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Chaos, Solitons and Fractals

NonlinearScience, andNonequilibrium andComplexPhenomena journalhomepage:www.elsevier.com/locate/chaos

Reservoir computing based on quenched chaos

Jaesung Choi, Pilwon Kim

^∗

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology(UNIST), Ulsan Metropolitan City 44919, Republic of Korea

a rt i c l e i n f o

Article history:

Received 27 August 2019 Revised 3 February 2020 Accepted 14 July 2020 Available online 23 July 2020 Keywords:

Reservoir computing Quenched chaos Chaos computing

a b s t r a c t

Reservoircomputing(RC)isabrain-inspiredcomputingframeworkthatemploysatransientdynamical systemwhosereactiontoaninputsignalistransformedtoatargetoutput.Oneofthecentralproblems inRCistofindareliablereservoirwithalargecriticality,sincecomputingperformanceofareservoiris maximizednearthephasetransition.Inthiswork,weproposeacontinuousreservoirthatutilizestran- sientdynamicsofcoupledchaoticoscillatorsinacriticalregimewheresuddenamplitudedeathoccurs.

This“explosivedeath” notonlybringsthesystemalargecriticalitywhichprovidesavarietyoforbitsfor computing,butalsostabilizesthemwhichotherwisedivergesooninchaoticunits.Theproposedframe- workshowsbetterresultsintasksforsignalreconstructionsthanRCbasedonexplosivesynchronization ofregularphaseoscillators.Wealsoshowthattheinformationcapacityofthereservoirscanbeusedas apredictivemeasureforcomputationalcapabilityofareservoiratacriticalpoint.

© 2020 Elsevier Ltd. All rights reserved.

1. Introduction

Recently,reservoircomputinghasemergedasapromisingcom- putationalframeworkforutilizingadynamicalsystemforcompu- tation. While an inputstream perturbsthe transientintrinsicdy- namicsofamedium(“reservoir”), areadoutlayeristrainedtoex- tract features out of such perturbations to approximate a target output.Dueto itscomplexhigh-dimensionaldynamics,thereser- voir serves asa vast repertoire ofnonlinear transformations that can be exploited by the readout. The major advantage of reser- voircomputingistheirsimplicityintrainingprocesscomparedto other neural networks. Anotheradvantage istheir universalityin that they can be realized using physicalsystems, substrates, and devices[12,14,34].

There is the hypothesis that a system can exhibit maximal computational power at a phase transition betweenordered and chaotic behavioral regimes [18,19]. It hasbeen observed that the brainoperatesnearacriticalstateinordertoadapttoagreatva- rietyofinputsandmaximizeinformationcapacity[3–5].Perturba- tions occurringinacriticalregimeneitherspreadnordieouttoo quickly, providing the most flexibility to the system [10,16].This concept of“computationat theedge ofchaos” may alsohave an implication to material computation, whereby a material hasthe most exploitable properties [27]. More extensive review on this subjectcanbefoundinMunoz[28].

∗ Corresponding author.

E-mail address: pwkim@unist.ac.kr (P. Kim).

InRC,designingareservoir whichhasa largecriticalityisim- portanttoperformcomplextasks.Incaseofa reservoirbasedon continuousdynamical systems, one can create criticality by tun- ing intrinsicparameters sothat the reservoiroperates ata bifur- cationpointacrosswhich thedimensionoftheattractorabruptly declines.Wecallsuchsystemacriticalreservoir.Asystemofcou- pled oscillator exhibits a first order transition from incoherent state to synchronized state that occurs under a specific relation between the coupling strength and connectivity, which is called explosive synchronization. In the previous work [7], we showed that a reservoir of coupled Kuramoto oscillators near explosive synchronizationforms a critical reservoir and performs excellent computations.

Amplitude death(AD) is another way to create a criticality in coupledoscillatory units.It indicates complete cessationof oscil- lationsinducedfromchangeinintrinsicparametersofthesystem.

