pu and inpu s a es. The idea o OBC
comes om he 1950s in he ield o logic [20], [21], whe e, ins ead o ol age le els, logical
alues a e encoded in he phase di e ences be ween an oscilla o y signal and a e e ence. This is
o signi ican in e es o se e al easons, no ably ha he swi ching be ween logical alues does
no equi e, in p inciple, ene gy expendi u e, and ha noise immuni y is also imp o ed compa ed
o he encoding wi h logical le els [22].
OBC encompasses a wide a ie y o ope a ing p inciples and a chi ec u es. Fi s , he e is a
dis inc ion be ween sys ems ha wo k wi h oscilla o s o ideally iden ical equency, whe e
p ocessing co esponds o ob aining a s able phase pa e n o phase synch oniza ion (PSK, Phase
Shi Key), and sys ems ha wo k wi h oscilla o s o di e en equencies, whe e p ocessing
co esponds o equency synch oniza ion. (FSK, F equency Shi Key) [23].
2 Chap e 1: In oduc ion
The connec ion o a mul i ude o oscilla o ci cui s, ia elec ical elemen s ha ac as synapses,
c ea es an in elligen collec i e sys em also e e ed o in he li e a u e as an oscilla o y neu al
ne wo k (ONN) [8], [24], [25], [26]. ONNs mimic he unc ioning o ce ain neu ons in speci ic
a eas o he b ain, which, hanks o hei synch ony, can p ocess and ansmi in o ma ion quickly
and e icien ly. This p ocessing by synch onism is no exclusi e o he human body and can be
ound elsewhe e in na u e: o example, in he cen al pa e n gene a o s and synch onized
locomo ion o animals [27], [13], [14], [28], [29], in he lashing hy hms o i e lies [30], [31],
o in he in e ac ion be ween coupled pendulum clocks [30].
In ecen yea s, due o ad ances in ma e ial echnology, ONNs ha e ecei ed signi ican in e es
and ha e become an ac i e a ea o esea ch hanks o he eme gence o de ices ha ope a e on
he basis o e y di e en physical phenomena, wi h he abili y o implemen compac oscilla o s
wi h e y low powe consump ion [17], [32], [33]. Vanadium dioxide (VO2) de ices, in pa icula ,
s and ou o he hys e esis in hei cha ac e is ic I-V cu e, which enables he easy
implemen a ion o elaxa ion oscilla o s [34], [35], [36].
This Thesis is amed in he con ex o he Eu opean p ojec "Neu ONN: Two-Dimensional
Oscilla o y Neu al Ne wo ks o Ene gy E icien Neu omo phic Compu ing" [37], [38], [39],
whose objec i es aimed o de elop ONN pla o ms using VO2 me al-insula o ansi ion (MIT)
de ices and 2D mem is o s, as well as, o deli e an ONN design me hodology oolbox co e ing
aspec s om ONN a chi ec u e design o algo i hms, acili a ing adop ion by use s o unleash he
po en ial o ONN echnology. Ou ONNs ope a e eso ing o he PSK ope a ing p inciple, whe e
he in o ma ion is encoded in he phase o he oscilla ions, and compu a ions a e ca ied ou
h ough he dynamic in e ac ions and synch oniza ion o hese oscilla o s.
The Thesis explo es di e en ha dwa e implemen a ions o ONNs applied o pa e n ecogni ion
asks and o sol ing combina o ial op imiza ion p oblems (COPs). The inal demons a o s o his
Thesis a e buil using VO2 oscilla o s.
The es o he Chap e is s uc u ed as ollows: Sec ion 2 desc ibes he implemen a ion o he
oscilla o using he VO2 de ice. Sec ion 3 in oduces he sys em o coupled oscilla o s, he ONNs.
Sec ion 4 and Sec ion 5 a e dedica ed o he applica ions o pa e n ecogni ion and COP sol ing,
espec i ely. The objec i es o he Thesis a e ou lined in Sec ion 6, and inally, he s uc u e o
he Thesis is highligh ed in Sec ion 7.
VO2-based oscilla o
VO2 s ands ou as a ascina ing phase- ansi ion ma e ial (PTM), exhibi ing ema kable p ope ies
ha enable he implemen a ion o highly e icien and compac oscilla o s wi h minimal ene gy
consump ion based on physical phenomena. I s dis inc i eness a ises om a empe a u e-d i en
phase ansi ion, a phenomenon in which VO2 unde goes a apid change in i s c ys al s uc u e.
Figu e 1.1a illus a es his phase- ansi ion beha io . A empe a u es below a c i ical poin o
a ound 68ºC (340 K), VO2 exis s in a semiconduc ing s a e, demons a ing insula ing p ope ies.
Howe e , as he empe a u e su passes his c i ical poin , a swi ansi ion occu s, ans o ming
VO2 in o a me allic s a e wi h signi ican ly al e ed elec onic and op ical beha io . This ansi ion
is accompanied by d ama ic al e a ions in conduc i i y (by 3 o 5 o de s o magni ude), allowing
VO2 o seamlessly shi om being an elec ical insula o o a conduc o . This p ocess is
accompanied by a s uc u al ansi ion om a monoclinic o a u ile c ys al s uc u e.
1.2. VO2-based oscilla o 3
These insula o -me al ansi ions can be d i en by elec ical s imuli. Thus, i can also be desc ibed
as a wo- e minal de ice wi h he I-V cha ac e is ic shown in Figu e 1.1b [40]. Unde no elec ical
s imuli, VO2 ends o s abilize in he insula ing phase. When he applied ol age inc eases and he
cu en densi y lowing h ough i eaches JC-IMT, insula o - o-me al ansi ion (IMT) occu s. Once
in he me allic s a e, when he ol age dec eases and he cu en densi y d ops below JC-MIT, me al-
o-insula o ansi ion (MIT) akes place. VIMT (VMIT) is he ol age a which he insula o - o-me al
ansi ion, (IMT) (me al- o-insula o ansi ion, MIT) ansi ion occu s. RINS (RMET) is he esis ance
in he insula ing (me allic) s a e. MIT and IMT ansi ions a e ab up bu no ins an aneous. The
hys e esis in he I-V cha ac e is ic allows he implemen a ion o a compac oscilla o .
Figu e 1.2a shows a VO2-based oscilla o [36], [42], [43]. No e ha he esis o and he VO2 can
be exchanged. To achie e consecu i e, sel -sus ained IMT and MIT in he de ice, i is necessa y
o se i s wo king poin in he nega i e-di e en ial egime o he VO2 I-V cu e (Figu e 1.2b). By
ca e ully selec ing an app op ia e esis ance alue, such ha he load line in e sec s he uns able
egion de ined by he ansi ion poin s, he sys em can swi ch pe iodically om a high (insula o )
o a low (me allic) esis i e s a e [36], [44]. This beha io can also be achie ed by connec ing he
VO2 de ice in se ies wi h a ansis o [42].
Figu e 1.1: (a) Ma e ial esis i i y o VO2
agains empe a u e. The phase change p esen s a s uc u al
ansi ion: monoclinic ( u ile) c ys al when he ma e ial is in insula ing (me allic) s a e. Rep oduced om
[41]
. (b) I
-
V elec ical cha ac e is ic o he wo
-
e minal VO
2
de ice.
a)
b)
4 Chap e 1: In oduc ion
Figu e 1.2c depic s wa e o ms o he oscilla o ou pu . The s a e o he VO2 is also shown o
be e illus a e he ci cui beha io . VO2,STATE=‘INS’ means he de ice is in he insula ing s a e,
and VO2,STATE=‘MET’ co esponds o he de ice in he me allic s a e. Assuming he VO2 is in an
insula ing s a e (poin “A” in Figu e 1.2c), he oscilla o ou pu is discha ged h ough he
ansis o . This inc eases he ol age d op ac oss he VO2 (VDD – VOUT), causing he cu en
h ough i o inc ease. Once enough cu en densi y ci cula es (JC-IMT), i swi ches o he me allic
s a e. Equi alen ly, once he VO2 ol age eaches VIMT, he swi ching o he me allic s a e occu s.
The ou pu is hen cha ged h ough he VO2. This cha ging is e y as because o he low RMET
alue. The ol age seen by he VO2 dec eases un il i eaches VMIT, a which poin he MIT occu s.
In e ms o ene gy, VO2-based elaxa ion oscilla o s ha e shown good pe o mance. P ojec ions
in se e al pape s epo ed hei po en ial o educe ene gy pe cycle and o be compe i i e wi h
ing oscilla o s and o he non-con en ional implemen a ions [17], [34], [41], [44].
Consequen ly, di e en g oups ha e been explo ing he capabili ies o VO2-based oscilla o s o
ONN implemen a ions [34], [45], [46], [47], [48], [49], [50], [51], [52], [53]. Wi hin he
Neu ONN p ojec , ab ica ion echniques we e imp o ed o enhance he pe o mance o VO2
ma e ial, and c ossba con igu a ions we e explo ed o hei g ea e ad an ages o e o he
ab ica ion me hods, such as plana , in e ms o powe consump ion and scalabili y [41], [54].
V V
DD MIT
-
V
OUT
ime
MET
INS
VO
2,STATE
V V
DD IMT
-
Figu e 1.2: (a) Schema ic o he VO2-based elaxa ion oscilla o . (b) I-V and oscilla ion-
enabling nega i e
esis ance egion de e mined by he load line. (c) Ou pu ol age wa e o m o he VO2-
based oscilla o and
in e nal s a e o he VO
2
de ice.
b)
a)
c)
1.5. ONNs o sol ing COPs 11
This is a simpli ied e sion o he Hamil onian, whe e in e ac ion wi h an ex e nal ield is omi ed.
Sol ing an Ising model co esponds o de e mining he spin con igu a ion which minimizes H
(g ound s a e) [70], [71]. Figu e 1.8 depic s a gene ic Ising model oge he wi h a ypical
Hamil onian shape.
Figu e 1.8: (a) Ising model ep esen a ion. (b) Hamil onian ene gy landscape.
A special ype o compu e , called Ising machine (IM) o Ising sol e , sol es Ising models. Thus,
i has been pu o wa d as an e icien way o pe o ming op imiza ion. An IM sol es a class o
op imiza ion p oblems by mapping hem o he Ising Hamil onian o a spin glass sys em and
inding i s g ound s a e solu ion.
The e a e many e o s o de elop e icien Ising sol e s o IMs. In [70], IMs a e classi ied in o
h ee main ca ego ies acco ding o hei ope a ing p inciple: classical annealing, quan um
annealing, and dynamical sys em e olu ion, al hough di e en ope a ing p inciples can be
combined. Examples o digi al dedica ed anneale s a e he Fuji su digi al anneale [72], [73], he
Toshiba bi u ca ion machine [74] o he FPGA-based solu ion epo ed in [75]. In he analog
domain, i has been p oposed o exploi he in insic andomness o nanode ices o a oid he a ea
and ene gy cos associa ed wi h he pseudo- andom numbe gene a o s equi ed in hei digi al
coun e pa s, also imp o ing he quali y o andomness. Fo example, using magne ic unnel
junc ions [61], [76]. In [77] a quan um anneale buil by D-Wa e Sys ems is epo ed. Howe e ,
i su e s om a ious p oblems. These included he need o ope a e a a empe a u e close o
absolu e ze o as well as he di icul y o scaling up he machine size, a consequence o i s use o
quan um echniques.
The hi d ype o IM is based on he e olu ion o a dynamical sys em in which he s a e o he
sys em is na u ally d i en owa ds he lowes ene gy s a e o he Ising model. In pa icula ,
coupled oscilla o s exhibi such beha io and se e al oscilla o -based IMs ha e been p oposed
using e y di e en ypes o oscilla o s, including op ical pa ame ic oscilla o s [78], [79], [80],
elec onic con en ional ones, such as disc e e LC oscilla o s [81], [82], [83], ing oscilla o s [59],
[60], di e en ial oscilla o s [84] o Schmi T igge oscilla o s [85], as well as o he buil based
on non-con en ional de ices like PTMs [46], [86], e oelec ic ansis o s [87], magne ic- unnel
junc ion o spin- o que de ices [61], [88].
I is con enien o explain ha he e is a signi ican di e ence be ween he wo explo ed
applica ions o ONNs. Bo h AMs and IMs ely on he sys em dynamics na u ally e ol ing o
s a es wi h less ene gy, bu he sys em can be apped a a local minimum. F om he poin o iew
o op imiza ion p oblems, his impedes ob aining he op imum solu ion. Unlike his, in he AM
his local minimum can co espond o he s o ed pa e n closes o he applied es pa e n. In he
IM case, being able o escape om i is posi i e, as i could imp o e he quali y o he solu ion,
bu i is nega i e in he AM since a w ong pa e n could be e ie ed, educing accu acy.
a)
b)
12 Chap e 1: In oduc ion
Max-Cu
The well-known unweigh ed Max-Cu p oblem can be mapped o an Ising model. A maximum
cu o a g aph is a pa i ion o he e ices o he g aph in o wo complemen a y se s such ha he
numbe o edges be ween hem is he la ges possible. The p oblem o inding a maximum cu in
a g aph is known as he Max-Cu p oblem. Fo mally, i can be o mula ed as ollows. Gi en an
unweigh ed g aph G = (V, E), whe e V a e he e ices and E a e he edges be ween hem, he
solu ion o he Max-Cu p oblem p o ides a disjoin pa i ion V1 ∪ V2 = V such ha he ca dinal
o E’ ={(u, ) ∈ E, u ∈ V1, ∈ V2} is maximized.
Assuming:
𝑥
=
1
𝑖𝑓
𝑖
∈
𝑉
−
1
𝑖𝑓
𝑖
∈
𝑉
(1.5)
he alue o he cu can be w i en as:
𝐶
=
𝑤
,
(
)
=
𝑤
,
−
𝑤
,
(1.6)
whe e wi,j = 1 i (i,j) ∈ E and 0 o he wise. Thus, he ela ionship wi h he Ising model is e iden .
C can be w i en in e ms o he Hamil onian in (1):
𝐶
=
12
𝑤
,
−
12
𝐻
(
𝑥
)
(1.7)
wi h Ji,j = -wi,j. No e ha he i s e m is cons an and ha by minimizing H, he alue o cu , C,
is maximized.
Figu e 1.9 shows a ou -node g aph as an example o he Max-Cu p oblem and he elec ical
simula ion o ou -oscilla o IM sol ing i . In his case, he maximum cu alue is 4 and i can be
ob ained wi h whiche e combina ion o subg oups con aining wo e ices. The wa e o ms show
ha he oscilla o phases e ol e so ha OUT1 (OUT3) is in phase wi h OUT2 (OUT4). Thus, he
p oblem solu ion can be ob ained om he s able phase pa e n o he coupled oscilla o s.
1
4
3
2
OUT1
OUT2
OUT3
OUT4
Figu e 1.9: Example o (a) ou -node g aph showing one Max-Cu op imum solu ion. (b) Simula ed
wa e o ms o a 4-oscilla o oscilla o y IM sol ing he p oblem.
a)
b)
1.6. Objec i es 13
Objec i es
The pu pose o his Thesis is o con ibu e o he ad ancemen o he s a e o he a o ONN using
phase- ansi ion VO2 nano-oscilla o s and he de elopmen o di e en demons a o s ha
suppo he po en ial o his no el pa adigm, especially in esou ce-cons ained scena ios ha
demand as and ene gy-e icien solu ions.
The speci ic objec i es a e:
1) To de elop ONN demons a o s including a i s digi al p oo -o -concep demons a o , a
CMOS demons a o in which VO2 beha io is emula ed, and demons a o s ha use
manu ac u ed VO2 de ices.
2) To de elop app op ia e lea ning algo i hms o ONNs unc ioning as AMs o apply hem o
pa e n ecogni ion asks.
3) To explo e he use o ONNs as IMs o COP esolu ion.
S uc u e o he con en s
The es o he Thesis is o ganized as ollows:
In Chap e 2 he design o he i s ully digi al ONN is p esen ed and cha ac e ized on an FPGA,
ep esen ing a p oo o concep o phase-based OBC ha dwa e. Func ional alida ion was
pe o med on pa e n ecogni ion asks. Fu he mo e, he ONN-FPGA was expe imen ally
alida ed on a mobile obo ha pe o ms obs acle a oidance, being he ONN esponsible o he
in eg a ion and p ocessing o he in o ma ion om he dis ance senso s and, a e , commanding
he obo mo emen di ec ion.
Chap e 3 co e s he explo a ion o HNN li e a u e on lea ning ules o AMs, wi h he aim o
ansla ing compa ible algo i hms o ONN aining, enhancing i s s o age and noise obus ness
capabili ies while espec ing he ha dwa e implemen a ion cons ain s. A no el i e a i e
algo i hm, wi h ou s anding AM pe o mance and adequa e o ONN synapse limi a ions, is
p oposed.
Chap e 4 con ains he design o an analog CMOS-ONN applica ion-speci ic in eg a ed ci cui
(ASIC) demons a o ab ica ed in TSMC 65nm echnology. The demons a o in eg a es a
CMOS emula o o he VO2 ma e ial o allow he ea ly e alua ion o he esponse o he p ojec ed
VO2-based ONN. The ASIC e alua ion boa d and he whole se up a e also desc ibed and
expe imen al measu emen s and cha ac e iza ion o he analog ONN dynamics a e epo ed, along
wi h he demons a ion o he ONN p oblem-sol ing capabili ies as AM and IM.
Chap e 5 collec s di e en ONN expe imen s using ab ica ed VO2 de ices a IBM-Resea ch
Zu ich acili ies. The ab ica ed de ice and he expe imen al se up a e desc ibed, as well as he
implemen a ion o oscilla o y IMs, which a e e alua ed on well-known COPs. O he expe imen s
a e also epo ed as he sol ing o G aph Colo ing p oblems, he implemen a ion o di e en ial
ONNs, he phase-encoded mino i y logic ga e and he e alua ion o al e na i e synch oniza ion
injec ion echniques. Finally, he ac o s impac ing he ONN ope a ion as IM a e analyzed and an
ope a ion p o ocol is p oposed o achie e he bes esul s.
Las , Chap e 6 ga he s a gene al discussion o he de eloped con en s and he ele an
conclusions and highligh s he u u e esea ch di ec ion pa ed by he p esen ed wo k.
14 Chap e 1: In oduc ion
Chap e 2
2.Digi al ONN on FPGA
In oduc ion
This Chap e co e s he de elopmen o a ully digi al ONN: design, alida ion, implemen a ion,
and cha ac e iza ion. Inside he digi al ONN, neu ons a e buil wi h digi al oscilla o s, and
in o ma ion is encoded on he phase di e ences be ween hem. The aim o his design is he
demons a ion o an ONN sys em able o ackle in e es ing AI asks eso ing o phase-encoded
compu a ion in he digi al domain. I is easily econ igu able and se es o p o e he ONN
compu ing pa adigm and o demons a e di e en applica ions. Fo example, i has been used o
pa e n ecogni ion and a obo ic na iga ion ask on an FPGA e alua ion boa d.
Tha is, he p ima y goal o his sys em is o showcase he compu ing pa adigm and an icipa e he
beha io o he analog ONN based on PTM, in oduced in Chap e 1, which holds he po en ial
o es ablish an ene gy-e icien compu ing pla o m. The e o e, an FPGA implemen a ion was
chosen, as i o e s a as and cos -e ec i e p o o ype while main aining he in ended
unc ionali y.
This Chap e is s uc u ed as ollows. Ini ially, i p o ides a de ailed desc ip ion o he digi al
ONN design, o e ing eade s a comp ehensi e insigh in o i s in e nal s uc u e and ope a ional
mechanism. Subsequen ly, he design unde goes a ho ough alida ion and cha ac e iza ion
p ocess, in ol ing di e se case s udies. Addi ionally, signi ican imp o emen s in he design a e
highligh ed, such as enabling p og ammable weigh s and he in eg a ion o an online lea ning
ope a ion mode. P ac ical demons a o s a e also p esen ed, showcasing he ONN e sa ili y in
eal-wo ld scena ios. The Chap e concludes wi h a syn hesis o key indings, o e ing insigh ul
conclusions d awn om he explo a ion o he digi al ONN pa adigm.
Desc ip ion o he digi al ONN design
As a p oo o concep o he ONN pa adigm, a ully digi al e sion has been designed and
e alua ed. Following he ypical FPGA design low, a op-down app oach was employed. The
design, desc ibed in he ha dwa e desc ip ion language (HDL) Ve ilog, ensu es e sa ili y wi h
a ious pa ame e s. I d aws s ong inspi a ion om an exis ing design p oposed by Thomas
Jackson in [89] bu a oiding i s analog componen s. In Jackson's design, he synapses a e
implemen ed by a esis o ne wo k, and a c i ical analog compa a o is equi ed a he inpu o
each digi al neu on o connec o i . I is epo ed ha he compa a o demands ca e ul design. In
ou implemen a ion, he synapses block is an a i hme ic ci cui , and he e a e no analog
compa a o s in he neu ons. Se e al ONN e sions ha e been implemen ed, showing p og essi e
imp o emen in e ms o FPGA esou ces, accu acy, equency, and econ igu abili y.