TheoccurrenceofAD hasbeenfoundinthe caseofchemical re- actions[8,11],neuronal systems[13,29]andcoupledlasersystems [15,37].Ithasbeenalsoreportedthat ADcan occurabruptlyina systemofcoupled nonlinearoscillators [21,35,36,39].Such simul- taneouscessationofoscillationsisthefirstordertransitiontoAD andcalled“explosivedeath”(ED).

Inthiswork,wefocusoncomputingabilityofchaotic systems nearacriticalitycreatedintheformofED.Therehavebeenmany researches onchaos computing [2,22], evenin the context ofRC [17,23–25]; Chaoscomputingtakesadvantage ofan infinitenum- beroforbits/patternsinherentintheattractortobeusedforpar- ticularcomputationaltasks.Italsoutilizesthesensitivitytoinitial conditionsofchaoticsystemstoperformrapidswitchingbetween https://doi.org/10.1016/j.chaos.2020.110131

0960-0779/© 2020 Elsevier Ltd. All rights reserved.

Fig. 1. Schematic of RC of this work. The nodes in the reservoir are chaotic os- cillators and their internal dynamics is aroused by the input signal u(t). Only the weight matrix W connecting the reservoir state R ( t ) needs to be trained to generate a desired output v ( t ).

computationalmodes.However,chaoscomputingoftenhasacon- trolproblemtostabilizeparticularorbit.

Our majorgoal isto constructa chaos basedreservoir witha largecriticalityinducedfromexplosivedeath.Wefocusonhowto createaneasily-adjustablecriticalityinreservoirs,since,aswewill seeinSection4,acriticalityprovidesareliablecriteriontochoose theparameter valuesformaximizedcomputingperformance. We usethe coupledchaoticoscillatorsandadjustacouplingstrength sothat theyremainnearthestageofED.Areservoirinsuchcrit- icalregimeprovidesalargevarietyoforbitsintransientdynamics whichcanbeusedincomputationaltasks.Differentfromprevious chaoticcomputingmethods,thereservoirsstillremaininaregular regimeduringcomputation,butcloseenoughtochaostoenjoyits richness.We alsoinvestigatehowsuch quenchedchaos enhances thecriticality inRC.Toshow thecontributionofchaosincritical- ity,wecomparethecomputationperformanceofachaoticcritical reservoirwithanon-chaoticcriticalreservoir.

2. Model

2.1.Setupforreservoircomputing

TheschemeforRCofchaoticoscillatorsisillustratedinFig.1. The setup is divided into three parts: an input layer, a reser- voir,anda readout layer. The reservoir inthe middleconsists of interconnectednodesofchaoticoscillators,someofwhicharealso connectedtothe input signal.The nodesin thereservoir evolves dynamicallyinreactiontothetemporalinputsignalsu(t)andtheir collectivestatesinthereservoirformthereservoir stateR(t).The role of the reservoir can be viewed as a nonlinear transforma- tion of the input u(t) to a space represented by R(t). The third partofRC isthereadoutlayer wherethe trainingprocess is car- ried out. In the readout layer, a set of sampled reservoir states R_k=R(^tk),k=1,2,... are used togenerate thefinal desired out- putv(t)throughalineartransformationas

v ₍

t

)

=WR (1)

whereR isthe reservoir state vector whichconsists ofa tempo- ralcollectionofR_k,andW isaweight matrixtobe found inthe trainingprocess.Since trainingthe RCsystem onlyinvolves solv- ingalinearEq.(1)fortheweightmatrixW,thetrainingcostcan besubstantiallyreducedcomparedtoconventionalneuralnetwork approaches.

is the meanfield of the state variable x.The parameter K is the strength ofcouplingand0≤Q ≤1,isthe intensityofthemean field.HereweuseQ=0.7,following[35].

EachsinglenodeexactlycoincideswiththeconventionalLorenz systemifK=0withw_i=1.Ifthefrequenciesofthenodesw_iare identical,thenthesysteminEq.(2)undergoesED uponachange oftheparameters[35].Inthefollowingsections,weconfirm that EDpersistentlyoccursinEq.(2)withasuitabledistributionofthe frequencyw_i. Whenrunning thesystemin Eq.(2)asa reservoir, we thereforeset theparameters for thesystemto be posedin a criticalregimewherethephase transitionoccurs. Theadjustment of the parameters according to an order parameter will be dis- cussedinSection3.

2.3. Readoutandtraining

The chaotic reservoirs are applied to supervised tasks of which training data comes in the form of (u, v) where u(^t)=(^u1(^t),...,u^p(^t))∈R^p is an input signal and

v

(^t)= (

v

¹₍t),. . .,

v

^q₍t))∈R^qisatargetoutput.Weassignpnodesofthe reservoirasinputnodes.Beforethetrainingprocessstarts,werun the network until it reachesan amplitude death state. Then the input stream u(t) is fed to the reservoir,in a waythat the value ofx(t) intheinputnodesareperturbedby addingu(t).Allevolu- tionaryactivitiesofthenodesaremeasuredtocomputetheoutput function fout=(^fout¹ ,...,f_out^q )∈R^q.

We construct a linear output function fthat maps the reser- voir state R(t) to a desired output v(t) as in the Eq. (1). In the readout process, it is reasonable to use the past values of the nodes as well asthe current ones, to exploit the rich dynamics ofthechaoticreservoirs.Herewesettheoutputfunctionftotake past ssampled values ofthe frequencyx_i= ^dx_dtⁱ atdiscrete times t−^t,t−2^t,...,t−s^{tasaninternalstateofthereservoir.}

We definethe outputfunction fout=(^fout¹ ,...,f_out^q )∈R^q of(s, ^t)-typeas

f_out^l

(

^t

)

= N

i=1

s

j=1

w^l_i,jx_i

(

^t−

(

^j−1

)

^t

)

, l=1,...,q (3)

NotethatEq.(3)essentiallyimplementsthelinearreadoutprocess ofRCinEq.(1).Here w^l_i,j areweights tobedeterminedfromthe trainingprocess foreach computational task,so that f_out(t) is as closetov(t)aspossible.Forexample,iftheoutput dataisatime series

v

(^t1),

v

(^t2),. . .,

v

(^tM),themean-squareerror

M

i=1

v ₍

t_i

)

−fout

(

^ti

)

²

_M

i=1

v (

^ti

)

^{2 (4)}

can beused todetermine theweights w^l_i,j.Note that minimizing theerrorinEq.(4)withrespecttotheweightsw^l_i,jinEq.(3)cor- respondstoalinearleastsquaresproblem.

Fig. 2. (a) The phase diagram of order parameter r varaccording to the coupling strength K and Lorenz parameter ρ. The dotted vertical line is drawn at ρ= 24 to denote separation of non-chaotic dynamics(left) and chaotic dynamics(right) in the case K = 0 . (b) Cross-sectional figures of two order parameters, r varand A , according to K with ρ= 28 fixed. (c) Cross-sectional figures of two order parameters, r varand A , according to ρwith K = 2 . 2 fixed.

3. Creatingacriticalitybytheexplosivedeath

Benefit of using the coupled chaotic systems in Eq. (2) as a reservoiristhatonecaneasilycreatealargecriticalitywithafirst orderphasetransitioninthesystem.Acrossacriticalpointofthe coupling force, the compound oscillatory motions of the system collapseintoanequilibriumpoint.Thisfixedequilibriumstatenear acriticalpointisusedasthegroundstateforreservoircomputing, wherethesystemalwaysreturnstoaftereverycomputation,eras- ingunnecessaryinformationfrompreviousevaluationsandprepar- ingforthenextinputs.