16 Chap e 2: Digi al ONN on FPGA
Sys em desc ip ion
The digi al sys em is composed o he ollowing blocks: ‘neu on’, ‘synapses’ and he con ol
modules. Figu e 2.1 depic s he a chi ec u e. Ex e nal po s a e indica ed wi h a pad symbol and
in e nal signals a e deno ed wi h a ows.
As inpu s, he sys em akes an ex e nal clock signal (clk), an asynch onous ese , and wo
dedica ed po s o ead he ini ializa ion signals: load and da a_inpu . As ou pu s, he neu on
ou pu s a e p o ided along wi h ano he wo ou pu signals, s eady_check and inconsis en _check,
indica ing whe he he ONN s a e was able o con e ge o no o a s able s a e. Addi ionally, he
ONN s a e is copied in o a se ial accessible egis e .
Figu e 2.1: Fully digi al ONN a chi ec u e.
Neu on uni
Figu e 2.2 depic s he block diag am o he neu on. I has one 1-bi inpu , Ninpu , and one 1-bi
ou pu , Nou pu , along wi h clocks and con ol signals o be in oduced la e . The ou pu signal
ep esen s he oscilla ion o he neu on and he inpu signal, gene a ed by he ‘synapses’ block,
ep esen s he weigh ed sum o he ou pu o he o he neu ons and de e mines he e olu ion o
he neu on s a e. The neu on wo ks such ha Nou pu is aligned in phase wi h Ninpu . Fo ha ,
he phase equi ed o Nou pu is compu ed by he ‘phase calcula o ’ sub-block, upda ing he
neu on s a e which is p o ided as inpu o a ‘phase-con olled oscilla o ,’ as shown in Figu e 2.2.
The s a e alue de e mines he phase o Nou pu , which is he ou pu o he la e sub-block.
2.2. Desc ip ion o he digi al ONN design 17
Figu e 2.2: En i y desc ip ion and ope a ing p inciple o single oscilla o y neu on.
The implemen a ion o he phase-con olled oscilla o is simila o Jackson’s design [89]. A
ci cula shi egis e and a mul iplexe a e used. Th ough he selec ion bi s o he mul iplexe ,
co esponding o he neu on s a e alue, any o he shi egis e s ages can be selec ed as he
neu on ou pu . The shi egis e has 16 s ages. The 16-bi pa e n {1111111100000000}
con inuously cycles h ough i , so ha 16 squa e signals wi h dis inc phases a e a ailable and can
be selec ed o become he neu on ou pu . Figu e 2.3a depic s he logic diag am o he ‘phase-
con olled oscilla o ’ and Figu e 2.3b, he wa e o ms co esponding o s age 0 and s age 8. No e
hey a e 180° apa om each o he . The egis e con olling he mul iplexe s o es he neu on
s a e o equi alen ly he phase o he neu on ou pu . No e di e en clocks a e used o con ol he
s a e egis e (d i en by he sys em clock) and he shi egis e . The la e is d i en by a slow
clock gene a ed om he sys em clock so ha he mul iplexe ou pu will cycle, as long as i s
con ol egis e emains immu able, wi h a pe iod, Tosc = 16 · Tslowclk. Ini ially, he ela ion
be ween he slow clock and he sys em clock gi en by a equency di ide was Tslowclk = 32 ·
Tclk, co esponding o a 5-bi di ision. La es explo a ions on he impac o his ac o showed
ha he design can be sped up using an equi alen equency di ide o 2 bi s, hus, he esul ing
slow clock pe iod is Tslowclk = 4 · Tclk.
The ‘phase calcula o ’ block is composed o wo edge de ec o s and a Fini e-S a e Machine
(FSM). The associa ed s a e diag am is shown in Figu e 2.4. One edge de ec o is used o Ninpu ,
gene a ed by he synapsis block, and he o he o he neu on ou pu , Nou pu . A con en ional
implemen a ion wi h h ee se ially connec ed lip- lops and a simple logic ope a ion on he lip-
lops ou pu s is used. The FSM measu es he ime di e ence be ween he ising edges o he
neu on inpu (ou pu o synapses block) and he neu on ou pu by using a coun e which s o es
he numbe o clock cycles. The o de o he edges is de ec ed and egis e ed using wo di e en
s a es ( i s Ninp and i s Nou ) and he a iable sig. When he second ising edge is de ec ed
(secondEdge), he phase di e ence is ob ained by scaling he coun e alue wi h he same ac o
om he slow clock, which d i es he ci cula shi - egis e in he phase-con olled oscilla o .
The neu on s a e is upda ed wi h he con ibu ion o he compu ed phase di e ence, so ha he
ou pu signal phase is aligned wi h he inpu signal phase. No e he inpu signal phase is
de e mined by he phase o he emaining neu ons and he weigh s associa ed o he co esponding
in e ac ion be ween hem.
18 Chap e 2: Digi al ONN on FPGA
The s a e egis e s o all he neu ons a e se ially connec ed, o ming a scan pa h ha allows he
s a e o he ONN o be loaded and ead in se ies. This is done wi h he signals se _s a e_in and
se _s a e_ou . Also, i has an ou pu signal, s a e_changed, ha goes high when he neu on s a e
is modi ied wi h espec o i s alue du ing he p e ious clk cycle.
Figu e 2.3: (a) Logic diag am o implemen ed phase-con olled oscilla o . (b) Ou pu wa e o m o
in e nal shi - egis e s ages 0 and 8.
Figu e 2.4: Logic diag am o FSM o he phase calcula o .
-0
di <= coun e >> clk_di _ ac o +1
01
sig <= 1
11
01
11
0-
00, 11
coun e <= 0
secondEdge
upda e
idle
i s Nou
coun e <= +1
10
sig <= 0
10
i s Ninp
coun e <= +1
s a e <= sig ? s a e + di : s a e -di
2.2. Desc ip ion o he digi al ONN design 19
Synapses block
The ‘synapses’ block is in cha ge o gene a ing he neu on inpu signals, as illus a ed in Figu e
2.5. As i has been abo e men ioned, unlike Jackson’s design, an a i hme ic logic ci cui , ins ead
o a esis o ne wo k, is used in ou ully digi al implemen a ion o he ONN. Inpu signal o i- h
neu on is gene a ed as:
𝑁𝑖𝑛𝑝𝑢𝑡
[
𝑖
]
=
𝑠𝑖𝑔𝑛
(
𝑤
−
𝑤
)
(2.1)
whe e j ex ends o hose neu ons wi h Nou pu [j]=1 and k o hose wi h Nou pu [k]=0. Oscilla o y
ou pu s om he neu ons de e mine he signs o he weigh s when hey a e summed in he
‘a i hme ic ci cui s.’ The sign o he summed weigh s in each ci cui p o ides he 1-bi inpu
signal used by ‘neu on’ block.
This is he cos lies componen in e ms o esou ces. Depending on he size o he ONN, he
spa si y o he weigh ma ix, and he esolu ion o he weigh s, i will be possible o i a
combina ional implemen a ion o his block wi hin he FPGA o a sequen ial implemen a ion will
be equi ed. The la es e sion adap ed he synapses block using a syn hesis- iendly app oach
ha allows a educ ion in he o al add ope a ions achie ing bo h a educ ion o equi ed esou ces
and a highe maximum equency ope a ion.
Figu e 2.5: Block diag am o digi al ONN showing in e nal sub-blocks.
Con ol block
An FSM in he ‘se ial2s a es’ module manages he ini ializa ion o he ONN s a e. Upon de ec ing
a ising edge o he load signal, he da a_inpu bi is ansmi ed se ially, h ough he scan pa h
con igu ed wi h he s a e egis e s o he neu ons, loading he ini ial s a e o each neu on in he
ONN. Remembe ha he inpu o be p ocessed is encoded in he ini ial s a e o he ONN. Once
he end o he da as eam is eached, de e mined by moni o ing an in e nal coun e ha inc emen s
a he posi i e edge o clk, he in e nal ull_ ick signal is aised. This ac ion igge s he oscilla ions
o he neu onal ne wo k, allowing he s a es o e ol e.
20 Chap e 2: Digi al ONN on FPGA
The ‘ eq_di ide ’ module scales he sys em clock equency o p o ide he signal slowclk ha
con ols he neu ons´ oscilla ion pe iod. The sys em de e mines he con e gence o he ONN
eso ing o he N_s a e_changed signal, ha indica es i any neu on s a e has changed, and a
ime which is ese each ime he p e ious signal a ises. I he s a es don´ change in a de e mined
ime, s eady_check is ac i a ed while inconsis en _check emains a low alue, indica ing
s abili y. Meanwhile, an addi ional wa chdog ime is cons an ly coun ing a each clock cycle,
unless a ese o load is asse ed, in such a way ha i p oduces he ac i a ion o bo h sys em
ou pu signals when he coun su passes an es ablished limi .
Design alida ion and cha ac e iza ion
A pa ame e ized Ve ilog egis e - ans e -le el desc ip ion o he ONN desc ibed in he p e ious
sec ion has been alida ed, syn hesized, and e alua ed using Xilinx’s Vi ado Design Sui e 2018.2
[90] and he XC7Z020-1CLG400C as he a ge de ice [91]. Design pa ame e s include
he numbe o neu ons, synapses esolu ion (numbe o bi s o he weigh s), phase esolu ion
(numbe o neu on s a es), and oscilla ion pe iod.
Func ional alida ion
As pa o he digi al design low, es benches a e used o simula e he ope a ion o he digi al ONN.
Expe imen s wi h a 5x3 ONN and a 10x6 ONN con igu ed as AMs ha e been ca ied ou . I means
ha a e he aining s age, he ONN s o es pa e ns as s able a ac o s. This is done o line,
compu ing he synap ic weigh s h ough he applica ion o a lea ning ule. Then, du ing he in e ence
s age, he ONN s a e e ol es om a ypically co up ed ini ial s a e o one o he s o ed pa e ns.
Figu e 2.6 illus a es he ope a ion o he 5x3 digi al ONN. The weigh s a e selec ed o s o e he
ep esen a ion o he digi ‘0’ (‘image0’) as one o i s s able pa e ns. I is associa ed wi h a 5x3
ma ix o pixels ep esen ing g ey-coded images. Each g ey alue pixel is encoded in o he phase
(o s a e) o he neu on associa ed o i . Whi e is encoded as s a e 0 and black as s a e 8. Ope a ion
s a s wi h he img_load pulse. The inpu pa e n is loaded in o he ONN wi h he signal se ial_ou
connec ed o po da a_inpu . The ONN s a e is se ially ini ialized h ough his po ( he egion in
Figu e 2.6 whe e se ial_ou exhibi s ac i i y). No e ha once his p ocess inishes (se ial_ou
s ops changing), he s a e o he ONN (signal s a e, conca ena ion o he s a es o he neu ons)
s o es he inpu image (noisy ‘image0’ à888808800800088). Then, he phase o each neu on is
upda ed acco ding o he ou pu o he o he ones and he weigh s. The ONN s a e e ol es un il a
s able one is eached. T aining pa e ns a e s able, and ideally, he one ha mo e closely esembles
he inpu image is e ie ed. As p e iously said, he 5x3 ep esen a ion o digi ‘0’ (‘image0’ à
888808808808888) is one o he s o ed pa e ns. When he ONN is ini ialized wi h an incomple e
e sion o i , he digi ‘0’ is eco e ed. T ansi ions be ween s a es a e ma ked in he igu e be o e
eaching he co ec alue.
2.3. Design alida ion and cha ac e iza ion 27
ONN in e nal s a e. The s a e o each neu on is ep esen ed wi h 4 bi s. I can be obse ed ha
bo h he “0” and he “1” pa e ns a e ecognized. No e he ac i a ion o s eady signal. Howe e ,
when pa e n “2” is applied, he closes aining pa e n is e ie ed (“0”).
Finally, Figu e 2.11c depic s he addi ion o pa e n “2” as addi ional aining pa e n. A e wa ds,
when pa e n “2” is applied as inpu o in e ence, i is ecognized and i does no con e ge o “0”
as i happened be o e being lea ned by he ONN.
Digi al ONN and HNN
As he ONN is a single-laye design, i is un ai o di ec ly compa e i o s anda d benchma k se s
based on o he Deep Neu al Ne wo k implemen a ions. The closes a i icial neu al ne wo k o
compa e wi h he ONN is he HNN. I is c ucial o cla i y o he eade ha compa ing he digi al
ONN wi h a digi al implemen a ion o he HNN [95], [96], [97], in e ms o powe consump ion,
esou ces, o speed is no meaning ul. The p ima y goal o he p esen ed digi al ONN is o se e
as a p oo -o -concep o he phase-based oscilla o y compu ing pa adigm. This concep will la e
be ansla ed in o highly e icien analog compu ing pla o ms buil wi h PTMs.
Howe e , i is in e es ing o assess he beha io o he HNN model in he a ge applica ions wi h
which he digi al ONN is e alua ed. Conside ing ha bo h ne wo ks a e ene gy-based ully-
connec ed neu al ne wo ks o a single laye , he aining me hods and ne wo k-s a e dynamics align
easily. In pa icula , Table 2.6 summa izes he compa ison o esul s ob ained o he 5x3 and 10x6
expe imen s wi h he digi al ONN designs and he HNN ma hema ical model. The able highligh s
a high a ini y in success ul in e ences. Mo eo e , ou ou o he i e inco ec ly e ie ed pa e ns
in he 10x6 case s udy ained wi h Hebb a e common o bo h he ONN and he HNN.
No ably, he equi ed cycles o he ONN ep esen he in e ence ime di ided by he oscilla ion
pe iod, while he i e a ions o HNN indica e he numbe o equi ed synch onous upda es o he
s a es. The ONN a chi ec u e allows s a es o con inuously e ol e and upda e when equi ed,
achie ing co ec in e ences in ewe cycles han he numbe o i e a ions equi ed by he HNN.
Table 2.6. Success ul in e ences and a e age ime o ONN designs and he HNN model.
5x3-Hebb 5x3-S o key 10x6-Hebb 10x6-S o key
ONN
( #success / # es ) 11 / 12 11 / 12 15 / 20 20 / 20
HNN
( #success / # es ) 12 / 12 12 / 12 15 / 20 20 / 20
ONN (a e . cycles) 1.15 2.00 1.20 1.12
HNN (a e . i e a ions) 2.08 2.00 2.00 2.05
28 Chap e 2: Digi al ONN on FPGA
Figu e 2.11: Illus a ion o lea ning ope a ion mode: (a) T aining he ne wo k wi h pa e ns ‘1’ and ‘2’.
(b) Recogni ion o h ee es pa e ns. (c) Second aining s ep wi h addi ional pa e n and ecogni ion es .
Weigh ma ix o “0”
W. ma ix o “0” and “1”
Apply “0” Apply “1” Apply “2”
Re ie e “0”
Re ie e “0”
Re ie e “1”
Lea n “2”
Re ie e “2”
a)
b)
c)
Apply “2”
2.4. Demons a o s and applica ions 29
Demons a o s and applica ions
Di e en demons a o s o ONN applica ions we e de eloped in coope a ion wi h o he pa ne s
in he Neu ONN p ojec . Pa icula ly, hey we e implemen ed on FPGA a he Labo a o y o
Compu e Science, Robo ics and Mic oelec onics o Mon pellie (LIRMM-CNRS), using ou
digi al ONN. Also, we we e in cha ge o aining he ONNs o he di e en applica ions. We
b ie ly desc ibe wo o hem.
Digi s ecogni ion
The i s demons a o does digi s ecogni ion om a came a s eam and p in s he esul on an
ex e nal sc een. Inpu images a e p esen ed by a phone o he came a. The came a is connec ed o
he de elopmen boa d and sends images o he ONN. When he compu a ion ime is o e , he
ou pu pa e n is p in ed on an ex e nal sc een.
The came a is a Pcam 5c came a and communica es ia MIPI CSI-2 (MIPI Came a Se ial
In e ace 2) wi h he Zybo-Z7 de elopmen boa d. The ex e nal sc een is connec ed ia HDMI
(High-De ini ion Mul imedia In e ace). The image s eaming om he came a o he HDMI
sc een comes om a Digilen Gi hub p ojec named Zybo-Z7-Pcam-5c, compa ible wi h Vi ado
so wa e. The digi al ONN implemen a ion is in eg a ed inside he p ocessing low. The image
om he came a is con e ed in g eyscale, bina ized in black/whi e pixels, and scaled down o
10x6 pixels image aking he majo colo be ween black and whi e pixels. The ONN ou pu is
es uc u ed in o a 1280x720 pixels image o be p in ed on he sc een as shown in Figu e 2.12.
Figu e 2.12: Design o he digi ecogni ion applica ion.
Fo his applica ion, he ONN is ained wi h a se o 5 pa e ns ep esen ing digi s om 0 o 4
ollowing he Hebbian ule. The es se is composed o he 5 ained pa e ns and 20 co up ed
images like o design alida ion. As expec ed, esul s a e equal o he simula ed es s, wi h 5
images no co ec ly ecognized. Few es pa e ns along wi h he co esponding in e ed pa e n
a e shown in Figu e 2.13.
This i s demons a o shows he usabili y o he ONN inside a comple e image ecogni ion
applica ion.
30 Chap e 2: Digi al ONN on FPGA
Figu e 2.13: Noisy es pa e ns and ONN in e ed pa e ns o digi s ecogni ion applica ion.
Obs acle a oidance: 3 dis ance senso s
The second demons a o deals wi h a obo ic applica ion. I uses he ONN as a obo b ain o
a oid obs acles. The 10x6 ONN is embedded inside a small mobile obo om A duino equipped
wi h 3 p oximi y senso s, see Figu e 2.14.
To in eg a e he ONN inside a obo , i has been implemen ed in o a CMOD-A7 de elopmen
boa d om Digilen which is smalle han he Zybo-Z7. The A duino obo uses an o - he-shel
A duino boa d o con ol he h ee p oximi y senso s and mo o s o he obo , see Figu e 2.15.
Each senso can measu e a dis ance be ween 0 cm and 20 cm. Measu emen s a e disc e ized o be
encoded in o an image o 10x6 pixels. A he mome e encoding is used wi h a 6- alue
disc e iza ion ( om 0 o 5), as exempli ied in Figu e 2.16. Fo each senso , he highe he whi e
le el is, he u he away he obs acle is. Eigh e e ence pa e ns ha e been selec ed o ain he
ONN, ollowing he Hebbian ule, and a d i ing command is associa ed o each, as shown in he
Figu e 2.17.
A ideo a ailable on [98] shows he capabili y o he obo o mo e in a close a ea wi hou
ouching any wall o obs acle.
Figu e 2.14: Schema ic o he obo obs acle a oidance applica ion.
2.4. Demons a o s and applica ions 31
Figu e 2.15: Pic u e o he A duino obo equipped o obs acle a oidance applica ion.
Figu e 2.16: En y image example o he ONN o obs acle a oidance applica ion.
Figu e 2.17: S o ed pa e ns and associa ed commands o obs acle a oidance.
CMOD
-
A7
A duino
Senso s
32 Chap e 2: Digi al ONN on FPGA
Obs acle a oidance: 8 dis ance senso s
On he basis o he esul s ob ained in he 3-senso demons a o , a mo e complex scena io was
add essed wi h 8 p oximi y senso s. The i s s ep was designing he ONN i sel . Tha is, o
de e mine he size o he ONN ha we a e going o use, as well as i s weigh o aining ma ix.
The la e equi es selec ing a se o pa e ns o s o e in he ONN ( aining pa e ns) ha a e
ele an o ou applica ion. The ONN will map he pa e ns o images gene a ed by he senso s in
his se o pa e ns which will be ansla ed in o con ol commands o he obo mo emen .
Ex ending ou expe ience om he h ee-senso sys em, a 40-neu on ONN was selec ed. Each senso
in o ma ion is encoded using i e neu ons (o pixels). So, we isualize pa e ns as 5x8 images. The
weigh s ma ixes we e gene a ed applying he Hebbian ule o he se o aining pa e ns. The ole o
he ONN is e ie ing one o hose pa e ns om any possible inpu image. Tha is, in o ma ion om
posi ion senso s is in eg a ed, abs ac ing de ails in o de o simpli y aking decisions.
Di e en se s o aining pa e ns ha e been de i ed and e alua ed by simula ion in o de o
iden i y hose wi h highe po en ial. Table 2.7 summa izes some explo ed solu ions. Rele an
c i e ia o assess po en ial include:
Numbe o ou pu pa e ns. The lowe his numbe is, he mo e abs ac ion o in o ma ion
is ca ied ou by he ONN. A he limi , i migh wo k wi h a e y simple and educed se
o con ol commands.
Con e gence a e. I is de ined as he ac ion o inpu images which con e ges o a s able
ou pu pa e n. Ideally i should be 100%. I his is no he case, he con ol algo i hm
mus moni o he signal gene a ed by he digi al design indica ing his non-con e gence
a e a easonable amoun o ime and mus ake a sui able decision.
Accu acy. F ac ion o inpu images which con e ges o a s able ou pu pa e n a
minimum dis ance.
Ideally, a educed numbe o ou pu pa e ns wi h 100% con e gence a e and 100% accu acy
would be desi able.
F om Table 2.7 i can be obse ed ha W_1 is e y compe i i e in e ms o con e gence a e and
accu acy bu gene a es a la ge numbe o ou pu s pa e ns han he o he solu ions. Thus, we
conside ed using a second ONN o u he abs ac in o ma ion and e en ually simpli ying he
con ol algo i hms. Fo ha , an ONN wi h eigh neu ons (one pe senso o pe column in he 40-
neu on ONN) was ained o ecognize he se o pa e ns in Figu e 2.18 plus bo h whi e and black
backg ounds. Table 2.8 summa izes ele an in o ma ion o his aining.
Table 2.7: Al e na i e 40 neu on ONN aining con igu a ion e alua ion.
T aining Se Desc ip ion # ou pu
pa e ns
Con e gence
a e Accu acy
40_W_1
Ex ending he 3-senso solu ion.
All possible combina ions o black and whi e
columns in he 5x8 image.
256 100% 100%
40_W_2 Ins ead o conside ing single columns, se s o wo
columns a e used like in W_1. 16 48% 48%
40_W_2_nbg Same as W_2 bu wi hou aining B/W
backg ounds. 16 94% 79%
40_W_4 Se s wi h ou columns g oup being sliced by 1
column and backg ounds. 42 98% 52%
40_W_4_nbg Same as W_4 bu wi hou aining backg ounds. 8 93% 69%
2.4. Demons a o s and applica ions 33
Figu e 2.18: T ained pa e ns o he second 8-neu on ONN s age.
Table 2.8: 8-ONN aining con igu a ion e alua ion.
Desc ip ion # pa e ns
# aining pa e ns 16
# inpu pa e ns 256
# ou pu pa e ns 80
# inpu pa e ns ha
e u ned a aining pa e n 192
Con e gence a e 100%
Accu acy 74%
F om Table 2.8, i can be obse ed ha he e a e many mo e ou pu pa e ns han aining ones.
Tha is, he e a e many s able ou pu pa e ns which a e no aining ones. They a e called spu ious
pa e ns. Howe e , i can be also seen ha mos inpu pa e ns con e ge o aining pa e ns.
Because o his we also conside his o be a solu ion wi h highe po en ial o ou applica ion.
No e he ac ual sui abili y o a aining can depend on many di e en ac o s such as he obo
i sel and he applica ion in which he OAV sys em is o be in eg a ed.
Figu e 2.19 illus a es he wo-cascaded ONNs scheme. I was inco po a ed in o a unicycle mobile
obo wi h wo wheel-d i e connec ed o wo di ec cu en mo o s and eigh in a- ed p oximi y
senso s, as shown in Figu e 2.20. The ou pu pa e n ob ained wi h he second ONN was pos -
p ocessed on A duino o ind a ee di ec ion which is commanded o he obo o ollow. A ideo
is a ailable on [99].
Figu e 2.19: Two-cascaded ONNs o obs acle-a oidance sys em imp o emen .
34 Chap e 2: Digi al ONN on FPGA
Figu e 2.20: Final obo demons a o doing obs acle a oidance wi h an embedded digi al ONN.
Conclusions
As a p elimina y s ep owa ds he analog ONN using PTMs, a ully digi al ONN has been
de eloped. The design has been syn hesized in o an FPGA echnology o as p o o yping and
alida ion o sui able applica ions. This FPGA implemen a ion has been comp ehensi ely
cha ac e ized and e alua ed as AM o di e en ONN sizes and lea ning ules. I demons a es
ha he sys em is able o con e ge o a s able phase pa e n wi hin e y ew oscilla ion cycles,
suppo ing he po en ial o he compu ing pa adigm in e ms o ene gy e iciency. Also,
signi ican imp o emen s in he design ha e been de eloped, such as he enabling o
p og ammable weigh s and he in eg a ion o an online lea ning ope a ion mode.
The expe imen s ha e also been e alua ed using a ma hema ical model o he HNN o assess i s
sui abili y as an analy ical ool o p edic ing he po en ial esul s ob ained wi h he ONN and he
quali y o he compu ed weigh s. The high deg ee o simila i y in he esul s sugges s ha
le e aging he ex ensi e li e a u e on lea ning ules wi h he Hop ield model can p o ide aluable
insigh s in o de eloping algo i hms compa ible wi h he ha dwa e cons ain s o he ONN.
Finally, demons a o s ha e been de eloped o showcase he capabili y o using he designed
digi al ONN o comple e applica ions. Speci ically, implemen a ions o a digi ecogni ion
applica ion and a esponsi e obs acle a oidance sys em in a mobile obo ha e been comple ed.
Chap e 3
3.Lea ning o Oscilla o y AM
In oduc ion
As i was al eady shown in p e ious Chap e s, a single ully connec ed ONN laye can implemen
an AM compa able o ha o a HNN. I possesses he abili y o s o e and e ie e pa e ns, hus
dis inguishing be ween wo di e en s ages in i s ope a ion: he lea ning o he pa e ns, and he
in e ence phase, whe e he s a e o he sys em e ol es un il i ma ches one o he s o ed pa e ns
[10].
Lea ning is he p ocess o ob aining app op ia e alues o weigh s, he eby encoding he
in o ma ion o be s o ed in he ela ionships be ween he weigh alues. The weigh s, which can
be g ouped in o a ma ix, de e mine he synap ic s eng hs, he e o e, de e mining he in e ac ions,
be ween neu ons. This p ocess is o en e e ed o as aining o he neu al ne wo k [100], [101],
[102].
T aining algo i hms ound in Machine Lea ning a e ypically classi ied as supe ised o
unsupe ised, on he basis o whe he o no knowledge o he a ge ou pu is used du ing
lea ning. Depending on he condi ions in which he lea ning is execu ed, dis inc ions can also be
made be ween one-sho o i e a i e, inc emen al o non-inc emen al, and local o non-local
lea ning. A lea ning me hod is inc emen al when i only uses in o ma ion om he new pa e n o
lea n and he cu en weigh ma ix o compu e he new one, whe eas i is i e a i e i i p esen s
se e al epe i ions o he same in o ma ion o he aining algo i hm un il i is lea ned. Locali y
occu s i he inc emen in he weigh be ween wo neu ons only depends on he desi ed s a e o
hose wo neu ons o hei hidden po en ial. This is an a ac i e p ope y because i makes
lea ning biologically plausible [64], [103], [104]. Lea ning can also be conside ed o line o
online depending on whe he i equi es knowledge o he ne wo k esponse be o e upda ing he
weigh s.
As sugges ed in Chap e 2, Lea ning o ONNs can exploi hei simila i y wi h HNNs. An
ex ensi e amoun o li e a u e is a ailable abou lea ning in HNNs. Howe e , no all o hese
algo i hms a e use ul o ONN aining due o he cons ain s imposed by hei physical
implemen a ion. Gi en ha ONN synapses comp ise elec ical couplings, he compu ed weigh s
canno be di ec ly ansla ed o elec ical conduc ance pa ame e s: a mapping s ep is equi ed
[105]. Fu he mo e, physical implemen a ion imposes es ic ions on he weigh ma ix:
bidi ec ional coupling in oscilla o connec ions equi es he ma ix o be symme ic; null diagonal
elemen s ep esen ing no sel -in e ac ions; weigh p ecision is limi ed by how many elec ical
s eng hs he sys em can dis inguish; and layou pa asi es can impac he pe o mance by
modi ying he e ec i e synap ic s eng h and ope a ing equency. The o iginal HNN model also
includes a bias pa ame e ep esen ing equi alen ly he con ibu ion o one ixed-s a e neu on.
Since he ONN a chi ec u e does no include such a con ibu ion, i canno appea in he lea ning
algo i hm.
36 Chap e 3: Lea ning o Oscilla o y AM
Weigh ma ices de i ed by he applica ion o he well-known Hebb’s ule a e symme ic, wi h
null diagonal and wi hou bias pa ame e , and hence Hebbian lea ning ule is he mos widely
adop ed me hod o con igu ing ONN-based AMs in he e y li le li e a u e add essing his opic,
despi e i s well-known limi a ions [19], [26], [50], [51], [89], [105], [106], [107], [108]. Also,
weigh ma ices de i ed by he S o key lea ning ule sa is y hose condi ions. In ac , i was
applied in some applica ions desc ibed in Chap e 2 showing i s supe io i y wi h espec o
Hebbian ule.
The e a e many o he lea ning algo i hms wi h a be e pe o mance han he wo ules
men ioned abo e. This Chap e explo es he HNN li e a u e on lea ning ules wi h he aim o
de eloping algo i hms sui able o ONN aining, enhancing i s pe o mance beyond wha can be
achie ed wi h he wo simple ules applied hus a .
The es o he Chap e is s uc u ed as ollows: Sec ion 2 e iews a ious HNN lea ning ules
om he li e a u e. In Sec ion 3, we ou line ou app oach o enhance he pe o mance o he
S o key ule. Sec ion 4 in oduces a no el i e a i e ule ailo ed o ONNs. A compa ison o
di e en lea ning ules is p esen ed in Sec ion 5. Finally, Sec ion 6 summa izes he conclusions.
Lea ning ules o AMs
Chap e 1 in oduced wo well-known lea ning ules (Hebb and S o key, [63], [64]), which a e
e iewed he e o acili a e he unde s anding o his Chap e .
As in oduced in Chap e 1, o a se o a ge P pa e ns o leng h N, 𝜉𝜖{−1,+1}, o be s o ed,
he coupling s eng h be ween neu ons co esponds o he co ela ion o hei s a e alues and can
be de e mined as:
𝑤
=
𝜉
𝜉
(3.1)
whe e componen 𝜉 ep esen s he desi ed s a e o neu on i when pa e n k shall be e ie ed.
Memo y capaci y limi a ions [109] s o ing highly co ela ed pa e ns s imula ed u he esea ch
in o lea ning mechanisms. The wo k in [64] p oposed an al e na i e ule ha conside s cu en
knowledge o he ne wo k du ing he aining s eps. Tha is o say, he weigh upda es a e ob ained
conside ing he p e ious weigh s.
𝑤
=
𝑤
+
1
𝑁
𝜉
𝜉
−
1
𝑁
𝜉
ℎ
+
ℎ
𝜉
ℎ
=
𝑤
𝜉
,
,
(3.2
)
This algo i hm se es o emo e lowe -o de noise associa ed wi h he in e ac ion o he di e en
a ac o s, whe e sub ac ion is applied by compu ing p e-synap ic and pos -synap ic o ms o he
local ield: ℎ
and ℎ
. I has been demons a ed ha while he absolu e capaci y o he Hebbian
algo i hm (no e ie al e o s allowed) is gi en by C=
(), S o key’s ule achie es C=
() [64], [109], [110].
Bo h lea ning ules ea u e a one-sho , local, inc emen al, unsupe ised aining me hod.
3.3. Imp o ing AM capabili ies wi h S o key lea ning ule 43
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a)
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Figu e 3.4: Noisy A-J pa e ns om he noise es se wi h (a) 10, (b) 20, (c) 30 and (d) 40 co up ed pixels.
The accu acy signi ican ly imp o es wi h he applica ion o epe i ion in all cases. No ably, he
ad an ages a e mo e p onounced when ansi ioning om one epe i ion o wo, compa ed o he
p og ession om wo o h ee. I is no ewo hy ha epea ing he na u al o de wice yields
accu acy le els no much poo e han hose achie ed wi h he bes -pe o ming la ge andom-
o de ials. Fo ins ance, in he digi s example, he accu acy was 88% wi h epe i ion compa ed
o 90.1% wi h he bes andom-o de ial, and in he cha ac e s case s udy, i was e en be e ,
78.5% e sus 60.4%, espec i ely. Addi ionally, i is essen ial o highligh he compu a ional
e iciency o epe i ion compa ed o explo ing nume ous andom o de s. Gi en his e iciency,
epe i ion eme ges as he p e e ed app oach o enhancing S o key ule pe o mance.
44 Chap e 3: Lea ning o Oscilla o y AM
Table 3.5: Accu acy esul s o digi se 10x6.
Lea ning Rule Success Coun (#) Max. Accu acy (%)
A e . Accu acy (%)
Random O de (100) 1 80.7 80.7
Random O de (200) 39 87.6 80.9
Random O de (300) 53 90.1 81.7
Repea ed x2 T ainSe 1 88.0 88.0
Repea ed x3 T ainSe 1 90.4 90.4
Random & Rep. x2 (100) 96 91.9 87.2
Random & Rep. x2 (200) 193 92.3 87.4
Random & Rep. x2 (300) 285 92.3 87.2
Random & Rep. x3 (100) 100 94.3 91.0
Random & Rep. x3 (200) 200 94.6 91.1
Random & Rep. x3 (300) 300 94.1 91.0
Table 3.6: Accu acy esul s o cha ac e se 12x8.
Lea ning Rule Success Coun (#) Max. Accu acy (%)
A e . Accu acy (%)
Random O de (500) 2 56.2 56.1
Random O de (1000) 3 60.4 59.6
Repea ed x2 T ainSe 1 78.5 78.5
Repea ed x3 T ainSe 1 85.1 85.1
Random & Rep. x2 (100) 85 82.4 78.2.
Random & Rep. x2 (500) 398 82.9 77.7
Random & Rep. x2 (1000) 800 83.9 77.6
Random & Rep. x3 (100) 100 88.0 86.2
Random & Rep. x3 (500) 500 87.7 86.0
Random & Rep. x3 (1000) 1000 88.1 86.0
Impac o weigh p ecision
Gi en he es ic ions on he weigh s o be compa ible o analog ONNs, an e alua ion o ou
app oach conce ning he impac o quan iza ion was pu sued. I has been explo ed s o ing om
‘0’ o ‘4’ digi -pa e ns o he digi s expe imen , since his is he la ges se success ully handle
by he simple S o key solu ion. S o key solu ion as well as hose ob ained wi h he epe i i e
lea ning app oaches using he na u al o de o he pa e ns ha e been e alua ed wi h weigh s
quan ized om 5 o 2 bi s.
The applied quan iza ion me hod is based on:
1) No maliza ion wi h espec o he maximum absolu e weigh alue o ob ain 𝑤≤1.
2) Scaling by he la ges absolu e alue which can be ep esen ed wi h he a ge numbe o bi s (nbi ).
3) Rounding.
No e ha he MSB bi is dedica ed o he sign, while he emaining bi s ep esen he in ege alue, up
o a maximum: 2−1. Tha is, he magni ude alue o nbi p ecision, 𝑤
∗, is ob ained as:
𝑤
∗
=
𝑟𝑜𝑢𝑛𝑑
2
−
1
𝑤
max
abs
(
𝑊
)
(3.11)
3.3. Imp o ing AM capabili ies wi h S o key lea ning ule 45
Table 3.7 ga he s ob ained accu acy using he same se -up as he p e ious expe imen s. These
da a a e also g aphically displayed is Figu e 3.5.
Accu acy is no epo ed in hose cases o which he i e pa e ns a e no s o ed. I is wo h o
poin ou ha al hough he cases wi h 2-bi weigh s a e able o keep he a ge pa e ns as ixed
poin s, hei e ie al capabili y as a ac o is eally poo . The e y low accu acy alues achie ed
indica es hei uselessness as AM.
Resul s e inces ha epea ed aining app oach allows o e ain he a ge numbe o pa e ns
while keeping sa is ac o y noise obus ness in spi e o he loss o p ecision due o quan iza ion.
These esul s p o ide he insigh ha he ob ained solu ions a e compe i i e when ew bi s
p ecision is equi ed, showing sco es no a om he FP p ecision e en quan izing up o h ee
bi s. Mo eo e , accu acy ob ained by epea ing ou imes he aining and quan izing o ou bi s
(96.8%) is la ge han he accu acy exhibi ed by he single S o key aining wi h ull p ecision
(90%).
Table 3.7: Quan iza ion impac on digi se 10x6 – 5 pa e ns.
Accu acy (%)
P ecision S o key Repea ed x2 Repea ed x3 Repea ed x4
FP 90.0 96.2 99.2 99.8
5-bi 89.6 94.4 99.6 99.8
4-bi - - 96.2 96.8
3-bi - 87.8 93.2 91.6
2-bi 9.4 7.2 6.8 6.8
Figu e 3.5: Quan iza ion impac on accu acy o 60-neu on ne wo ks ained o s o e 5 digi pa e ns (‘0’-‘4’).
46 Chap e 3: Lea ning o Oscilla o y AM
The pe o mance o he simple S o key ule on he cha ac e se expe imen was so poo (only he
i s ou pa e n whe e s o ed when using he na u al o de ), ha we conside also en pa e ns
o he weigh quan iza ion expe imen . Table 3.8 and Table 3.9 depic ob ained esul s o ou
and en pa e ns espec i ely. Fo he en pa e ns expe imen , we ha e been able o imp o e he
accu acy achie ed wi h ull p ecision weigh s and wice aining i e a ions wi h na u al o de s
(78%) using jus 4 bi s by inc easing he numbe o epe i ions (83.1%).
Table 3.8: Quan iza ion impac on cha ac e se 12x8 – 4 pa e ns.
Accu acy (%)
P ecision S o key Repea ed x3 Repea ed x5
FP 83.8 92.7 92.5
5-bi 85.9 91.9 92.2
4-bi - 84.8 84.9
3-bi - 55.3 80.9
2-bi - - -
Table 3.9: Quan iza ion impac on cha ac e se 12x8 – 10 pa e ns.
Accu acy (%)
P ecision S o key Repea ed x3 Repea ed x5
FP - 85.1 86.5
5-bi - 84.4 86.2
4-bi - 80.0 83.1
3-bi - - -
These esul s show ha he ad an ages o epe i i e S o key app oach a e kep a e quan iza ion.
So, i is conside ed as a p ac ical solu ion o enhance he pe o mance o analog ONNs which
ha e a se e e limi ed weigh p ecision.
I e a i e Random Pa ial Upda e Symme ic Hebbian (IRPUSH)
We ha e also p oposed an i e a i e ule ha is based on D&O’s Rule I [111], desc ibed in Sec ion
3.2, and hence o h e e ed o as he I e a i e Hebbian (IH) ule. Figu e 3.6 shows he low cha
o he no el lea ning algo i hm deno ed I e a i e Random Pa ial Upda e Symme ic Hebbian
(IRPUSH). I consis s o wo nes ed loops, which a e epea ed un il he algo i hm con e ges o a
solu ion ha sa is ies condi ion (Equa ion 3.3) (upd_ lag=0) o un il a limi numbe o i e a ions
is eached. The ex e nal loop is o pa e ns and he inne loop o neu ons. This is exac ly he
same as in he IH ule. The key di e ence is he weigh upda ing p ocedu e.
Fi s , in o de o o ce symme y in D&O’s algo i hm, i is imposed ha he new algo i hm mus
ha e 𝑤=𝑤. In he IH ule, weigh upda es caused by neu on 𝑖 no sa is ying (Equa ion 3.3) ha e
no impac on he compliance o non-compliance s a us o he emaining neu ons because only
𝑤 ∀ 𝑗∈{1,…,𝑁 / 𝑗≠𝑖} a e modi ied. Due o he symme y cons ain , howe e , modi ica ion
o 𝑤 implies modi ica ion o 𝑤, and so o he neu ons a e also a ec ed. F om a di e en poin o
iew, IH does no modi y weigh s 𝑤 i neu on 𝑗 sa is ies (Equa ion 3.3) ( o a gi en pa e n), bu
his is no he case when symme y is o ced. This could nega i ely impac he pe o mance o he
lea ning ule since some neu ons which al eady sa is ied (Equa ion 3.3) migh no ul ill i a e he
weigh upda e o ced by he symme y cons ain . In o de o somehow coun e ac his e ec , we
3.4. I e a i e Random Pa ial Upda e Symme ic Hebbian (IRPUSH) 47
p opose ha only a andomly selec ed ac ion o he weigh s be upda ed a each s ep. This is
mo i a ed in [119], which i s a es ha a educed numbe o connec ions (weigh s) while applying
he i e a i e lea ning leads o a edis ibu ion o he los in o ma ion o he emnan ones. Possibly,
i can con ibu e o he esul ing weigh ma ix being mo e sui able o he quan iza ion.