Tolookforapossiblephase transitioninEq.(2),we definean orderparameterrvarintermsofthevariationofamplitudes,as

rvar= 1 N

N

j=1

exp

(

^−cvarj

)

^{, c>0 (5)}

where var_j is the temporal variance of the frequency x_j(^t) ^[7]. Thisisameasurefordesynchronythatsensitivelyshowsadegree ofdeviationof oscillatorsfroma steadyfrequency.Inthe ground state, thetemporal variance ofthefrequency shouldbe kept low for reliable computations.Note that, foreach oscillator, the tem- poral variance of the frequency becomes 0 if a strong coupling strength holds oscillators in a phase-locked state, keeping their common frequencysteady. We also use anotherorder parameter basedonthenormalizedaverageamplitudeas

A

(

^K

)

=a

(

^K

)

a

(

⁰

)

^{, a}

(

^K

)

= N

i=1

(

^xi,max

^t−

^xi,min

^t

)

N , (6)

which is widely used forchaotic oscillators [30,31,35]. Note that A(^K)=0impliesacompletecessationofoscillationsandA(^K)=1 impliesnondepressedchaoticoscillations.

If the frequencies of nodes in Eq. (2) are identical, then the systemexhibitsexplosivedeath,thediscontinuoustransitionfrom theoscillatorystatetothecompletelyquenchedstate[35].Indeed, Eq.(2)withnonidenticalnaturalfrequenciesstillexhibitsthesame phenomena. Fig. 2(a) is thephase diagram ofthe order parame- ter rvar in the

ρ

−K parameter plane. We used N=100 oscilla- tors whosenaturalfrequencies followthe uniformdistributionin [1,1.3]. The dynamical states of system is obtained by backward continuation from a large value of K [35]. The order parameter r_var wasaveraged betweent=[3900,4000].The diagonal line in Fig.2(a) clearlysplits theparameter planeaccordingto thevalue of rvar, indicating that the discontinuousphase transition occurs withrespect toboth

ρ

^{andK.The graphsinFig. 2(b)and(c)in-}

dicate cross-sectionalfigures oftwoorderparameters inK and

ρ

directions,respectively.Itisverifiedthatbothoftheorderparam-

etersrvarandAexhibitextremelyabruptjumpatthesamecritical point.

4. Results 4.1. Numericaltests

Inthe followingnumerical examplesto test thelearningabil- ityoftheoscillatornetworks, we usethe (¹⁰,0.1)−type readout.

Thatis,theoutput functionf_outatt isobtainedfrom10previous sampledvaluesoftheoscillatorfrequenciesx_i(^t),...,x_i(^t−0.9)ⁱⁿ Eq.(3).

Weset up twotypes oftasks,inferring missingvariables(Task 1andTask2)andfilteringsignals(Task3),allofwhichrequirethe presenceoflong-termmemoryforproperexecution.InTask1,RC is used to reconstruct evolutionary value of the variable y(t) (or z(t))oftheRösslersystem

dx

dt =−y−z dy

dt =x+ay dz

dt =b+z

(

^x−c

)

(7)

fromobservationofasinglevariablex(t).Thatis,u(^t)=x(^t)^isthe inputsignaland

v

(^t)=y(^t)^{isthetargetoutputofRCinthetask.}

Weusea=0.2,b=0.2andc=5.7whicharecommonlyusedfor studyofchaotic behavior[20].Similarly,Task2istoinfermissing variablesoftheChua’scircuit

dx

dt =

α (

^y−

φ (

^x

))

dy

dt =x−y+z dz

dt =−

β

^y

(8)

where

φ

(^x)=m₁x+1/2(^m0−m₁)(

|

^x+1

|

−

|

^x−1

|

)^{. RC istrained} toreconstructa desiredoutput

v

(^t)=y(^t) ^{fromthe inputu}(^t)= x(^t)^{. The parameters are set as}

α

=15.6,m0=−8/7,m1=−5/7 and

β

=28whichleadtochaoticbehavior.