Thus, he main di e ences, poin ed ou wi h le e s inside a ci cle box in Figu e 3.6, a e:
A. Symme y is imposed.
B. The pa ial ac o de e mines he numbe o connec ions (PF) ha a e andomly selec ed om
he cu en upda e se . A e he upda e s ep, he chosen ones (candida ePF) a e emo ed om he
upda e se . All he weigh s a e es o ed back in he upda e se once i s ca dinali y is unde PF,
imposed by he pa ial ac o .
C. The algo i hm e mina es when he NP inequali ies a e sa is ied o a maximum numbe o
aining i e a ions is eached.
I is in e es ing o poin ou ha he ob ained ma ix solu ion is always he same, in case o null
pa iali y du ing he upda e s ep. Wi h he pa iali y app oach, a andom ac ion o he weigh s
associa ed o he cu en e alua ed neu on is modi ied wi h Hebbian lea ning. The andomness in
his p ocedu e allows o explo e u he di e en solu ions in he weigh s assignmen ha o ally
complies wi h he embedding condi ions o he aining se when he algo i hm con e ges. The
limi on aining i e a ions is implemen ed o p e en p olonged execu ion, which can occu when
lea ning a challenging aining se .
To e alua e he pe o mance o he AM ained wi h IRPUSH in he pa e n ecogni ion ask, he
expe imen wi h he p e ious 12x8 cha ac e da ase was ca ied ou . Di e en 96-neu on
ne wo ks ained wi h ou IRPUSH algo i hm and using di e en alues o pa ame e 𝑇 and
pa ial ac o we e e alua ed o s o ing he p oposed se o 10 cha ac e s. The pa ial ac o ,
which de e mines he numbe o weigh s ha a e modi ied when a neu on is upda ed, was i s
cons ained o 100%: i.e., all 95 weigh s associa ed wi h one neu on being changed. Table 3.10
summa izes he capaci y esul s ob ained in his analysis and assesses he use o weigh s wi h
educed p ecision. The quan iza ion me hod is he same as desc ibed in Equa ion 3.11.
Fo pu poses o compa ison wi h a compe i i e candida e, he IH algo i hm was also e alua ed
unde simila condi ions. Since upda e inc emen s a e no scaled by N, la ge T alues we e chosen.
Thus, case 𝑇=1 in he o iginal wo k co esponds o 𝑇=N in ou implemen a ion.
Table 3.10: Capaci y wi h di e en T alues and weigh p ecision.
IRPUSH, Capaci y (#) IH, Capaci y (#)
T FP 5-bi 4-bi 3-bi FP 5-bi 4-bi 3-bi
0 10 10 9 6 10 8 7 0
10 10 10 10 5 10 10 9 7
50 10 10 10 8 10 10 10 8
110 10 10 10 9 10 10 10 9
150 10 10 10 10 10 10 10 9
200 10 10 10 10 10 10 10 9
400 10 10 10 10 10 10 10 8
As can be seen in Table 3.10, e e y ne wo k e ained he 10 pa e ns using FP weigh s. Howe e ,
IRPUSH ou pe o med he IH capaci y wi h 3-bi p ecision, being able o s o e he 10 pa e ns o he
la ges alues o 𝑇. IRPUSH also ound solu ions a lowes alues o 𝑇 wi h 5-bi and 4-bi p ecision.
48 Chap e 3: Lea ning o Oscilla o y AM
Figu e 3.6: P oposed IRPUSH lea ning ule.
3.4. I e a i e Random Pa ial Upda e Symme ic Hebbian (IRPUSH) 49
To u he explo e he capabili ies o he IRPUSH algo i hm o s o e pa e ns, a la ge case s udy has
been e alua ed. A aining se o P=26 pa e ns (Figu e 3.7) wi h 32x32 (N=1024) pixels ha e been used.
Figu e 3.7: Cha ac e s A-Z, bina y pa e ns wi h dimension 32x32 used as aining pa e ns.
Table 3.11 compa es IRPUSH wi hou pa ial upda e and IH o di e en T alues. Two di e en
pa ame e s a e shown. The minimum weigh p ecision able o s o e he 26 cha ac e s and he
numbe o lea ning i e a ions equi ed by each algo i hm.
Table 3.11: Minimum numbe o bi s and numbe o lea ning i e a ions wi h di e en T alues o IH and
IRPUSH wi hou pa ial upda e o he 26 pa e ns example.
# bi s equi ed # i e a ions
T IH IRPUSH
(100%) IH IRPUSH
(100%)
0 13 10 9,844 8,093
10 10 8 10,137 8,380
50 8 7 11,186 9,123
110 7 7 13,025 10,513
150 7 7 14,121 11,356
200 7 7 15,188 12,594
400 7 6 20,856 17,181
As in he p e ious simple example, i can be obse ed ha inc easing he h eshold alue used
o de i e he weigh ma ix allows educing he weigh p ecision. Also, he supe io i y o IRPUSH
is clea . A solu ion wi h jus 6 bi s was ound o IRPUSH bu no o IH in ou expe imen . IH
equi es a la ge numbe o i e a ions han IRPUSH.
Table 3.12 compa es IRPUSH wi h no pa ial upda e o IRPUSH wi h 33% pa ial ac o .
Ad an ages o IRPUSH (33%) in e ms o numbe o bi s equi ed is clea ly obse ed. I uses a
highe numbe o i e a ions han IRPUSH wi h 100% pa ial ac o o he la ge T alues when
he andom upda e is applied. This is due o a educed inc emen o he neu on po en ial, ℎ, pe
each upda e since a smalle numbe o weigh s a e modi ied, he e o e equi ing mo e i e a ions
o each he h eshold T.
Table 3.12: Minimum numbe o bi s and numbe o lea ning i e a ions wi h di e en T alues o
IRPUSH using a pa ial ac o o 33% and 100% (no pa ial upda e).
# bi s equi ed # i e a ions
T IRPUSH (33%) IRPUSH (100%) IRPUSH (33%) IRPUSH (100%)
0 7 10 7,007 8,093
10 5 8 7,467 8,380
50 5 7 9,293 9,123
110 5 7 12,171 10,513
200 5 7 16,450 12,594
400 5 6 26,107 17,181
50 Chap e 3: Lea ning o Oscilla o y AM
Noise obus ness esul s
Noise obus ness was e alua ed o hose weigh ma ixes ha s o ed he 10 aining pa e ns om
he 96-neu on cha ac e da ase (shaded cells in Table 3.10). As p e iously desc ibed, he e ie al
accu acy o his expe imen was measu ed as he pe cen age o es pa e ns which we e co ec ly
in e ed. Tha is, he pe cen age o cases whe e he closes aining pa e n, indica ed by he
minimum HD, is e ie ed. Figu e 3.7 shows he e ie al accu acy o ne wo ks ained wi h
IRPUSH e sus noise le el. I is e iden ha inc easing he alue o 𝑇 p oduced la ge basins o
a ac ion, pushing o wa d he numbe o noisy pixels ha he ne wo k could ole a e be o e he
numbe o e oneous in e ences began o ise. Howe e , he bene i s o inc easing 𝑇 we e quickly
educed o 𝑇>50, as in ui i ely obse ed in Figu e 3.7.
Figu e 3.7: Re ie al accu acy e sus noise le el o ne wo ks ained wi h IRPUSH wi h di e en T alues.
The accu acy esul s ob ained wi h IRPUSH and IH o di e en 𝑇 alues a e epo ed in Table
3.13, oge he wi h hei pe o mance wi h limi ed p ecision. The ad an ages o IRPUSH we e
g ea ly educed as 𝑇 inc eased, displaying e y simila pe o mance o IH. IRPUSH accu acy loss
a 4-bi p ecision emained unde 3% wi h espec o FP, while accu acy in 3-bi cases was
deg aded by a ound 20%. On he o he hand, he 4-bi IH p esen ed accu acy deg ada ion o up
o 6% wi h espec o FP.
Table 3.13: Re ie al accu acy wi h di e en T alues and weigh p ecision.
IRPUSH, Accu acy (%) IH, Accu acy (%)
T FP 5-bi 4-bi 3-bi FP 5-bi 4-bi 3-bi
0 63.4 57.7 - - 37.1 - - -
10 76.2 70.5 77.1 - 57.8 53.4 - -
50 87.8 86.9 85.6 - 83.6 83.0 80.5 -
110 89.7 89.6 87.9 - 89.5 87.7 86.7 -
150 90.6 90.2 88.3 67.0 90.1 89.0 84.6 -
200 91.2 90.4 89.2 71.3 91.0 90.9 86.3 -
400 91.5 90.3 89.5 72.0 91.6 91.1 89.3 -
3.4. I e a i e Random Pa ial Upda e Symme ic Hebbian (IRPUSH) 51
Keeping in mind ha IRPUSH was used wi h a pa ial ac o equal o 100%, he only di e ence
wi h espec o IH was ha IRPUSH cons ained he weigh ma ix o be symme ic. The esul s
show ha imposing symme y did no penalize he accu acy, as indica ed in Sec ion 3.4, whe e
he en o cemen o symme y was obse ed o po en ially hinde he pe o mance o IRPUSH. In
ac , ou app oach ob ained be e esul s o lowe T alues. The be e accu acy esul s desc ibed
abo e we e p obably a side e ec o he la ge numbe o upda e ope a ions esul ing om ha ing
imposed symme y. Unlike he o iginal asymme ic upda e in IH,
𝑤
(and
𝑤
) could bo h be
modi ied when p ocessing neu on i o neu on j. O e en g ea e in e es , howe e , was he success
o ou app oach in s o ing he en pa e ns wi h 3-bi weigh s, as poin ed ou in he capaci y
e alua ion, o T ≥150.
The nex s ep was o e alua e IRPUSH o pa ial ac o s below 100% so ha he p oposed
andom pa ial upda e mechanism was ac ually applied. Tha is, he ne wo ks we e also e alua ed
by applying he philosophy o upda ing a educed numbe o weigh s, each ime an upda e s ep
ook place. They had a common T alue o 95 (co esponding o se he Rule I h eshold o uni y).
The pa ial ac o was educed om 100% o 1% in s eps o 25%, plus he selec ed ac o s o 33%
and 10%. All he ne wo ks s o ed he aining se pe ec ly.
I is in e es ing o no e ha he algo i hm e u ns a di e en weigh ma ix solu ion each ime i is
un, because he connec ions o be upda ed a e andomly selec ed. A ba e y o 100 expe imen s
o each case was he e o e ca ied ou . Figu e 3.8 epo s hei espec i e maximum, a e age,
and s anda d de ia ion alues. No e ha he case in which all neu ons we e upda ed (100% pa ial
ac o o PF equal o 95), an iden ical solu ion was he e o e e u ned e e y ime. I was ound
ha educing he numbe o upda ed connec ions esul ed in an inc ease in accu acy wi h ega d
o IRPUSH wi h a pa ial ac o o 100%.
Figu e 3.8: Accu acy e sus PF o IRPUSH wi h T=95.
The impac o 𝑇 was also explo ed o wo pa icula cases: pa ial ac o s o 33% and 25%. Table
3.14 shows he accu acy esul s o di e en 𝑇 alues and weigh p ecisions. No e ha accu acy
inc eases wi h T o FP and 5-bi weigh s; howe e , o 3-bi p ecision, bo h inc emen s and
dec emen s a e obse ed. This is explained because o he la ge impac o ounding when
p ecision is signi ican ly educed.
52 Chap e 3: Lea ning o Oscilla o y AM
Table 3.14: Accu acy o IRPUSH s T wi h pa ial upda e.
Accu acy, 33% ac o (%) Accu acy, 25% ac o (%)
T FP 5-bi 4-bi 3-bi FP 5-bi 4-bi 3-bi
0 39.0 40.4 35.0 - 30.5 30.5 - -
10 80.7 82.0 80.9 - 82.4 81.0 78.2 -
50 91.7 91.4 90.8 84.4 90.8 90.9 89.5 84.0
110 92.6 92.4 91.9 90.2 93.1 92.7 91.9 87.5
150 92.7 93.0 92.5 90.9 92.8 92.4 93.5 75.0
200 93.2 93.0 92.2 83.3 92.9 92.7 92.2 84.6
400 93.4 93.2 93.3 - 92.3 92.5 93.3 82.6
I is wo h no ing ha IRPUSH wi h he andom pa ial upda e mechanism was qui e sa is ac o y
no only in he numbe o cases s o ing he 10 pa e ns. Compa ing he esul s ob ained o he
mos educed numbe s o bi s wi h pa ial upda e (Table 3.14) and wi hou (Table 3.13), i can be
obse ed ha much be e esul s a e achie ed wi h pa ial upda e. Fo example, accu acy up o
90.9 wi h 33% o pa ial ac o and h ee bi s is ob ained while he bes esul s wi hou pa ial
upda e and ha p ecision is only 72%. Tha is, he esul s suppo ou hypo hesis conce ning he
bene i s o he pa ial upda e p ocedu e.
To u he compa e he in e ence obus ness o he IRPUSH algo i hm agains noise, Figu e 3.9a
depic s he e ie al pe o mance o IRPUSH wi h pa ial ac o s o 100% and 33% agains
di e en noise le els o he same T alue o 150. Simila esul s we e ob ained wi h FP and a
pa ial ac o o 100% and wi h 3-bi p ecision and a pa ial ac o o 33%. Bo h ou pe o m 3-bi
p ecision wi h a pa ial ac o o 100%. This clea ly illus a es he ad an age o he andom pa ial
upda e mechanism using limi ed p ecision and demons a es i s abili y o a oid se ious
deg ada ion o he e ie al capabili ies unde low weigh p ecision, when compa ed wi h he FP
p ecision case. Figu e 3.9b compa es he maximum numbe o noisy pixels o which he
e alua ed ne wo ks achie ed an accu acy g ea e han 95%. I shows he bes esul ob ained
among di e en T alues in each case. The pa ial upda e s a egy o 3-bi weigh s had a majo
impac , aising his igu e o me i om 7 o 27.
4.2. Analog CMOS-ONN demons a o desc ip ion 59
Figu e 4.1: (a) Schema ic o a di e en ial oscilla o . (b) Schema ic o he emula o ci cui o he VO2 de ice.
Table 4.1: Singled-ended oscilla o s design pa ame e s.
Pa ame e Value
C 50 F
R1 35kΩ
R2 40kΩ
WSHIL, LSHIL 120nm, 480nm
WEMU, LEMU 1µm, 500nm
Figu e 4.2: Physical layou o he di e en ial oscilla o included in he ASIC.
Synapse
As an analogy o he Whea s one b idge, Figu e 4.4a shows he schema ic o he synap ic ci cui ,
which is capable o p o iding posi i e, nega i e, and ze o weigh s. Being a ou - e minal ci cui
makes i app op ia e o di e en ial s uc u es. Depending on he ga e ol ages, i is possible o
ha e posi i e (VP>VN), nega i e (VN>VP), o ze o weigh (VP=VN). The wo PMOS ansis o s a e
used o con olling he cu en be ween he neu ons. T ansis o s be ween he posi i e b anches
can ans e cu en when (VP-V1+)< VTH, because VP is applied o hei ga e. In addi ion,
ansis o s be ween he nega i e b anches can ans e cu en when (VN-V2+)<VTH, because VN is
applied o hei ga e. The e o e, cu en be ween he neu ons is con olled using hese PMOS
ansis o s. Figu e 4.4b depic s he opology o he ab ica ed di e en ial ONN (DONN). All
ansis o s a e equally sized wi h W=360nm and L=120nm. The physical layou o he synap ic
ci cui is shown in Figu e 4.5. The a ea o he synap ic uni y cell is 63.2m2.
Each synapse con ol ol age can be connec ed o six di e en ol ages by means o
p og ammable swi ches con olled by p og amming egis e s, simila ly o he oscilla o
calib a ion shown in Figu e 4.3a.
60 Chap e 4: ASIC CMOS-ONN Demons a o
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Q
Q
G RB
CLR
D
Se ial
Pa allel
Da aReady
Se ial da aSe ial da a
CTRLP
CTRLN
Vosc1 Vosc2 Vosc3
Vcal
Oscilla o
Oscilla o OSCi
Vosc4 Vosc5 Vosc6
CLK
Figu e 4.3: (a) Schema ic o oscilla o calib a ion ol age selec ion. (b) Scheme o a di e en ial oscilla o .
b)a)
Figu e 4.4: (a) Schema ic o he synapsis. (b) DONN implemen a ion om [124].
Figu e 4.5: Physical layou o he synap ic ci cui .
a)
b)
4.2. Analog CMOS-ONN demons a o desc ip ion 61
ASIC desc ip ion
The ASIC consis s o he ollowing blocks:
- ONN o 9 di e en ial oscilla o s (9-neu on DONN) ully in e connec ed wi h each o he h ough
36 synapses. The DONN includes a con ol sys em om which he ol ages de ining he synapse
weigh s can be selec ed, as well as he oscilla o calib a ion mechanism o imp o e ne wo k
synch oniza ion. The oscilla o ou pu s a e connec ed o digi al pads h ough Schmi -T igge
bu e s wi h con igu able hys e esis o egene a e ail- o- ail signal swing and o cope wi h he
wa e o m shape o he elaxa ion oscilla o s.
- Basic ci cui s. Speci ically, a single-ended oscilla o , a di e en ial oscilla o and wo di e en ial
oscilla o s connec ed h ough a synapse simila o he one used in he DONN ha e been included.
Thei ou pu s a e digi al as in he DONN ci cui . In addi ion, a di e en ial oscilla o has been
included whose ou pu s a e connec ed di ec ly o analog pads in o de o be able o obse e he
wa e o ms wi hou digi izing.
Figu e 4.6 shows he layou o he ab ica ed ci cui , showing he h ee ypes o ci cui s included
(3x3 ONN, simple ci cui s, di e en ial oscilla o wi h analog ou pu s) and hei connec ions o
he pad ing. The 3x3 DONN occupies a ec angle o 776µm · 747µm wi h some emp y a ea
inside and he comple e chip a ea including pad ing is 1710µm · 1710µm.
Basic ci cui s
3x3 ONN
Di e en ial oscilla o
wi h analog ou pu s
Figu e 4.6: Layou o he ab ica ed ci cui .
Desc ip ion o he signals/pads
Table 4.2 summa izes he main cha ac e is ics o he ASIC signals, di ided in o he ollowing ca ego ies:
- Pola iza ion and g ound signals. Supply and g ound ol ages o he co e (VDD, VSS) and he
pad- ing powe supply and g ound (VDDPST, 3.3V, and VSSPST, 0V, espec i ely) a e included.
- Oscilla o supply ol ages. Since di e en ial oscilla o s a e being used and he e a e 9 neu ons
in he DONN, 18 signals a e equi ed. These signals a e gene a ed ex e nally and applied o digi al
pads ha gene a e s ep signals be ween 0V and 1.2V. Con olling he ela i e iming on ini ial
phase selec ion s ep is essen ial: a delay be ween bo h signals co esponding o hal a pe iod
in ol es applying inpu s imuli wi h opposi e phases. The ini ializa ion signals o he i- h
oscilla o a e deno ed as VDDOSC<i> and VDDOSCx<i>.
62 Chap e 4: ASIC CMOS-ONN Demons a o
- Con ol sys em signals. Includes he clock signal (CLK), he signal ha codi ies he in o ma ion
o be loaded in o he calib a ion/p og amming ol age selec ion egis e s (DATA_CTRL), and he
signal ha indica es ha he in o ma ion has been loaded in o he se ial egis e s and se ial- o-
pa allel con e sion can be done (DATA_READY).
- Oscilla o calib a ion signals. These six signals can ake alues be ween 0V and 1.2V and allow
he oscilla o equency o be uned. They a e e e ed o as IN_OSC<1> o IN_OSC<6>.
- Synapse p og amming signals. These wel e signals a e used o se he weigh s o he synapses.
They ake alues be ween 0V and 1.2V and a e deno ed as IN_SYN<1> o IN_SYN<12>. The
IN_SYN<11> and IN_SYN<12> signals a e also used o se he h esholds o he DONN ou pu
Schmi -T igge bu e s.
- Synch oniza ion signal. I is he digi al signal VSYNC, wi h a equency double ha o he
DONN, used o enable he SHIL mechanism. This ex e nal inpu anges om 0V o 3.3V.
- Ou pu mul iplexe con ol signals. Named OUT1_CTRL, OUT2_CTRL and OUT3_CTRL. Used o
con igu e he mul iplexe -ou pu s age o allow access o he DONN ou pu s and he basic ci cui s ones.
- Digi al ou pu signals. These a e he signals OUT<1> o OUT<10> o he DONN ou pu s and
OUT_SIMPLE o he ou pu o he basic ci cui s. They show a ol age ange be ween 0V and 3.3V.
- Analog ou pu signals. F om he isola ed di e en ial oscilla o connec ed o analog pads,
e e ed o as OUT_OSC and OUT_OSCx. The ou pu ol age ange is be ween 0V and 1.2V.
Table 4.2: ASIC ex e nal signals.
Name Numbe o pins Pu pose Type Pad ype Vol age ange
VDD 2 Co e supply ol age I/O Supply 1.2V
VSS 2 Co e g ound I/O G ound 0V
VDDPST 1 Pad- ing supply ol age I/O Supply 3.3V
VSSPST 1 Pad- ing g ound I/O G ound 0V
VSSA 1 Analog g ound I/O G ound 0V
VDDOSC 9 Ini ializa ion Inpu Digi al 0V-3.3V
VDDOSCx 9
CLK 1
Con ol logic Inpu Digi al 0V-3.3V DATA_CTRL 1
DATA_READY 1
IN_OSC 6 Oscilla o calib a ion Inpu Analog 0V-1.2V
IN_SYN 12 Synapses p og amming Inpu Analog 0V-1.2V
VSYNC 1 SHIL signal Inpu Digi al 0V-3.3V
OUT1_CTRL 1
MUX selec ion Inpu Digi al 0V-3.3V OUT2_CTRL 1
OUT3_CTRL 1
OUT 10 Ou pu s DONN Ou pu Digi al 0V-3.3V
OUT_SIMPLE 1 Ou pu simple ci cui s Ou pu Digi al 0V-3.3V
OUT_OSC 1 Analog ou pu s Ou pu Analog 0V-1.2V
OUT_OSCx 1
4.2. Analog CMOS-ONN demons a o desc ip ion 63
Figu e 4.7 desc ibes he oo p in o he 64-pin quad la no-lead (QFN) package chosen o he
ASIC and he co espondence be ween i s pins and he inpu /ou pu signals o he ci cui .
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
OUT_SIMPLE
OUT<1>
OUT<2>
OUT<3>
OUT<4>
OUT<5>
OUT<6>
OUT<7>
OUT<8>
OUT<9>
OUT<10>
VDD
VDDPST
VSS
VSSPST
VDDOSC<9>
VDDOSCx<8>
VDDOSC<8>
VDDOSCx<7>
VDDOSC<7>
VDDOSCx<6>
VDDOSC<6>
VDDOSCx<5>
VDDOSC<5>
VDDOSCx<4>
VDDOSC<4>
VDDOSCx<3>
VDDOSC<3>
VDDOSCx<2>
VDDOSC<2>
VDDOSCx<1>
VDDOSC<1>
VDDOSCx<9>
VSYNC
DATA_CTRL
DATA_READY
CLOCK
VDD
VSS
CTRL1_OUT
CTRL2_OUT
CTRL3_OUT
VSSA
OSC_ANA
OSCx_ANA
IN_OSC<1>
IN_OSC<2>
IN_OSC<3>
IN_OSC<4>
IN_OSC<5>
IN_OSC<6>
IN_SYN<1>
IN_SYN<2>
IN_SYN<3>
IN_SYN<4>
IN_SYN<5>
IN_SYN<6>
IN_SYN<7>
IN_SYN<8>
IN_SYN<9>
IN_SYN<10>
IN_SYN<11>
IN_SYN<12>
Figu e 4.7: Foo p in o he QFN64 package and names o he signals.
Con ol logic o calib a ion and p og amming
This mechanism is desc ibed as ollows:
- Each oscilla o and each synapse ha e as many lip- lops as swi ches o be con olled. Tha is, 12
o each oscilla o (see Figu e 4.5b) and 12 o each synapse. They all a e con igu ed in a single
shi egis e , gene a ing, he e o e, a connec ion o 12·9+12·36=540 memo y elemen s. These
egis e s a e con olled by he clock signal. A con ol wo d is se ially loaded in he shi egis e . I
con ains he calib a ing and p og amming bi s indica ing which swi ches a e closed and which a e
no . Ob iously, o each oscilla o o synapse only one o i s swi ches should be closed.
- Once he con ol wo d is ully loaded, a signal ha indica es ha he da a is eady o be loaded
is ac i a ed and he in o ma ion con ained in he shi egis e s is loaded in pa allel o he lip-
lops di ec ly con olling he swi ches.
64 Chap e 4: ASIC CMOS-ONN Demons a o
Digi al Ou pu S age
Since he ou pu s o he 9 di e en ial oscilla o s (18 in o al) ange be ween alues away om
g ound and VDD, i is necessa y o ca e ully design an ou pu s age ha deli e s pe ec ly
econs uc ed signals o he digi al pads. The ou pu s age is u he di ided in o wo sub-s ages:
1. A digi iza ion o he analog signal om he oscilla o mus be pe o med. A equen p oblem
in he ou pu o hese oscilla o s is ha a ise and all can appea a ound he middle o VDD
which can cause gli ches i a con en ional bu e is placed a he ou pu . To sol e his
p oblem, a p og ammable Schmi -T igge bu e is used, as shown in Figu e 4.8. By means
o he VCTRLP and VCTRLN ol ages, he ansi ion h esholds o he Schmi -T igge can
be con olled, hus supp essing he gli ch and keeping he ansi ions wi hin he ange o he
oscilla o ou pu a ia ion ol ages. VCTRLP and VCTRLN signals a e in e nally connec ed
o signals IN_SYN<11> and IN_SYN<12>, espec i ely.
V
DD
VCTRLN
VCTRLP
V
DD
OUT
IN
Figu e 4.8: Schema ic o he designed Schmi -T igge bu e .
2. Once all he ou pu s ha e been p ope ly digi ized, 10 4-inpu mul iplexe s (con ol signals
CTRL1_OUT and CTRL2_OUT) ha e been used o selec di e en combina ions o ou pu s,
as shown in Table 4.3. In addi ion, one o he combina ions moni o s he DATA_CTRL signal
a he ou pu o he las shi egis e .
Table 4.3: Con igu a ion o he DONN ou pu mul iplexe s acco ding o he con ol bi s.
OUT
(CTRL1_OUT, CTRL2_OUT)
(0,0) (1,0) (0,1) (1,1)
OUTPAD,1
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,2
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,3
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,4
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,5
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,6
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,7
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,8
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,9
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
OUTPAD,10
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝑂𝑈𝑇
,
𝐷𝐴𝑇𝐴
_
𝐶𝑇𝑅𝐿
4.3. Expe imen al alida ion and cha ac e iza ion 65
Unlike he DONN oscilla o ou pu s, in he basic ci cui s i is no necessa y o use he Schmi -
T igge bu e . The e o e, a con en ional bu e is used o digi ize he signal. The obse able
signals a he ou pu a e 7, so i is necessa y o use a mul iplexe con olled by 3 signals
(CTRL1_OUT, CTRL2_OUT and CTRL3_OUT). The emaining ou pu has been used o display
he synch oniza ion signal. The co espondence be ween he con igu a ion o he con ol bi s and
he ou pu is summa ized Table 4.4.
Table 4.4: Con igu a ion o he ou pu mul iplexe o he se o simple ci cui s acco ding o con ol bi s.
CTRL1_OUT
CTRL2_OUT
CTRL3_OUT
OUT_SIMPLE
0 0 0 Single-ended oscilla o
1 0 0 Di e en ial oscilla o (pos)
0 1 0 Di e en ial oscilla o (neg)
1 1 0 Coupled oscilla o s (OSC1 pos)
0 0 1 Coupled oscilla o s (OSC1 neg)
1 0 1 Coupled oscilla o s (OSC2 pos)
0 1 1 Coupled oscilla o s (OSC2 neg)
1 1 1 Synch oniza ion signal
Expe imen al alida ion and cha ac e iza ion
The expe imen al e i ica ion o he ASIC has been pe o med using a cus om designed es
p in ed-ci cui boa d (PCB) o his pu pose. The block diag am o he expe imen al se -up is
shown in Figu e 4.9a, oge he wi h a pho og aph (Figu e 4.9b). Addi ionally, used es equipmen
is also depic ed. No e ha an FPGA is used o p og amming he DONN (selec ing he assignmen
o synapses and calib a ion ol ages) and o con olling he ime-delayed powe -on o he
di e en ial oscilla o s. A physical mic o-swi ch on-boa d allows o con igu e he ini ial s a e o
he sys em, on which basis he inpu pa e ns a e applied o he DONN.
The main ea u es o he expe imen al se -up a e summa ized below:
- The ini ializa ion o he oscilla o s is pe o med om wo digi al signals. They s ep in om low
o high le el wi h a delay equi alen o hal a pe iod o he oscilla o s, and a e applied o he
supply ol ages o he di e en ial oscilla o s. The o de in which hey a e applied is gi en by he
posi ion o he swi ch associa ed wi h each oscilla o (whi e box wi h colo ed swi ches loca ed a
he op o he PCB pic u e). These signals a e also gene a ed by he cus om digi al design in he
FPGA a e ha he Digi al Disco e y module igge s a START signal. Addi ionally, he supply
ol ages can be immedia ely u ned o wi h a RESET signal om he Digi al Disco e y module.
-The synch oniza ion signal is ob ained using he Tek onix AFG3102 unc ion gene a o , whose
ou pu is connec ed o he SMA connec o on he PCB.
- The gene a ion o he calib a ion and p og amming ol ages (pins 8 o 25 in Figu e 4.7) is ca ied
ou in he PCB wi h a simple ci cui consis ing o an ope a ional ampli ie , a po en iome e ,
esis o s and capaci o s.
- Digi al ou pu s can be obse ed using oscilloscope p obes o he logic analyze included in he
Digilen Digi al Disco e y module.
- Analog ou pu s can be obse ed using oscilloscopes p obes. A Keysigh DSOX4104A
oscilloscope has been used.
66 Chap e 4: ASIC CMOS-ONN Demons a o
Figu e 4.9: (a) Block diag am and (b) pho og aph o he expe imen al se up
b)
a)
4.3. Expe imen al alida ion and cha ac e iza ion 67
Figu e 4.10a depic s an al e na i e high-le el iew o his expe imen al se -up. The a ows wi h
di e en colo s indica e he ype o in e ac ion and he a ow ip, he des iny o he in e ac ion. A
MATLAB sc ip allows he au oma ed con igu a ion and ini ializa ion o e e y ins umen . A
GUI-app, shown in Figu e 4.10b, was de eloped on he Digi al Disco e y endo so wa e [126]
o ha e easy and simul aneous con ol o e he p og amming and he u n-on o he DONN and
he logic analyze measu emen s.
The digi al design in he FPGA eads he con ol signals (VDD_ON, LOAD_CFG and RESET)
and gene a es he p og amming and he ope a ing signals o he DONN: clock and bi s eam o
he p og amming, he SHIL igge and he oscilla o supply ol ages. Depending on he posi ion
o he FPGA swi ches, i selec s he ne wo k con igu a ion o be p og ammed and, also, i he
expe imen will be execu ed single o mul iple imes.
Once he expe imen is s a ed, he logic analyze acqui es he measu emen o he nine digi al
ou pu s as ime-s amped e en s ha a e sa ed in a comma-sepa a ed alue (CSV) ile. The CSV
iles a e la e pos -p ocessed wi h MATLAB sc ip s o ob ain he ele an in o ma ion: he
oscilla o s phase di e ence, oscilla ing pe iods, s abili y, ime o achie e a s able phase ela ion
(DONN s a e), e c.
Figu e 4.10: (a) Diag am o ins umen s in ol ed in he se -up. (b) F on panel o he GUI-app o
con olling he ONN expe imen s.
a)
b)
68 Chap e 4: ASIC CMOS-ONN Demons a o
E alua ing oscilla o beha io
Al hough he DONN sys em has only digi al ou pu s, an isola ed di e en ial oscilla o , iden ical
o he ones in he DONN, was also included in he chip and connec ed o analog pads in o de o
be able o obse e i s beha io . Figu e 4.11 depic s expe imen al wa e o ms o ha analog
oscilla o . Expec ed beha io is obse ed wi h he wo ou pu s being ou o phase. As p e iously
poin ed ou , oscilla ions a e no ull swing. Ou pu a e age ol age anges om 379mV o 763mV.
No e he small oscilla ion ampli ude jus i ies he equi ed ca e ully designed ou pu s age included
o digi aliza ion.
Figu e 4.11: Expe imen al wa e o ms o a di e en ial oscilla o wi h analog ou pu s.
Figu e 4.12 depic s he wo ou pu s o a di e en ial oscilla o s a e digi aliza ion wi h he ou pu
s age desc ibed in p e ious Sec ion. I can be also obse ed ha hey a e ou o phase showing
co ec ope a ion.
Figu e 4.12: Expe imen al wa e o ms o di e en ial oscilla o ou pu s a e digi aliza ion.
Figu e 4.13 illus a es he c i ical ole o he ou pu s age. I depic s ob ained wa e o ms o one
o he di e en ial oscilla o s o di e en con igu a ions o he ou pu bu e s age. I can be
obse ed ha he du y cycle o he digi al oscilla o ou pu s changes. In he las con igu a ion, an
undesi ed gli ch is obse ed. A oiding his beha io was he eason o include he Schmi -T igge
bu e ins ead o a simple one o con e ing he oscilla o ou pu o ull swing.
4.4. Ope a ing he CMOS-ONN as AM 75
- The pa e n in e ed in pos layou simula ion. This pos -layou simula ions ha e been ca ied
ou wi hou misma ching. SHIL signal is applied.
- Expe imen al e ie ed pa e n. Di e en expe imen s a e epo ed. They di e in he applied
synap ic ol ages, as well as in he ope a ion condi ions, as can be seen in he able. Fi e ials
ha e been ca ied ou o each inpu pa e n. The numbe o imes P1 is e ie ed ollowed by he
numbe o imes P2 is e ie ed is depic ed. Reading ope a ion is ca ied ou 100µs a e he
applica ion o he es pa e ns.
I can be obse ed ha simula ion esul s ma ch pe ec ly hose ob ained wi h he HNN model.
Fo he en pa e ns ha a e close o one o he s o ed pa e ns han o he o he s, ha one is
e ie ed, as expec ed. Fo he six es pa e ns which a e a equal dis ance om bo h s o ed
pa e ns, a s able ou pu pa e n which is nei he o he s o ed ones is ob ained.
Expe imen EXP1 co esponds o ep oducing he condi ions o he simula ion. Since misma ch was
no conside ed in he simula ions, and so, he equency o all he oscilla o s was iden ical,
calib a ion has been applied. SHIL was applied as in simula ions. SHIL equency was di e en
because ab ica ed oscilla o s we e as e han p edic ed by pos -layou analysis. Also no e ha a
sligh ly highe synapse ol age was equi ed in he expe imen . EXP1 esul s exhibi di e ences
wi h espec o simula ion esul s. Fi s , no spu ious pa e ns we e e ie ed. Ins ead, o hose es
pa e ns a equal dis ance om P1 and P2, one o hem was e ie ed. Howe e , no always he same
one. Tha is, a non-de e minis ic beha iou was obse ed. Secondly, o wo o he es pa e ns
close o P2 han o P1 (T3 and T8), one (T3) o wo ials (T8) ailed o e ie e he expec ed pa e n.
Howe e , o bo h es pa e ns, he co ec one was in e ed in he majo i y o he ials.
EXP2 di e ed om EXP1 only in he synapse ol age applied, which ha e been inc eased om
0.85V o 0.95V. No e, ha hen bo h T3 and T8 co ec ly e ie ed P2 in he i e ials. Howe e ,
T7 did no . We applied en imes T7 and e ie ed eigh imes he expec ed pa e n. Al hough he
numbe o ials was oo low o allow o ex ac inal conclusions, ob ained esul s seemed o
indica e ha inc easing he s eng h o he coupling, he DONN pe o mance was imp o ed.
Addi ionally, wo cases whe e es e alua ion e u ned a spu ious pa e n wi h bo h he model and
pos -layou simula ion, esul ed in o uns able phases´ ela ionships (T11, deno ed as ‘x#’ in Table
4.8) when e alua ed wi h he eal ASIC.
In EXP3, oscilla o s we e no calib a ed. Resul s we e wo se in e ms o ole ance o noise. No e
ha only in h ee o he es pa e ns, he co ec expec ed pa e n was e ie ed in all he ials.
Finally, in EXP4, he oscilla o s we e calib a ed bu SHIL is no applied. Be e esul s han in
EXP3 we e ob ained.
No esul s we e epo ed o he DONN ope a ing wi hou calib a ion and wi hou SHIL because
his expe imen was no success ul.
76 Chap e 4: ASIC CMOS-ONN Demons a o
Figu e 4.19: (a) The lea ned pa e ns and (b) he es pa e ns selec ed o he 3x3 DONN AM.
Table 4.8: DONN expe imen esul s co esponding o he AM.
Expec ed
HNN
ASIC
Pos -Layou
Simula ion
ASIC
EXP1
ASIC
EXP2
ASIC
EXP3
ASIC
EXP4
SHIL - - 8MHz 14.3MHz 14.3MHz 14.3MHz No
Posi i e weigh s - - Vp = 0.8V
Vn = 0V
Vp = 0.85V
Vn = 0V
Vp = 0.95V
Vn = 0V
Vp = 0.95V
Vn = 0V
Vp = 0.95V
Vn = 0V
Nega i e weigh s
- - Vp = 0V
Vn = 0.8V
Vp = 0V
Vn = 0.85V
Vp = 0V
Vn = 0.95V
Vp = 0V
Vn = 0.95V
Vp = 0V
Vn = 0.95V
Ze o weigh s - - Vp = 0V
Vn = 0V
Vp = 0V
Vn = 0V
Vp = 0V
Vn = 0V
Vp = 0V
Vn = 0V
Vp = 0V
Vn = 0V
Calib a ion - - - Yes Yes No Yes
P1 P1 P1 P1 5/0 5/0 5/0 5/0
P2 P2 P2 P2 0/5 0/5 0/5 0/5
T1 (4) P1, P2 Spu ious Spu ious 1/4 2/3 3/2 2/1/x2
T2 (3) P1 P1 P1 5/0 5/0 4/1 5/0
T3 (3) P2 P2 P2 1/4 0/5 1/4 0/5
T4 (4) P1, P2 Spu ious Spu ious 2/3 5/0 2/3 0/1/x4
T5 (3) P2 P2 P2 0/5 0/5 0/5 0/5
T6 (4) P1, P2 Spu ious Spu ious 1/4 4/1 1/4 1/2/x2
T7 (2) P2 P2 P2 0/5 2/3 1/4 5/0
T8 (3) P2 P2 P2 2/3 0/5 2/3 0/5
T9 (3) P1 P1 P1 5/0 5/0 5/0 5/0
T10 (2) P1 P1 P1 5/0 5/0 3/2 5/0
T11 (4) P1, P2 Spu ious Spu ious 2/3 3/0/x2 0/5 2/0/x3
T12 (3) P1 P1 P1 5/0 5/0 5/0 5/0
T13 (4) P1, P2 Spu ious Spu ious 3/2 5/0 2/3 3/0/x2
T14 (3) P1 P1 P1 5/0 5/0 4/1 5/0
T15 (3) P2 P2 P2 0/5 0/5 1/4 0/5
T16 (4) P1, P2 Spu ious Spu ious 4/1 5/0 1/4 0/1/x4
4.4. Ope a ing he CMOS-ONN as AM 77
In o de o measu e he epea abili y and he s abili y o he DONN ope a ing as AM, ano he se
o expe imen s wi h a la ge numbe o ials was ca ied ou . Addi ionally, he ex e nal SHIL
gene a o was synch onized wi h he powe -on o he oscilla o s o ensu e ha he si ua ion o he
SHIL signal and he s a o he oscilla ions was iden ical o each ial. In p e ious desc ibed
expe imen s we we e using a SHIL signal ha was comple ely independen om he DONN
con ol signals. Table 4.9 depic s he ob ained esul s o 100 ials. Synapses con ol ol ages
co espond o hose o EXP2. Fo each ial, he eading ope a ion was ca ied ou a di e en
ime ins an s a e he applica ion o he es pa e ns, conc e ely a 3, 10 and 720 oscilla ion cycles
om he beginning. Conside ing ha he oscilla ion pe iod was app oxima ely 140ns, hese
measu emen s co esponded o 42ns, 1.4µs and 100.5µs.
Pa icula ly, T7 was he only es pa e n closes o a single s o e pa e n ha did no con e ge o
he expec ed pa e n in 90 ou o 100 a he i s eading a 3 cycles, bu he DONN quickly in e ed
and s abilized in he co ec pa e n om he second eading a 10 cycles, 1 us la e . Addi ionally,
i can be obse ed om he hi d ead a 720 cycles ha he e ie ed pa e n is kep . E en o he
es pa e n a equal dis ance om bo h s o es pa e ns one o hem is e ie ed in he hi d eading
wi hou spu ious. So, i is concluded ha he DONN success ully s o es he wo pa e ns.
Table 4.9: Summa y o expe imen al esul s co esponding o he AM.