Task3istolearntogeneratethefilteredscalaroutput

v ₍

t

)

⁼_m¹ m

k=1

au

(

^t−k

)

^+bu

(

^t−k

)

^2+cu

(

^t−k

)

³

(9) which is determined fromthe past m valuesof an input stream u(t). Here a, b andc are some nonzeroparameters. Ifm=1, the

Fig. 3. Test errors according to the coupling K and lorenz system parameter ρ. (a) inferring a missing variable of the Rössler system. (b) inferring a missing variable of the Chua’s circuit. (c) filtering the Mackey-Glass equation.

task is simply to implement a polynomial function of the cur- rentvalueofthe input.The taskbecomesmore challengingasm increases,requiring long-term memory to evaluate averaged val- ues.In ourtasks,we usetheparameters m=20,a=1,b=3and c=5, and have the input u(t) generated from the Mackey-Glass equation

dx

dt =

β

^x

⁽

^t−

τ )

1+x

(

^t−

τ )

ⁿ⁻

γ

^x,

β

^=0.2,

γ

^=0.1,

τ

^=17andn=10.

(10)

which provides standard benchmark task for chaotic series han- dling[6].

In each task, the continuous input signal u(t) and the tar- get signal v(t) are generated for t ∈ [0, 6000]. To make sure that the system is positioned in the reliable ground state, we skip first 1000 time steps of the output. The training process is applied to match fout to v(t) over the 4000 discrete time steps,t=1001,1002,. . .,5000.Thatis,thereadoutweightsw^l_i,jin Eq.(3) are determined to minimize the relative errorin Eq. (4). Then we evaluate the relative error between fout to v(t) as the performance measure over 1000 discrete sampled time steps for t∈(5000,6000]. Throughthe threetasks, weagainuse N=100 oscillatorsandtheuniformdistribution in[1,1.3]fortheir natural frequencyw_i.

Fig. 3 depicts the errors in three tasks with respect the pa- rameters K and

ρ

^{. One can see in each task that minimum er-}

ror occurs along a diagonal line. The line forms a clear border acrosswhichthe errorjumps fromthelow errorregime (red)to higherrorregime(yellow).Itshouldbenotedthatthethreelines areidenticalandthesameasthealignedcriticalpointsinFigure 2.(a)wheretheexplosive death ofthenodesoccurs. Thisassures thatthecomputationalperformanceofthereservoirsismaximized nearthefirstorderphasetransition.

As a final remark,it should be pointedout that thereservoir in Eq. (2) provides decent performance as a learning algorithm compared to other types of RC, for example, an echo state net- work(ESN).WeappliedastandardESNinLukoševiˇciusetal.[26]to the above three tasks for comparison. The parameters

ρ

^{and K}

in Eq. (2) are chosen on the diagonal line in Fig. 3 so that the reservoirenjoysa criticalityattheborderbetweentwo phases.It turnedoutthatthereservoirinEq.(2)showscompetitive(Task1) orbetter(Task2 and3) resultsascompared to a ESN with1000 nodes.

4.2. Informationcapacityofregularandchaoticreservoirs

In the previous work [7], a reservoir that consists of regular phaseoscillatorswaspresentedas

θ

i=

ω

i+

λ|

^wi

|

k_i N

j=1

A_{i j}sin

( θ

j−

θ

i

)

^{, i=1,...,N, (11)}

where

λ

^{isthecouplingstrengthof}oscillatorsandA_ijistheentry oftheadjacencymatrixofthenetwork.Here A_{i j}=1ifi= j,oth- erwise0.ThemodelinEq.(11) isknowntohavea simultaneous synchronizationata certain couplingvalue

λ

^{[38]. Beingusedas}

a reservoir,it showsgreat performance improvementacross such criticalpoint.

ThissectioninvestigateshowchaosenhancesacriticalityinRC.

We compare the performance of the forementioned two critical reservoirs,chaoticoneinEq.(2)andregularoneinEq.(11),when both are being poised at the first order phase transition. From here on, we call the former QC(quenched chaos) and the latter ES(explosive synchronization). When comparing the performance ofthesereservoirs, itisnecessarytoconsiderthatthenumberof equationsrequiredtoimplementasinglenodeisdifferent:ifthey have the same numberof nodes, the computational cost for the reservoirofEq.(2)isgreaterthanthatofthereservoirofEq.(11), roughly,byafactorofthree.