Pa e n Expec ed Read
(@ 3 cycles)
Read
(@ 10 cycles)
Read
(@ 720 cycles)
P1 P1 100/0 100/0 100/0
P2 P2 0/100 0/100 0/100
T1 (4) P1, P2 99/1 99/1 99/1
T2 (3) P1 100/0 100/0 100/0
T3 (3) P2 0/100 0/100 0/100
T4 (4) P1, P2 53/25/22sp 54/43/3sp 54/46
T5 (3) P2 0/100 0/100 0/100
T6 (4) P1, P2 97/1/2sp 97/2/1sp 98/2
T7 (2) P2 0/10 0/100 0/100
T8 (3) P2 0/100 0/100 0/100
T9 (3) P1 100/0 100/0 100/0
T10 (2) P1 100/0 100/0 100/0
T11 (4) P1, P2 0/75/25sp 0/96/4sp 0/100
T12 (3) P1 100/0 100/0 100/0
T13 (4) P1, P2 0/69/31sp 0/95/5sp 0/100
T14 (3) P1 100/0 100/0 100/0
T15 (3) P2 0/100 0/100 0/100
T16 (4) P1, P2 1/99 0/100 0/100
The exe cise o s o ing h ee pa e ns was also add essed. Figu e 4.20 illus a es hem. No e ha
he coupling ma ix associa ed wi h his con igu a ion equi ed a g ea e weigh p ecision since i
used mo e han one unique alue o posi i e and nega i e couplings. S ong posi i e (nega i e)
couplings we e coded wi h VP=1.2V and VN=0V (VP=0V and VN=1.2V), while he weak posi i e
(nega i e) couplings wi h VP=0.48V and VN=0V (VP=0V and VN=0.48V) and uncoupled wi h
VP=VN=0V.
Se e al expe imen s wi h 100 ials o each un we e pe o med o p o e he s o age capabili ies
o he ne wo k. The injec ed SHIL signal was ope a ing a 14.2MHz.
78 Chap e 4: ASIC CMOS-ONN Demons a o
Figu e 4.20: T aining pa e ns o he h ee-pa e ns AM e alua ion.
The a e age o he success ul ials o he i s and he hi d pa e n we e 96.8% and 92%.
Howe e , he second pa e n ended o con e ge o a spu ious s a e. I was only e ie ed in 13.6%
on a e age while he spu ious was achie ed in he 84% o he ials. Al hough pos -layou
simula ion esul s showed he DONN o s o e hese h ee pa e ns, simula ion se up was oo much
ideal. Thus, i is no su p ising ha , a leas in ou es ic ed expe imen s, we we e no able o
achie e i . These esul s sugges ed ha u he expe imen s wi h he calib a ion and p og amming
ol ages could be equi ed.
Ope a ing he CMOS-ONN as IM
I was desc ibed in he in oduc o y Chap e ha an ONN can be used o implemen an IM able
o ob ain he g ound s a e o an Ising model. I was also in oduced ha he Max-Cu op imiza ion
p oblem can be o mula ed as an Ising model. In his sec ion we desc ibed he esul s o using ou
DONN o ob aining he Max-Cu o di e en 9-node g aphs. Figu e 4.21 depic s he h ee 9-
node g aphs ha ha e been explo ed.
Acco ding o Sec ion 1.5, in o de o sol e Max-Cu p oblems, he ONN mus be p og ammed
such ha edges a e ansla ed o nega i e couplings. The co esponding synapses we e coded wi h
VP=0V and VN=0.85V while he uncoupled had VP=VN=0V. I was equi ed o apply SHIL a
14MHz o implemen he bina y spins o he IM.
The op imum cu o he di e en g aphs is depic ed in Table 4.10 along wi h he esul s o he
expe imen s ca ied ou wi h he ASIC. As explained, he cu is he numbe o edges sha ed
be ween wo nodes ha belong o he wo di e en subse s (de e mined by he assigned spin).
Pa icula ly, in his IM implemen a ion, he Ising spin was ep esen ed wi h he bina ized
oscilla o phase, wi h espec o a selec ed phase e e ence. The s a e o he sys em co esponded
o he whole collec ion o spins.
Each g aph ins ance was e alua ed ga he ing 100 ials each ime. The du a ion o each ial was
100μs. Du ing he pos -p ocessing wi h he de eloped MATLAB sc ip s, he oscilla o s a es o
each ial we e e alua ed in segmen s o 5μs, which led o 20 measu ed s a es ha we e e alua ed
in e ms o cu s. Tha is, he bina y s a es we e in oduced as inpu in one o he sc ip s o compu e
he ob ained cu alue. So, o each un, 2000 s a es we e ead (20 measu emen s · 100 ials).
The numbe o he s a es p o iding an op imum cu alue is epo ed in Table 4.10 (‘Op imum
Cases’ column). Addi ionally, he di e en numbe s o cu alues ob ained a e also epo ed.
No e ha in mos cases, he expe imen s we e conduc ed wi h all he swi ches in an iden ical
posi ion (OFF). This co esponds, as was al eady explained, o all he oscilla o s being ini ialized
in he same s a e o phase. Fo g aph G9A, a di e en ini ial s a e wi h only one swi ch (‘Swi ch
1’) ON was also ied. I can be clea ly obse ed ha he ob ained esul s a e signi ican ly
di e en . The op imum solu ion is mo e equen ly ead om he DONN s a e wi h he second
ini ial s a e. The dependence o he ob ained esul s on he ini ial DONN s a e is no su p ising.
In ac , in he p e ious sec ion, his was exploi ed o implemen he AM unc ionali y. Di e en
es pa e ns con e ged o di e en s able s a es. The ne wo k e ol es o minimize ene gy bu
4.5. Ope a ing he CMOS-ONN as IM 79
dis inc minimum ene gy s a es a e eached depending on he s a ing poin . In he IM applica ion,
a non-op imum solu ion o he associa ed Max-Cu p oblem co esponds o a local minimum o
ene gy, while a global one is he op imum solu ion.
Figu e 4.21: G aphs wi h 9 nodes p oposed o he Max-Cu ha d-p oblem: (a) G9A, (b) G9B, (c) G9C.
Table 4.10: ASIC esul s o sol ing Max-Cu p oblem associa ed o he 9-node g aphs.
G aph Ins ance Op imum Cu Ini ial s a e Op imum
Cases (#) Ob ained Cu alues
G9A 8 All swi ches OFF 1619 2, 4, 6, 8
G9A 8 Swi ch 1 ON 1905 6, 8
G9B 9 All swi ches OFF 1262 2 o 9
G9C 11 All swi ches OFF 1640 5 o 11
Figu e 4.22 shows he empo al e olu ion o he numbe o ials ha achie ed he op imum cu
in each expe imen . Tha is, o each o he 20 measu ed poin s, he numbe o ials ha ob ained
he op imum cu alue is depic ed. Figu e 4.22 shows a quick e olu ion o a la ge ac ion o ials
o G9A and G9B g aphs o he op imum solu ion in he i s 5μs. The g aph G9C has a highe
numbe o edges. The impac o his mo e complex con igu a ion was ha he g ow h o he
numbe o solu ions ha ound an op imal solu ion is slowe in compa ison wi h he o he g aphs.
I can also be obse ed ha he DONN beha ed di e en ly o G9B compa ed o he o he wo.
A high numbe o ials ound an op imum solu ion quickly, bu hey we e no s able h ough ime.
Figu e 4.23 depic s he numbe o ials e sus he numbe o op imum eadings and he numbe
o op imum eadings o each ial is collec ed in he inse ables. Fo example, o G9A, he 64%
o he ials achie ed an op imum solu ion in 19 o he 20 eadings and 16% o he ials ailed o
ob ain an op imum solu ion. The la e numbe educes o 0 and 3 o G9A wi h swi ch on and
G9C espec i ely.
The i s poin o no e is ha , despi e ha ing he SHIL signal synch onized, no all he ials a e
iden ical. This also occu ed o some es pa e ns a equal dis ance om he s o ed ones in he
AM applica ion. I is due o he impac o dis inc sou ces o noise. Howe e , he e is much mo e
a iabili y among ials in he IM expe imen s han in he AM. The le el o ha noise sensi i i y
depends on many ac o s including he shape o he ene gy landscape o he oscilla o sys em. In
his sense, i is in e es ing o no e he di e ences be ween he wo-pa e n AM example and he
Max-Cu s expe imen s. Fo ha , we look he sys ems om he poin o iew o he cons ain s
associa ed o hei weigh ma ixes. The AM example has weigh s +1, -1 and 0. A +1 (-1)
cons ain is sa is ied i he wo associa ed oscilla o s a e in phase (ou o phase). The e is no
cons ain associa ed o null weigh . The sys ems implemen ing he Max-Cu p oblem ha e -1 and
0 weigh alues. The wo s o ed pa e ns in he AM applica ion sa is y all he cons ain s.
Howe e , o he h ee Max-Cu expe imen ed sys ems, none ne wo k s a e sa is ies all he
cons ain s. I is said ha he e is con en ion in he sys em.
O1
O2
O3
O4
O5
O6
O7
O8
O9
1
1
1
1
1
1
1
a)
b)
c)
80 Chap e 4: ASIC CMOS-ONN Demons a o
Figu e 4.22: Tempo al e olu ion o he numbe o ials ha achie ed he op imum cu alue o each
g aph: (a) G9A,
(b) G9B, (c) G9C.
0
10
20
30
40
50
60
70
80
90
100
110
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Op imum Cases (#)
Measu emen ime (μs)
G9A (sw1=0)
G9A (sw1=1)
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Op imum Cases (#)
Measu emen ime (μs)
G9B
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Op imum Cases (#)
Measu emen ime (μs)
G9C
a)
b)
c)
4.5. Ope a ing he CMOS-ONN as IM 81
Figu e 4.23: Numbe o measu ed op imum s a es upon each ial. (a) G9A, (b) G9B, (c) G9C. Inse ables
depic he pe cen age o ials wi h he same numbe o measu ed op imum.
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Op imum Cases (#)
T ial (#)
G9A (sw1=0)
G9A (sw1=1)
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Op imum Cases (#)
T ial (#)
G9B
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Op imum Cases (#)
T ial (#)
G9C
a)
b)
c)
82 Chap e 4: ASIC CMOS-ONN Demons a o
Conclusions
The analog CMOS-ONN ASIC demons a o in he Neu ONN p ojec and i s se -up desc ip ion
ha e been p esen ed in his Chap e , epo ing he oscilla o y dynamics and he e alua ion on wo
a ge applica ions. I consis s o a 9-neu on CMOS-ONN esembling a VO2-based ONN. I uses
a CMOS sub-ci cui emula ing he I-V cha ac e is ic o VO2 de ices o build an a chi ec u e based
on di e en ial oscilla o s. The synapse is implemen ed wi h a 6- ansis o b idge opology
enabling esis i e coupling among oscilla o s. Bo h posi i e and nega i e weigh s can be ealized.
The ab ica ed ONN-ASIC is p og ammable, including all- o-all connec ions, wi h a la ge deg ee
o con ollabili y and obse abili y o be able o deep in o he synch oniza ion dynamics o
coupled non-linea oscilla o s, on which basis he ONN compu a ion is ca ied ou . Se e al
expe imen s ha e been ca ied ou wi h di e en coupling and calib a ion con igu a ions, along
wi h a equency cha ac e iza ion o he oscilla o s. The impac o SHIL and he coupling s eng h
has been explo ed wi h espec o he synch oniza ion o he oscilla o s.
Di e en expe imen s ope a ing he CMOS-ONN as AM and IM ha e also been epo ed,
showing he u ili y o he demons a o o showcase eal scena ios o analog compu a ion wi h
oscilla o s. Bo h applica ions equi ed SHIL o enhance he s abili y o he measu emen s since i
has been obse ed ha , despi e a good synch oniza ion could be achie ed wi h he ol age
calib a ion, some sudden phase shi s occu ed due o non-con ollable noise sou ces, al e na ing
he ONN s a e be ween di e en s able si ua ions.
In con as wi h he digi al ONN desc ibed in Chap e 2, he insigh s collec ed du ing he
de elopmen and es o his demons a o ha e a meaning ul added alue, helping wi h he
planning o he asks and goals o he inal VO2 demons a o s epo ed in Chap e 5.
Chap e 5
5.Expe imen s wi h Coupled VO2-based
Oscilla o s
In oduc ion
P e ious Chap e s desc ibes wo ONN demons a o s buil wi h comme cial echnologies.
Chap e 2 p esen s a digi al implemen a ion p o o yped in an FPGA, while Chap e 4 ocuses on
an analog ONN in eg a ed in a 65nm CMOS echnology. Bo h AM and IM unc ionali ies we e
explo ed expe imen ally using he second demons a o . The acqui ed expe ience was applied o
build he inal demons a o s using ab ica ed VO2 de ices o he oscilla o s. These
demons a o s we e ca ied ou as a join collabo a ion in he con ex o he Neu ONN p ojec a
he IBM-Resea ch Zu ich acili ies, whe e hey ha e he capabili y o ab ica e he VO2 ma e ial.
As men ioned in Chap e 1, IMs wi h coupled oscilla o s ha exploi hei dynamics o e olu ion
owa ds a minimal ene gy s a e ha e been p oposed and expe imen ally alida ed in di e en
echnologies. No ably, expe imen al implemen a ions o an Ising sol e using coupled phase-
ansi ion nano-oscilla o s wi h VO2 de ices ha e been epo ed [46], [86], [128]. Expe imen ally
calib a ed nume ical simula ions show hey exhibi high ene gy e iciency and po en ial o
imp o ing by se e al o de s o magni ude o e CPU, GPU, quan um, and pho onic app oaches
[86]. Fu he mo e, in [86] he pe o mance o he VO2-based IM is imp o ed by emula ing
classical annealing by applying a SHIL signal wi h an inc easing ampli ude.
Va ious expe imen al es s we e conduc ed he e o explo e he ope a ion o he VO2-based
ONNs. Emphasis was placed on he VO2-based IMs o sol ing Max-Cu op imiza ion p oblems.
Also, he sol ing o Max-3Sa p oblems we e add essed. In e es ingly, he G aph Colo ing
p oblem associa ed o he same g aphs can also be sol ed wi h he ONNs when ope a ing hem
wi hou o cing he bina iza ion o he phases. No e ha hese applica ions equi e implemen ing
an ONN wi h only nega i e weigh s, which can be success ully ealized h ough capaci i e
coupling o single-ended VO2-based oscilla o s.
E o s we e also made o e alua e i s applica ion as AM o pa e n ecogni ion. Howe e , his
ypically equi es bo h posi i e and nega i e weigh s. This ansla es in o bo h esis i e and
capaci i e coupling in he ONN using single-ended oscilla o s, bu esis i e coupling p o ed o
be qui e challenging because o he pa asi ic e ec s o he se up. This limi a ion can be o e come
using di e en ial oscilla o s. Ne e heless, due o he educed numbe o VO2 de ices a ailable
o p obe simul aneously, and since he di e en ial oscilla o uses wo o hem, only a small-scale
DONN could be expe imen ed wi h.
O he expe imen s we e also explo ed as he phase-encoded mino i y ga e o di e en
synch oniza ion injec ion echniques. In addi ion, mo i a ed by he annealing epo ed in [86], a
se o simula ion expe imen s has been designed o gain insigh in o how o ope a e ONNs o
imp o e hei pe o mance as IMs.
84 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
The emainde o he Chap e is o ganized as ollows. Sec ion 5.2 con ains he desc ip ion o he
CMOS-compa ible VO2-based oscilla o s and he expe imen al se up in which di e en expe imen s
explo ing VO2-based ONNs we e ca ied ou . Sec ion 5.3 desc ibes he esul s o applying he ONNs
o sol e he Max-Cu and he Max-3SAT op imiza ion p oblems. Sec ion 5.4 epo s on o he
expe imen s ca ied ou wi h he ONNs. Sec ion 5.5 co e s he simula ion expe imen s conduc ed o
de i e a sui able ope a ion p o ocol o IMs. Finally, Sec ion 5.6 is de o ed o conclusions.
Desc ip ion o ab ica ed VO2-based oscilla o s and expe imen al
se up
The VO2 de ices we e ab ica ed on a silicon pla o m wi h a ha nium oxide in e laye o c ea e
a VO2 g anula ilm comp ised be ween wo me allic (pla inum) elec odes whose c oss-sec ion
de ine an a ea whe e cu en can low and gene a e he mal ilamen s h ough Joule hea ing. This
manu ac u ing p ocess ollowed semiconduc o indus y s anda ds, acili a ing he in eg a ion o
he de ices a he Back-End-o -Line while ensu ing hei compa ibili y wi h CMOS echnology.
The de ailed ab ica ion p ocess and he basics on undamen al ope a ion can be ound in [54].
Each ab ica ed die held 4 a ays wi h 13 ields, whe e each ield was composed o 9 wo- e minal
VO2 c ossba de ices, as Figu e 5.1 illus a es. The e we e de ices wi h di e en c oss-sec ion sizes.
The VO2 hickness was kep cons an a 60nm. The numbe o de ices simul aneously accessible
was delimi ed by he 18 elec ical p obes a ailable, which co esponded o a single ield. The
measu ed VO2 samples we e con ained in a acuum chambe , shown inside he o al ma k in Figu e
5.2a, a con olled empe a u e ( ypically 220K). The p obes we e wi ed o an ex e nal cus om PCB
(squa e ma k in Figu e 5.2a and Figu e 5.2b) whe e he disc e e componen s we e moun ed and he
ins umen s o he gene a ion and he measu emen o he signals we e connec ed. The in e ace
PCB had a s acked design, so he uppe PCB included he holes o place he disc e e coupling
elemen s in he so-called coupling ma ix. I con ained wo iangula -ma ix con igu a ions ha
co esponded wi h 36 possible in e connec ions o each con igu a ion. Also, a p og ammable
esis o module (ci cle ma k in Figu e 5.2a) was inco po a ed in he ins umen s o he se ies
esis ance equi ed o implemen he oscilla o s. The con igu a ion o he ins umen s was ca ied
ou wi h a cus om applica ion buil wi h LabView, allowing o de ine all he equi ed pa ame e s
and a bi a y wa e o ms.
Figu e 5.1: Illus a ion o ab ica ed die s uc u e.
5.3. Expe imen al IM wi h VO2-based oscilla o s 91
Figu e 5.6 p esen s he esul s o epea ed Max-Cu expe imen s on g aphs C, D, E, and G, using
six o nine in e connec ed VO2 oscilla o s. The ac ion o ials pe numbe o ob ained cu s is
shown. In he case o g aph C, which has he highes connec ion densi y (η > 0.7), he ne wo k
a ain s abili y in he g ound s a e in o e 62% o he ial uns. Fo all g aphs excep D, he bes
solu ion is ound in he majo i y o cases, exceeding 42% o he la ges g aph employing nine
oscilla o s.
Simila o he indings demons a ed in Figu e 5.5, he ONNs shown in Figu e 5.6 consis en ly
each hei s able s a es wi hin 15 oscilla ion cycles. This highligh s he po en ial o as e ime
execu ion o massi e pa allel compu a ions h ough coupled oscilla o s [18]. I should be no ed
ha mos g aphs in Figu e 5.5 and Figu e 5.6 a e ei he plana , i.e. hey can be d awn on a plane
wi hou edges in e sec ing, o nea ly plana [132]. Al hough algo i hms exis o sol e he Max-
Cu p oblem in polynomial ime o plana g aphs [132], ou esul s in Figu e 5.5 and Figu e 5.6
demons a e no able e iciency by a aining solu ions wi hin 15 oscilla ion cycles. I will be
essen ial o ee alua e his le el o pe o mance when dealing wi h dense non-plana g aphs
in ol ing a g ea e numbe o VO2 oscilla o s.
Figu e 5.6: Dis ibu ion o he cu s a ained o g aphs in ol ing 6 o 9 coupled VO2 oscilla o s o sol e he
Max-cu p oblem. The op imal solu ion is ound in mos ins ances o g aphs C, E, and G. Ma e ial
ep oduced wi h pe mission om [62].
92 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Max-3SAT p oblem
The 3SAT p oblem consis s in inding a Boolean combina ion o a iables in a way ha sa is ies
a gi en o mula ℱ [133]. This o mula is made up o smalle pa s called clauses 𝐶, and each one
con ains h ee speci ic pieces o in o ma ion called li e als [133].
ℱ
=
𝐶
∧
𝐶
∧
…
∧
𝐶
∧
𝐶
(
5.2
)
Whe e 𝐶 is he inclusi e disjunc ion o h ee li e als 𝑥, such ha :
𝐶
=
𝑥
∨
𝑥
∨
𝑥
(5.3)
Ha ing a dedica ed ha dwa e sol e o 3SAT p esen s a signi ican po en ial o accele a e he
compu a ion o NP-comple e p oblems, as hese p oblems can all be educed o 3SAT [133]. The
NP-ha d Max-3SAT p oblem seeks he maximum numbe o sa is ied clauses 𝐾. I he solu ion
o Max-3SAT says ha all he clauses a e sa is iable, hen he co esponding 3SAT p oblem is
sa is iable [69], [134].
The e a e di e en o mula ions o encode his NP-ha d p oblem in o he Ising pa adigm [69].
Typically, he Max-3SAT p oblem add essed wi h IM equi es he addi ion o an ex e nal ield
ha in e ac s wi h all he spins in one di ec ion. Thus, he Ising Hamil onian associa ed o he
ene gy-s a e landscape is o mula ed as:
𝐻
=
−
𝐽
𝑠
𝑠
−
ℎ
𝑠
(5.1)
Whe e 𝐽 is he coupling coe icien o he in e ac ion be ween uni s 𝑖 and 𝑗, 𝑠 is he spin (up ↑
o down ↓) o uni 𝑖. Also, ℎ is he ex e nal bias, ha ep esen s he in e ac ion o uni 𝑖 wi h an
ex e nal uni wi h ixed spin alue. The ex e nal bias is capaci i ely injec ed as an addi ional
signal wi h he same equency as he coupled oscilla o s. This is simila o applying SHIL bu
he injec ed signal is he i s ha monic and he esul ing impac is ha all he oscilla o s´ phases
a e pushed o be in-phase oge he . Bo h SHIL and he i s -ha monic injec ion locking (FHIL)
we e p o ided h ough hei co esponding capaci o s connec ed o he ou pu s.
De eloping he Maximum Independen Se o mula ion [134], [135] o cons uc he equi alen
g aph, each li e al is mapped o a node (VO2 oscilla o ), while each clause co esponds o an
in e connec ion (capaci o ), o ming he iangles such as he ones shown in he h ee examples
in Figu e 5.7. Fu he mo e, all complemen a y li e als a e in e connec ed (blue connec ions in
Figu e 5.7), as hey canno simul aneously sa is y he wo co esponding clauses.
Figu e 5.7 also p esen s he esul s o epea ed Max-3SAT expe imen s on g aphs in ol ing six
(𝓕𝟏) and nine (𝓕𝟐 and 𝓕𝟑) oscilla o s. No e ha he maximum alue o 𝑲 is de e mined by he
numbe o clauses. The alues o he elec ical ci cui pa ame e s chosen o each g aph a e
epo ed in Table 5.2.
ℱ
=
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
∧
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
(5.4)
ℱ
=
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
∧
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
∧
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
(5.5)
ℱ
=
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
∧
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
∧
(
𝑥
1
∨
𝑥
2
∨
𝑥
3
)
(5.6)
5.4. Addi ional expe imen s 93
Figu e 5.7: Dis ibu ion o he numbe o sa is ying assignmen s 𝐾 a ained o g aphs in ol ing 6 o 9
coupled VO2 oscilla o s o sol e he Max-3SAT p oblem. Ma e ial om [62].
Table 5.2. Ci cui pa ame e s alues used o sol e he Max-3SAT p oblems in Figu e 5.7.
G aph VDD
(V)
RS
(kΩ)
CL
(nF)
Cc
(nF)
C2-HIL
(pF)
C1-HIL
(pF)
V2-HIL
(V)
𝑽
𝒉
(V)
𝒇
(kHz)
Ac i e a ea
(nm3)
𝐹
9.0 45 10.0 0.68 470 270 6.0 4.0 3.10 300 × 300 × 60
𝐹
6.75 48 10.0 1.0 470 270 9.0 4.0 2.47 150 × 150 × 60
𝐹
6.75 46 10.0 1.0 270 470 6.0 8.0 2.33 150 × 150 × 60
Figu e 5.7 shows he sys em capaci y o iden i y he Max-3SAT in all ins ances 𝓕𝟏, 𝓕𝟐, and 𝓕𝟑,
wi h a success a e eaching up o 75% in he case o six coupled oscilla o s. Ac oss all
expe imen al ials, he ne wo k consis en ly achie ed s abili y wi hin 23 oscilla ion cycles, a
signi ican imp o emen compa ed o digi al me hods. To po en ially enhance he sys em abili y
o sol e Max-3SAT, an app oach could in ol e balancing and adap ing be e he pa ame e s o
each speci ic g aph. Fo example, ensu ing he coupling coe icien s 𝐽 a e highe han he bias ℎ,
which ansla es in o hei espec i e capaci ance alues, would p e en ge ing apped in local
ene gy minima and ensu e oscilla o s ep esen ing complemen a iables s abilize wi h opposi e
spins. The chosen ampli udes (Vh and V2-HIL) and capaci ances (C1-HIL and C2-HIL) o he injec ed
signals also impac signi ican ly he e en ual con e gence s a e o he ne wo k. Fu he
explo a ion o he in luence o hese alues is equi ed o maximize he likelihood o success ully
sol ing he Max-3SAT p oblem. None heless, he op imal solu ion is ound in all ins ances and
he esul s p esen ed in Figu e 5.7 se e as a p oo o concep , p o ing 3SAT sol e s can be
implemen ed wi h a ne wo k o capaci i ely-coupled VO2 oscilla o s.
Addi ional expe imen s
This sec ion co e s a b oad ange o expe imen s ca ied ou wi h he coupled VO2-based
oscilla o s in addi ion o he IM applica ion desc ibed in p e ious sec ion: g aph colo ing
p oblems; he di e en ial oscilla o con igu a ion o emula e a di e en ial neu on; he
demons a ion o a mino i y ga e using disc e e componen s and VO2 oscilla o s; and al e na i e
me hods o SHIL injec ion.
94 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
G aph Colo ing p oblem
The G aph Colo ing p oblem consis s in assigning a colo o he nodes in a g aph using he
minimum numbe o colo s, such ha no connec ed nodes sha e he same colo [53], [136]. Wi hin
he oscilla o y pa adigm, he oscilla o s ep esen he nodes and he capaci i e couplings ma ches
he edges. The colo o each node is assigned om he phase o de ing once a s eady s a e is
eached. S a ing wi h one e e ence oscilla o , he closes oscilla o phase is checked i i is
connec ed wi h he e e ence, and i is assigned a di e en colo in he posi i e case. This
p ocedu e is i e a i ely done un il all he oscilla o s ha e been assigned o a colo .
In Figu e 5.8, he expe imen al esul s o ou g aph colo ing p oblems in ol ing h ee o six nodes
a e shown. The inpu geog aphical p oblem is mapped on o a ne wo k o VO2 oscilla o s, whe e he
coupling capaci o s ep esen bo de s be ween indi idual coun ies. The ci cui apidly con e ges
o a s eady s a e wi hin 10 oscilla ion cycles, a signi ican ly as e p ocess compa ed o es ing all
po en ial combina ions. Fo he mos complex g aph, he sys em con e ges o he solu ion wi hin
10 oscilla ion cycles, whe ein he phase o de de ines a colo assignmen o each node.
Table 5.3 p o ides de ails ega ding he di e en pa ame e alues used in each expe imen al se ,
along wi h co esponding wa e o ms showcasing he s able s a e in Figu e 5.8. The pa ame e
se s should be conside ed as possible examples. O he combina ions whe e he supply ol age
(VDD) and he se ies esis ance (RS) a e adjus ed wi h he de ices ac i e a ea could wo k as well.
An adap ion o he pa ame e s o g aphs wi h di e en numbe o nodes and edges is equi ed as
he capaci ances a y. This makes i e iden ha he e is a conside able challenge in es ablishing
any o m o me ic o selec ing alues o colo ou map and con e ge apidly owa ds a solu ion
[53]. In ou case, he ca e ul choice o hese alues esul ed in a clus e diame e , ha is ela i ely
small [53]. Fo he Cen al Eu opean and Sou h Ame ican g aphs, he clus e diame e a e aged
33.5° and 30.0°, espec i ely.
The inhe en spa si y o hese g aphs wi hou all- o-all connec i i y makes colo ing mo e
challenging [53], esul ing in la ge clus e diame e s compa ed o he No he n Eu ope and Eas
Asia g aphs. In he Sou h Ame ica g aph, cha ac e ized by nonuni o m connec i i y wi h a ying
deg ees o connec ions on each node, he combined and unbalanced epelling e ec o he
coupling capaci ances es ablishes a phase o de ing among he oscilla o s [53]. This phase o de ing
can only app oxima e he minimum e ex colo ing, causing une en clus e spacing in he
solu ion [53], [137]. Mo e e ec i e mapping echniques employing a ci cula o de ing in colo
assignmen could be employed o omi he impac o he clus e diame e in he eading o he
solu ion, pa icula ly o la ge -scale g aphs [53].
Table 5.3: Ci cui pa ame e s alues used o he G aph Colo ing p oblems in Figu e 5.8.
G aph VDD (V) RS (kΩ) CL (nF) Cc (nF) Ac i e a ea (nm3)
No he n Eu ope 5.5 40 10.0 2.2 100 × 50 × 60
Cen al Eu ope 9.0 40 10.0 0.68 300 × 300 × 60
Eas Asia 7.0 30 10.0 2.2 300 × 80 × 60
Sou h Ame ica 5.5 42 11.5 1.0 100 × 50 × 60
5.4. Addi ional expe imen s 95
Figu e 5.8: Expe imen al esul s in ol ing 3 o 6 nodes o sol e he G aph colo ing p oblem: a) Inpu
p oblem, b) Oscilla o g aph, measu ed c) Wa e o ms and d) Phase ela ionships o he ONN ou pu s. e)
Numbe o cycles equi ed o ge o he s able s a e.
96 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Di e en ial Oscilla o y Neu al Ne wo k
The CMOS ONN desc ibed in Chap e 4 used di e en ial oscilla o s ins ead o single-ended ones
in o de o enable implemen a ion o bo h posi i e and nega i e coupling using only esis i e
coupling elemen s. The implemen a ion and coupling o VO2-based di e en ial oscilla o s we e
also expe imen ally e alua ed. This subsec ion con ains elec ical measu emen s o capaci i ely
coupled di e en ial oscilla o s.
Fi s , posi i e and nega i e couplings we e demons a ed using only capaci o s. Gi en ha he
capaci o be ween wo oscilla o s imposes an an i-phase beha io , he posi i e coupling was
implemen ed h ough he in e connec ion o he di e en ial ou pu s in a c ossed manne . Tha is,
he posi i e ou pu o one oscilla o was connec ed o he nega i e ou pu o he o he oscilla o
and ice e sa. The nega i e coupling was done connec ing he co esponding posi i e ou pu s
oge he (and he same wi h he nega i es). Bo h couplings a e illus a ed in he schema ics shown
in Figu e 5.9 along wi h he measu ed wa e o ms. The pa ame e s o hese expe imen s a e:
ac i e a ea: 200×200×60nm3, VDD = 8V, Rs = 50kΩ, CL = 10nF, CD = 2.2nF, CC = 150pF. The
posi i e ou pu o he oscilla o s has been ma ked wi h c oss symbols. I can be obse ed ha in
bo h cases, he ou pu s o each oscilla o a e ou -o -phase as i is expec ed. Also, posi i e ou pu s
a e in-phase when he oscilla o s a e posi i ely coupled (Figu e 5.9a) and ou -o -phase when
nega i ely coupled (Figu e 5.9b).
Following he desc ibed app oach, a DONN wi h 4 di e en ial oscilla o s was e alua ed. No e i
implied he use o 8 single-ended oscilla o s. The coupling elemen s we e chosen o encode he
ollowing:
𝑊
=
0
−
1
−
1
0
−
1
−
1
−
1
−
1
−
1
−
1
−
1
−
1
0
−
1
−
1
0
(5.7)
This ma ix con igu es an IM o he Max-Cu p oblem o he ully-coupled ou -node g aph. An
op imum solu ion o his g aph is wha e e di ision in g oups o wo ou pu s. I is wo h o
men ion ha his ma ix, when in e p e ed as an AM, s o es he pa e ns shown in Figu e 5.10.
Figu e 5.10 shows he measu ed wa e o ms o he desc ibed DONN. Again, he posi i e ou pu
o he oscilla o s has been ma ked wi h c oss symbols. Figu e 5.10 shows ha in his pa icula
case, he ob ained ela ionship is oscilla o 1 and 3 in one phase and 2 and 4 in he o he . No e
ha each one o he complemen ed ou pu s a e ound in a di e en g oup wi h espec o i s
co esponding ou pu .
5.4. Addi ional expe imen s 97
Figu e 5.9: Schema ic and measu ed wa e o ms o DONN. (a) Posi i e coupling. (b) Nega i e coupling.
b)
a)
98 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Figu e 5.10: Wa e o ms o he 4-neu on DONN showing a s able solu ion. S o ed pa e ns encoded in he
nega i e weigh ma ix (Equa ion 5.7).
Mino i y ga e ope a ion
A majo i y (mino i y ga e) ga e is a logic ga e whe e he ou pu eplica es he logic alue ( he
complemen o he logic alue) in he majo i y o inpu s. Table 5.4 depic s he logic unc ions
implemen ed by majo i y and mino i y 3-inpu s ga es.
Table 5.4: T u h able o 3-inpu majo i y and mino i y ga es.
X1 X2 X3 Majo . Mino .
0 0 0 0 1
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 0
5.4. Addi ional expe imen s 99
The disc e ized beha io o he oscilla ing phase o he VO2 oscilla o can be exploi ed o
implemen logic unc ions. Pa icula ly, au ho s in [67] p oposed he phase-encoded logic using
VO2-based oscilla o s o implemen he majo i y unc ionali y. In he phase-encoded pa adigm,
logic inpu alues a e ansla ed in o bina y phases in he oscilla o y signals: 0º o low- alue and
180º o high- alue. The phase ou pu signal o he oscilla o eplica es he mos common alue
among he inpu phases. Logically, he oscilla ing equency is he same o e e y inpu signal
and he VO2 oscilla o ou pu signal.
As a p oo -o -concep o he phase-encoded logic ga e implemen ed wi h VO2-based oscilla o ,
an expe imen al alida ion o a 3-inpu mino i y ga e was conduc ed wi h he ab ica ed VO2
de ices and disc e e elec onic componen s. The schema ic o he e alua ed ci cui is shown in
Figu e 5.11. Gi en ha ou pa icula implemen a ion employs a capaci o as he coupling
elemen , he co ec ope a ion akes place when he ou pu phase is he complemen a y o he mos
epea ed inpu phase. In ac , he capaci o cons i u es a nega i e in e ac ion be ween he phases
o he coupled signals, pushing hem apa in an i-phase (complemen ed) beha io .
The pa icula pa ame e s o his expe imen a e: ac i e a ea: 150×150×60nm3, VDD = 7V, Rs =
40kΩ, CL = 10nF, CC = 150pF, VINPUTp,p = 6V, CSHIL = 270pF, VSHILp,p = 4V.
Figu e 5.12 illus a es he ope a ion p o ocol. A synch oniza ion signal (SHIL) o ces he
disc e iza ion o he ou pu phase while a e e ence oscilla o y signal is conside ed o ha e a
common measu emen amewo k. The inpu signals a e kep inac i e be o e he e alua ion
phase. Once hey a e ac i a ed, hey oscilla e o ew cycles (10 cycles, 2ms), encoding in hei
phases he inpu o be e alua ed by he mino i y ga e. SHIL is u ned o while he ac i a ion o
he inpu s o ease he in e ac ion be ween he oscilla o and he inpu s. Once SHIL is ac i a ed
again, he ou pu phase is ozen and he esponse o he logic ga e is ead.
VO
2
R
VO
2
OUT
R
V
SHIL
C
V
DD
V
DD
V
INPUT-1
V
INPUT-2
V
INPUT-3
Figu e 5.11: Schema ic o he VO2-based mino i y ga e wi h he capaci i ely-coupled inpu s and SHIL.
100 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Figu e 5.12: Mino i y ga e ope a ion. Vol age wa e o ms o he SHIL and he inpu s signals.
The beha io o he implemen ed mino i y ga e has been exhaus i ely alida ed. Figu e 5.13 and
Figu e 5.14 depic wa e o ms associa ed o wo di e en ep esen a i e inpu combina ions: he
h ee inpu signals being in-phase and one inpu signal being ou -o -phase wi h he o he wo.
No e ha in o de o alida e he ci cui ope a ion, each inpu combina ion is e alua ed wice.
Each ime wi h a di e en ini ial ou pu oscilla ion phase. E alua ions a e ma ked as A and B in
Figu e 5.13a and Figu e 5.14a. The e a e inse s o each o hem o be e show he ope a ion
de ails.
In Figu e 5.13a, i can be obse ed ha he ga e ou pu (blue wa e o m) is ini ially in-phase wi h
he e e ence signal (g een wa e o m). All inpu s in-phase wi h he e e ence signal ( ed
wa e o m) a e ac i a ed (wi h SHIL o ). When SHIL is ac i a ed ( ead ope a ion mode) he
ou pu ozen in ou o phase wi h he e e ence signal as expec ed acco ding o he mino i y
unc ionali y. Second e alua ion (Figu e 5.13c) s a s wi h he ou pu ou o phase. I can be
obse ed ha i emains ou o phase a e e alua ion.
Simila ly, Figu e 5.14b depic s he same ini ial si ua ion being he ga e ou pu and he e e ence
signal in-phase. Fo he e alua ed inpu , one o he inpu phases is ou -o -phase wi h he o he
wo and wi h he e e ence signal. As in p e ious example, he ob ained esponse is ou -o -phase
wi h he e e ence signal, as long as he e alua ed inpu sha ed he same expec ed logic ou pu
alue. Figu e 5.14c shows he wa e o ms o he second e alua ion wi h he simila expec ed
beha io .
0
2
4
SHIL
0
5
Inpu 2
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5
Time (ms)
0
5
Inpu 3
0
5
Inpu 1
5.5. Ope a ion o ONNs as IMs 107
Figu e 5.19: A 4-node Max-Cu p oblem. (a) Op imal solu ion (4 cu s). (b) Ene gy landscape.
A key componen o ha dwa e IMs ha can be undamen al in e icien ly sol ing COPs is he
in e nal noise o he ci cui , which helps escape om subop imal solu ions (local ene gy
minimum). In ac , in digi al IMs, andom numbe gene a o s a e used o his [138], [139].
Mo eo e , di e en au ho s ha e s a ed he c i ical ole o inhe en s ochas ici y in he
implemen a ion o IMs, which is iden i ied wi h a p obabilis ic explo a ion [59], [86]. Because o
his, hey epo ha many ials a e conduc ed so ha he chance o ob aining he op imum
solu ion inc eases.
I is well-known and has been epo ed in he li e a u e ha ac ual VO2 de ices exhibi a iabili y
in he s a e ansi ion ol ages. In [140], i is epo ed ha VIMT exhibi s a iabili y mo e han
se en imes la ge han o VMIT. I is also epo ed ha i a ies s ochas ically o all cycles
wi hou a educ ion in he mean o ange and can be app oxima ed by a no mal dis ibu ion.
Addi ionally, he impac o he insula ing- o-me al a iabili y in he pe o mance o coupled
oscilla o sys ems is much g ea e han he one associa ed wi h he e e se phase ansi ion [141].
Thus, a cycle- o-cycle a iabili y o i s insula ing o me al ol age (VIMT) has been included in he
VO2 model. A no mal dis ibu ion is associa ed wi h i . Tha is, VIMT changes in ime ollowing a
no mal dis ibu ion a ound i s nominal alue, wi h a gi en s anda d de ia ion deno ed NOISE o
noise le el. I ansla es in cycle- o-cycle a ia ions in he oscilla ion pe iod (ji e ). The chosen
name e e s o he ac ha i , no only allows us o model he expe imen al a iabili y obse ed
in said pa ame e , bu also allows o cap u e he e ec o o he noise sou ces ha mani es as ji e
in he oscilla ion pe iod. No e ha he e is also ma hema ical noise associa ed wi h he in eg a ion
algo i hms and hei ole ances.
Figu e 5.20a shows simula ion esul s o he cu alue associa ed wi h he e olu ion o he 4-
oscilla o sys em ep esen ing he g aph o Figu e 5.19a wi h wo di e en noise le els. Iden ical
ini ial s a e and synch oniza ion signals ha e been used in bo h simula ions. I can be obse ed
ha he sys em wi h a e y small noise le el becomes apped in a local minimum wi h a cu -
alue equal o 2, no a maximum. The sys em wi h mo e noise can each he global op imum
(minimum ene gy spin con igu a ion).
Figu e 5.20b depic s he esul o simula ions wi h a e y small noise le el and wo di e en
ampli udes o he synch oniza ion signal. This expe imen shows how SHIL coun e ac s noise.
Reducing he ampli ude o he synch oniza ion signal allows escaping om he local minimum.
108 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Tha is, as al eady men ioned, he e is a ela ionship be ween he di e en ac o s in ol ed in he
dynamic e olu ion o he coupled oscilla o sys em.