Fig.4(a)–(c)depicttheerrorsofQCandESintask1to3,re- spectively, accordingto the number ofnodes used forthe reser- voirs. It is observed that the errors continuously decrease and reach the minimum at 150 nodes or less. In all three tasks, the errorofQCisatleast1000timessmallerthanthatofES.Thisindi- catesthatQCexcelsbyfarES,evenwhenconsideringtheforemen- tioned difference in computation complexitybetween two reser- voirs.

Oneofpossibleexplanationsonsuperiorityofthechaoticreser- voiristhatthecomputingcapabilityofcriticalreservoirsmayde- pendonthecollapseddimensionofattractorsofreservoirsacross thecriticalpoint.Thatis,theeffectofcriticalityoncomputingper- formancemayberelatedtohowmuchreductionoccursinthedi- mensionof thesynchronizationmanifold atthe phasetransition.

One canguessthat thecollapsed dimensionof Eq.(2)attheex- plosivedeath ismuchgreater thanthat ofEq.(11),fromthefact that an attractor of a single Lorenz system has a greater Haus- dorff dimension(~ 2.06), compared to one dimensional attractor of a phase oscillatorin Eq.(11).Computing the dimension of an attractorofalargecoupledchaotic systemis,however,extremely time-consumingandnotpractical.

Fig. 4. Comparison of test errors of QC and ES with respect to number of nodes. (a) inferring a missing variable of the Rössler system. (b) inferring a missing variable of the Chua’s circuit. (c) filtering the Mackey-Glass equation .

Fig. 5. Total information capacity of QC and ES with respect to number of oscilla- tors. The most of the parameters for evaluation are from [9] .

To overcome such difficulty in analyzing reservoir’s internal structure, we rather adopt a measure that focuses on external functionalcapacityofsystems.Here we usethetotalinformation capacity which is developed to compute the capacity of anyin- put drivendynamical systems [9]. Thetotal informationcapacity, roughlyput,isdefinedastheassessmentofreconstructingtheset of orthonormal functionswhich isa basis ofthe fadingmemory Hilbertspace.Wereferthereaderto[9]formoredetails.

Fig.5comparesthetotalinformationcapacityofQCandES.We confirm that thecapacityofQCcontinuously increaseseven near 150 nodesthendecreasein tendency,whileES onlyincreasestill about 50 nodes, which agrees with the results of the numerical tasksinFig.4.

5. Discussion

Inthiswork,weshowedthat thecoupledchaoticsystemscan be used for efficient reservoir computing. The chaotic reservoirs can createa large criticalityat thefirst orderphase transitionto createa ground state forcomputation. It notices inseveral com- putingtasksthatthechaoticreservoirsexceltheregularreservoirs, whichis alsoconfirmedfromcomparing theirinformationcapac- ity.

Theresultsimplythatusingchaoticnodesismorebeneficialin constructingreservoirs.Thisfindingisimportantinseveralaspects.

Firstofall,chaosiswidelyobservedinneuronalsystems,bothex- perimentallyandtheoretically [1].We confirmed that such ubiq- uityof chaos can be justifiedfrom the perspective ofcomputing performance.Thatis,aslongasitisproperlyquenchedinthecrit- icalregime,chaosisangoalworthpursuingratherthananunde- sirablestate tobeavoided.Chaoscomputingistheparadigmthat exploitsthe controlledrichnessofnonlineardynamicsto doflex- ible computations.This work showsanother theoretical direction ofchaoscomputingdifferentfromtheapproachusingchaoticele- mentstoemulatedifferentlogicgates[32,33].Basicunderstanding ofaroleofcriticality inregular andchaoticreservoirs canbeex- pectedtoshedlightonhowinformationisprocessedinquenched couplednonlinearsystems,potentially leadingto propositionofa broadrangeofreservoirs.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

JaesungChoi:Conceptualization, Methodology, Writing- orig- inaldraft. Pilwon Kim: Formal analysis, Visualization, Validation, Writing-review&editing.

Acknowledgments

Thisworkwassupported bytheNationalResearchFoundation ofKorea (NRF-2017R1D1A1B04032921).The funderhadnorolein studydesign,data collectionandanalysis, decisionto publish,or preparationofthemanuscript.

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