Ano he poin o conside is he SHIL signal scheduling. Di e en schedulings ha e been
p oposed in he li e a u e. In pa icula , an imp o emen in he success p obabili y o eaching he
op imum solu ion has been epo ed by linea ly inc easing SHIL ampli ude compa ed o a
cons an SHIL [128]. This has been compa ed o a he mal annealing p ocess in which he
educ ion o he empe a u e is associa ed wi h he educ ion o he s ochas ic noise in he sys em
as a consequence o he inc ease in he ampli ude o SHIL signal.
Figu e 5.21 illus a es he po en ial o using a SHIL wi h smoo hly inc eases i s ampli ude and i s
impac on he sys em. I can be obse ed ha he applica ion o he cons an ampli ude SHIL
signal eezes he sys em in a non-op imum phase ela ionship wi h h ee oscilla o s in one phase
(cu alue equal o 2). This is in ag eemen wi h [79] in which i is s a ed ha he eeze-ou e ec s
associa ed wi h he disc e iza ion o he oscilla o phases a e a limi a ion o he synch oniza ion
dynamics.
In [142], we showed ha he success p obabili y can also be inc eased by delaying he applica ion
o he synch oniza ion signal. In bo h cases, we a e gi ing ime o he sys em o e ol e owa ds
a minimal ene gy s a e be o e o cing a bina y disc e iza ion o he oscilla o phases, which occu s
o an enough la ge SHIL ampli ude. Con a y, i a s ong enough synch oniza ion signal is
applied om he e y beginning o oo ea ly, he sys em ge s s uck in a s able s a e in he
neighbo hood o he ini ial one which, in gene al, can be subop imal. Tha is a local minimum.
Figu e 5.20: E olu ion o he numbe o cu s wi h ime o (a) wo di e en noise le els and (b) wo
di e en SHIL ampli udes.
a)
b)
5.5. Ope a ion o ONNs as IMs 109
Figu e 5.21: Wa e o ms o wo SHIL schedules, a) cons an ampli ude and b) inc easing ampli ude.
IM ope a ion p o ocol
On he basis o he analysis o he expe imen s ca ied ou , we p opose he ollowing oscilla o y
IM ope a ing p o ocol.
SHIL scheduling acco ding o Figu e 5.22. The VMAX pa ame e is he signal ampli ude, and
KPER con ols he linea inc easing/dec easing a e o i s ampli ude. I is measu ed in cycles
o pe iod, TOSC. The ampli ude ises om 0 o VMAX in KPER cycles. The epe i i e scheme
o sweeping SHIL ampli ude allows u he explo a ion o he associa ed ene gy landscape
o he Ising model, such as he use o many ials epo ed in he li e a u e, and enables an
impo an escape mechanism om local minimum solu ions when SHIL s eng h is educed.
The solu ion is selec ed as he bes one om mul iple eadings. This aims o coun e ac he
possibili y o escaping a global minimum o wo sening he quali y o he solu ion. This could
occu because o excessi e noise in he sys em o as a consequence o SHIL s eng h
educ ion. Readings mus be ca ied ou wi h enough SHIL signal s eng h so ha phases a e
bina ized and so ha he eading p ocess is simpli ied. Readings a e ca ied ou a ound he
poin s a which he SHIL signal eaches i s maximum ampli ude. Thus, how many eadings
ake place depends on how long he IM is ope a ed (TCOMP) and KPER.
Figu e 5.22: The p oposed SHIL schedule and he pa ame e s ha desc ibe i s con igu a ion.
a)
b
)
110 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
To e alua e he p oposed ope a ion p o ocol, a second Max-Cu ins ance, depic ed in Figu e 5.23
has been used in he simula ion expe imen s. No e ha i exhibi s c i ical di e ence wi h espec
o ha p e iously analyzed (g aph in Figu e 5.19). The associa ed oscilla o sys em p esen s phase
con en ion. This means ha i is no possible o ul il all phase cons ain s imposed by he
capaci o s ha connec pai s o oscilla o s in he associa ed IM. In addi ion, i is a non-balanced
g aph in he sense ha nodes exhibi a di e en numbe o connec ions.
The IM associa ed wi h he g aph in Figu e 5.23 has been e alua ed wi h h ee di e en SHIL
schedules illus a ed in Figu e 5.24: an al eady in oduced inc easing ampli ude SHIL signal
(SHIL1), a SHIL signal pe iodically swi ching be ween wo di e en ampli udes (SHIL2) [143]
and he p oposed one (SHIL3). The mul iple eading app oach has been used in all he cases. Fo
each o hem, 20 Mon e Ca lo (MC) uns ha e been pe o med o each o he 64 possible ini ial
bina y phase combina ions. MC analysis is used because o he andom dis ibu ion included in
he VO2 model. This expe imen was ca ied ou o h ee di e en alues (no noise, 1mV and
10mV). Th ee TCOMP alues we e analyzed. TCOMP1 co esponds o 1.5 pe iods o he SHIL3 signal
(2 eadings), TCOMP2 co esponds o 9.5 pe iods and 10 eadings, and TCOMP3 o 17.5 pe iods and
18 eadings. Fo selec ed KPER, TCOMP1 is 12µs. VMAX = 0.4V has been chosen so ha i is s ong
enough o p oduce he desi ed disc e iza ion o he phases and i s ampli ude is simila o ha o
he oscilla o s hemsel es. KPER = 20 has been selec ed.
Figu e 5.23: 6-node Max-Cu p oblem wi h wo op imum solu ions wi h 8 cu s.
Figu e 5.24: SHIL schedules used o he e alua ion o he oscilla o y IM.
1
6
5
2
3
4
1
6
5
2
3
4
5.5. Ope a ion o ONNs as IMs 111
Table 5.5 shows he numbe o ini ial sys em s a es om which an op imum solu ion is ob ained
o di e en compu a ion imes. Tha is, a leas one o hei associa ed 20 MC simula ions
ob ained an op imum solu ion. No e ha a single simula ion is ca ied ou when no noise is
modeled. I is clea ha noise is bene icial in inc easing he numbe o success ul ini ial s a es,
al hough his igu e o me i alone does no comple ely suppo he abo e s a emen , bu he
imp o emen obse ed (especially o he SHIL1 and TCOMP1) be ween 1mV and 10mV o noise
also sugges s his. I can also be seen ha he numbe o SHIL1 esul s is much wo se han he
o he wo o he sho es compu a ion ime. This can be explained since he capabili y o
explo ing he ene gy landscape and so escaping om a local minimum educes once a s ong
enough SHIL is applied. Bo h SHIL2 and SHIL3 bene i om he educ ion o he SHIL s eng h.
When he compu a ion ime is inc eased, he chance o escaping due o noise is inc eased, and so
he numbe o success ul ini ial s a es o he SHIL1 expe imen also inc eases signi ican ly. Also,
i can be obse ed ha he noise le el has li le impac bu o he SHIL1 and o he sho es
compu a ion ime. This ag ees wi h ou abo e explana ion. The highe he noise le el inc eases
he p obabili y o escaping om local minima.
Table 5.5: Ini ial s a es ha con e ged o an op imum solu ion wi h SHIL1, SHIL2, and SHIL3, espec i ely.
NOISE TCOMP1 TCOMP2 TCOMP3
0mV (no noise) 9, 18, 26 18, 16, 48 21, 49, 58
1mV 31, 61, 64 58, 64, 64 62, 64, 64
10mV 41, 62, 64 55, 64, 64 62, 64, 64
Much mo e insigh in o sys em pe o mance can be ob ained i he ac ion o MC uns achie ing
he op imum solu ion is also analyzed. I can be conside ed as he success p obabili y. A alue o
1 o his igu e o me i means ha he 20 uns p oduce an op imum solu ion.
Figu e 5.25 shows such a ac ion o each o he 64 ini ial s a es e sus i s no malized HD o he
closes op imum solu ion. A HD equal o 0 means ha he ini ial s a e is an op imum solu ion o he
associa ed op imiza ion p oblem. The di e ences a e no o ious among he h ee SHIL schedules.
On he one hand, a ac ion o success ul uns is signi ican ly highe o he p oposed SHIL (black
poin s a e o e 0.8 o all he ini ial s a es) han o he o he wo. The wo s esul s a e ob ained
wi h SHIL1. The second in e es ing aspec o be poin ed ou is ha he esul s a e almos
independen o he ini ial s a e ( he 64 poin s ha e e y li le dispe sion) o he p oposed SHIL,
while his is no so o he o he SHILs, especially o SHIL1. In ac , e y ew black poin s appea
since many o hem occupy he same posi ion. This educed dependence is also a consequence o
educing he SHIL s eng h du ing he ope a ion, allowing a be e explo a ion o he solu ion space.
112 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Figu e 5.25: Success a io e sus HD be ween he ini ial s a e and he closes op imum solu ion o he
h ee SHIL schedules. Simula ion pa ame e s: VMAX = 0.4V, σNOISE = 1mV, TCOMP3.
I is also in e es ing o analyze he ac ion o success ul uns o di e en compu a ion imes. In
his analysis, he a e age and s anda d de ia ion o e he 64 ini ial s a es ha e been used. A alue
o 1 o he a e age means ha 1280 (64x20) uns p oduced an op imum esul . Figu e 5.26
depic s he a e age o he success a e including ba s o he s anda d de ia ion. No e ha he e
a e wo addi ional compu a ion imes wi h espec o p e ious expe imen s (TCOMP12 and TCOMP23).
The limi a ions o SHIL1 a e also e iden om his igu e. Bo h SHIL2 and SHIL3 achie ed
be e esul s in TCOMP1 han SHIL1 in TCOMP3, al hough TCOMP1 is a ound en imes sho e han
TCOMP3. The smalle s anda d de ia ion exhibi ed by SHIL3 also indica es ha he ini ial s a e is
less ele an o SHIL3, as al eady poin ed ou when desc ibing Figu e 5.25.
In [86], he Max-Cu p oblem depic ed in Figu e 5.27a was expe imen ally sol ed using VO2
oscilla o s and he SHIL schedule iden i ied in his wo k as SHIL1. They epo a Max-Cu alue
equal o 10 wi h a 96% success a e wi h KPER o e 600 oscilla ion cycles. We ha e simula ed he
same g aph wi h SHIL3. Compu a ion ime ha e been ixed o 600 cycles o a ai compa ison.
In pa icula , h ee SHIL3 cycles wi h KPER = 100 ha e been applied. 20 Mon e Ca lo simula ions
o each o 50 di e en ini ial s a es ha e been ca ied ou . Figu e 5.27b depic s ob ained esul s.
No e ha Max-Cu alues highe han 10 ha e been ob ained. In pa icula , he op imum solu ion
o he p oblem (maximum-cu equals o 12) ha e been ob ained in 46.4% o he ials. Mo eo e ,
maximum-cu alues equal o g ea e han 10 ha e been achie ed in 96.1% o all he simula ions.
The expe imen in [86] has been also simula ed wi h ou VO2 models. The op imum Max-Cu
alue is ob ained in 14.8% o he ials.
Figu e 5.26: A e age and s anda d de ia ion o he success a e o he h ee SHIL schemes e alua ed a
i e di e en compu a ion imes.
5.5. Ope a ion o ONNs as IMs 113
Addi ionally, hese esul s indica e ha he p oposed SHIL scheme could also ansla e in o
ad an ages in e ms o ene gy e iciency. In [86], as al eady men ioned, analysis esul s ha show
he po en ial o VO2 based oscilla o y IMs o d as ically inc ease ope a ions pe second and pe
wa wi h espec o pla o ms con en ionally used o sol e op imiza ion p oblems a e epo ed.
The abili y o SHIL3 o achie e highe success a es in sho e compu a ion imes could only
con ibu e o inc easing his igu e o me i wi h espec o [86] ha applied SHIL1.
Figu e 5.27: a) 8-node Max-Cu p oblem epo ed in [86] wi h 12 cu s (b) His og am ep esen ing he
p obabili y o success o ob aining a leas 10, 11 o 12 cu s.
114 Chap e 5: Expe imen s wi h Coupled VO2-based Oscilla o s
Conclusions
This Chap e collec s he expe imen s ca ied ou wi h he ab ica ed VO2 de ices du ing he join
collabo a ion a he IBM-Resea ch Zu ich acili ies. The implemen ed demons a o s aimed o
p o e he capabili y o he VO2-based ONNs o implemen IMs o sol e COPs.
I has been benchma ked di e en Max-Cu and Max-3-SAT p oblems wi h g aphs up o 9 nodes,
explo ing he use o capaci i ely injec ed SHIL o ensu e he synch oniza ion o he ab ica ed
oscilla o s. The in insic noise and a iabili y o he VO2 and he se -up helps he ONN o u he
explo e he Hamil onian ene gy landscape, con e ging wi hin e y ew cycles. The dependence
o he supply ol age wi h he size o he ab ica ed ma e ial is exposed, being necessa y a ine
selec ion o applied ol age and he alue o he passi e disc e e componen s o ease he
synch ony o he oscilla o s.
Mo eo e , addi ional expe imen s ha e been epo ed showcasing he g aph-colo ing sol e and
o he ci cui s implemen a ion as he VO2-based di e en ial oscilla o s o he phase-based mino i y
ga e, as well as, di e en echniques o injec ing he synch oniza ion signal o e he supply ol age
e minal o being p o ided by an addi ional and bu e ed wice- equency oscilla o .
Finally, a comp ehensi e s udy o he SHIL applica ion, o p o ocol, o enhance he e icacy o
he IM sol e has been p o ided. The in e play be ween noise a iabili y and he uning o he
s eng h o SHIL is assessed o inc ease he accu acy o he COPs sol ing. The simula ion wo k
in Sec ion 5.5 shows he po en ial o inc easing and dec easing he SHIL ampli ude o escape
om local minimum ene gy s a es in oscilla o y IMs and he ole o noise in his ene gy landscape
explo a ion.
Chap e 6
6.Conclusions
This Ph.D. Disse a ion has con ibu ed o he alida ion and unde s anding o he oscilla o y
compu ing pa adigm based on VO2-based ONNs, h ough se e al p oo -o -concep demons a o s
and p ac ical applica ions. I has es ablished a solid ounda ion o how elec ical ONNs should
be ope a ed and con igu ed while explo ing me hods o o e come he in insic cons ain s and
limi a ions o such a complex non-linea dynamical sys em. The main con ibu ions a e p esen ed
ollowing he o de o p esen a ion in his wo k:
The i s ully digi al ONN ha dwa e has been designed as a p oo -o -concep demons a o
o he oscilla o y compu ing pa adigm. I demons a ed he impo an ea u e o con e ging
o a s able phase pa e n wi hin e y ew oscilla ion cycles, suppo ing he po en ial o he
compu ing pa adigm in e ms o ene gy e iciency. Addi ionally, he design has been p o ided
wi h ully p og ammable weigh s and an online lea ning ope a ion mode.
The digi al ONN has been implemen ed on an FPGA, cons i u ing a as p o o ype o alida e
sui able eal applica ions. Speci ically, p o o ypes o a digi ecogni ion applica ion and a
esponsi e obs acle a oidance sys em in a mobile obo ha e been expe imen ally alida ed.
The ex ensi e HNN li e a u e on lea ning ules o AMs has been explo ed wi h he aim o
de eloping lea ning algo i hms compa ible wi h he cons ain s imposed by analog ONNs.
Di e en aining app oaches wi h he S o key lea ning ule ha e been epo ed. The limi s
in s o age capaci y we e ex ended wi h any o he p oposed app oaches, being no ewo hy
ha he a e o success ul solu ions when explo ing he andom o de in he aining se , was
boos ed o 100% when h ee epe i i e i e a ions we e applied. Speci ically, he epe i i e
aining showed a highly sa is ac o y impac on ob aining a success ul weigh ma ix using
educed weigh p ecision. Fo example, aining wi h h ee i e a ions o he S o key ule, he
60-neu on ne wo k s o ing 5 pa e ns ob ained a success o 93.2% a e ie al accu acy using
h ee bi s. The same case wi h FP p ecision achie ed 99.2% while he ained ne wo k wi h a
single i e a ion o he S o key ule only go 90%, no being able o succeed using a weigh
p ecision lowe han 5 bi s (89.6%).
A no el i e a i e algo i hm (IRPUSH) has been p oposed. I ea u es con igu able ma gin
s abili y o he s o ed pa e ns, allowing he ob en ion o mul iple success ul ma ix solu ions,
e en wi h educed weigh p ecision wi hou sac i icing AM capabili ies. I has shown o be
mo e compe i i e han any o he e alua ed algo i hm, e en when compa ed wi h he o he non-
cons ained i e a i e algo i hm. In pa icula , IRPUSH was he unique ule ha achie ed a 96-
neu on ne wo k success ully s o ing all he 10-cha ac e pa e ns using a 3-bi weigh p ecision
and ema kably wi hou a signi ican loss o e ie al accu acy (90.9% agains he 92.7%
ob ained wi h he FP p ecision). The bes sco e achie ed by ano he algo i hm, 88.8% applying
DEB wi h FP p ecision, did no o e passed 3-bi IRPUSH pe o mance.
116 Chap e 6: Conclusions
An analog 9-neu on CMOS-ONN ASIC has been designed and ab ica ed in TSMC 65nm
echnology. This demons a o in eg a ed a CMOS emula o o he VO2 ma e ial. Replica ing
he VO2 ol age-cu en cha ac e is ic enabled an ea ly e alua ion o he VO2-based ONN
esponse. I has been designed wi h a la ge deg ee o con ollabili y and obse abili y o be
able o deep in o he synch oniza ion dynamics o ONNs. As well, he ASIC e alua ion boa d
and he whole expe imen al se -up ha e been de eloped.
Measu emen s and cha ac e iza ion o he analog CMOS-ONN and he oscilla o dynamics
ha e been ca ied ou . The ope a ion o he ONN as AM and IM ha e been demons a ed,
showing he u ili y o he ASIC o showcase eal scena ios o analog compu a ion wi h
oscilla o s. Bo h applica ions p o ed he impo an ole o SHIL.
The AM demons a o s o ing wo pa e ns p o ed he capabili y o ecognize all he es
pa e ns in he i s h ee cycles. The esul s ma ched well wi h he ob ained wi h he HNN
model and he pos -layou simula ions.
The IM demons a o was applied o h ee di e en Max-Cu p oblems. Conc e ely, he
e alua ed g aphs showed an a e aged success o 67%, 82%, and 89% inding he op imum
case when measu ed a e 35, 70, and 245 oscilla ing cycles, espec i ely.
Expe imen s wi h ab ica ed VO2 de ices ha e been ca ied ou du ing he secondmen a
IBM-Resea ch Zü ich acili ies. The VO2-based ONN ope a ion has been explo ed wi h
an emphasis on sol ing COPs wi h he Ising o mula ion. ONNs up o 9 oscilla o s encoding
Max-Cu and Max-3-SAT COPs ha e been implemen ed and e alua ed. The mo e complex
9-node Max-Cu p oblem a ains s abili y in he g ound s a e ( he op imal solu ion) in o e
42% o he ial uns. All he IM demons a o s con e ged wi hin e y ew cycles.
Addi ionally, o he applica ions ha e been explo ed eso ing o he VO2-based oscilla o s.
A de ailed analysis o he ac o impac ing he ONN pe o mance as IM using elec ical
simula ion has been ca ied ou . I was e inced ha he in insic noise and a iabili y o he
VO2 helped he ONN o u he explo e he Hamil onian ene gy landscape. On his basis, an
ope a ion p o ocol o IMs has been de eloped which has been shown o imp o e he quali y
in he sea ch o op imal solu ion o he COPs esolu ion. A SHIL signal is epea edly and
g adually u ned on and o as a mechanism o escape om he local minimum.
While conside ing ha he speci ic objec i es o he Ph.D. Disse a ion ha e been sa is ac o ily
accomplished, he de eloped esea ch wi h ou ONN demons a o s lea es open ques ions ha
can lead o u u e esea ch e o s. As an immedia e example, a second CMOS-ONN ASIC could
be designed adding all he gained knowledge in di e en aspec s. Fo example, om he lea ning
aspec , he scalabili y challenge could be o e come: he IRPUSH algo i hm can be adap ed o
limi he numbe o connec ions, he e o e, educing he quad a ically-inc easing numbe o
connec ions in he all- o-all a chi ec u e. Also, design echniques o mi iga e he impac o noise
sou ces o layou pa asi es can be applied. I has been p o ed ha he analog ONN bene i s om
he calib a ion ol age o uning he equency o he compensa ion p o ided by capaci o banks.
In a simila way, subsequen ASIC designs could a ge a mo e modula app oach enabling he
elec ical in e acing and he he e ogeneous in eg a ion wi h o he eme ging de ices, as could be
mem is i e synapses o di e en oscilla o implemen a ions, ad ancing sepa a ely he design and
es o al e na i e a chi ec u es.
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