Deep Learning and AdS/CFT - CERN Indico

DeepLearningandAdS/CFT

KojiHashimoto(Osakau)ArXiv:1802.08313w/S.Sugishita(Osaka),A.Tanaka(RIKENAIP),A.Tomiya(CCNU)

KIAS,26March,2018CQuest,Sogangu.,29March,2018

MIT,CTP,4Apr,2018MPI,AEI,13Apr,2018

HETgroup,Osaka,30May,2018DLAP2018workshop,Osaka,1June,2018

ParisQCDworkshop,11June,2018Machinelearningworkshop,TSiMF,China,15June,2018

3

��AdS boundary

WatchhowmachinelearnsAdSblackhole

AdSradialdirec\on

4

Brane Brain(Superstringtheory) (Neuroscience)

DeepLearning

BlackholeCFT AdS

AdS/CFT

“cat”

[Maldacena‘97]

Cf.TalkbyYi-ZhuangYou

6

1.  Formula\onofAdS/DLcorrespondence

2.Implementa\onofAdS/DLandemergingspace

Solvinginverseproblem1-1

　AdS/CFT:quantumresponsefromgeometry

　Deeplearning:op=mizedsequen=almap

FromAdStoDL

Dic=onaryofAdS/DLcorrespondence

review

review

1-2

1-3


8

1974Yoneya,Scherk-Schwarz:String=quantumgravity.

1997Maldacena:DiscoveryofAdS/CFT.Aquantumgravityisnonperturba\velydefined.

2002HolographicQCD.

1976Hawking:Informa\onlossproblemofblackholes.Hawking,Phys.Rev.D14(1976)2460.

Yoneya,Prog.Theor.Phys.51(1974)1907.Scherk,Schwarz,Nucl.Phys.B81(1974)118.

Maldacena,Adv.Theor.Math.Phys.2(1998)231.

Briefhistoryofquantumgravity1-1

Karch,Katz,JHEP0206:043.Kruczenski,Mateos,Myers,WintersJHEP0405:041.Sakai,Sugimoto,PTP113(2004)843.

2008Holographicsuperconductor.Hartnoll,Herzog,Horowitz,PRL101(2008)031601.

2009Bulkreconstruc\on.Heemskerk,Penedones,Polchinski,Sully,JHEP0910:079.

Conven\onalholographicmodeling

Metric

Experimentdata

Model

gµ�

Predic\onPredic\on

Comparison

Experimentdata

Solvinginverseproblem1-1AdS/CFT

(Noproof,noderiva\on)

Classicalgravityind+1dim.space\me

Quantumfieldtheoryinddim.space\me(Strongcouplinglimit,

largeDoFlimit)

||


Metric

Experimentdata

Model

gµ�

Predic\onPredic\on

Comparison

Experimentdata

Solvinginverseproblem1-1Ourdeeplearning

holographicmodeling

Metric

Model

gµ�

Predic\on

Experimentdata

Experimentdata

11

AdS/CFT:quantumresponsefromgeometry

Classicalscalarfieldtheoryin(d+1)dim.geometry

S =�

dd+1x��det g

�(��)2 � V (�)

�

f � �2, g � const.f � g � exp[2�/L]AdSboundary():� � �

Blackholehorizon():� � 0

ds2 = �f(�)dt2 + d�2 + g(�)(dx21 + · · · + dx2

d�1)

SolveEoM,getresponse.Boundarycondi\ons:

��=0

= 0

AdSboundary():� � �

Blackholehorizon():� � 0

� = Je�� +1

�+ � ��O�e��+�

�O�J

review

[Klebanov,Wioen]

Deeplearning:op=mizedsequen=almap

F = fix(N)i

Layer1 Layer2 LayerN

“Weights”(variablelinearmap)

�(x)“Ac\va\onfunc\on”(fixednonlinearfn.)

1)  Preparemanysets:input+output2)  Trainthenetwork(adjust)bylowering“Lossfunc\on”

{x(1)i , F}

Wij

W (1)ij

x(1)i x(2)

i = �(W (1)ij x(1)

j ) x(N)i

review

E ��

data

�� fi(�(W (N�1)ij �(· · · �(W (1)

lm x(1)m ))))� F

��

13

FromAdStoDL

Discre\za\on,Hamiltonform

�(� + ��) = �(�) + ��

�h(�)�(�)� �V (�(�))

��(�)

��(� + ��) = �(�) + �� (�)

Neural-Networkrepresenta\on

BulkEoM �2�� + h(�)�� V [�]

��= 0

h(�) � ��

�log

�f(�)g(�)d�1

�metric

1-2

��

� � = 0� =��

��=0

= 0

14


AdS/CFT DeeplearningEmergentspace Depthoflayers

Bulkgravitymetric Networkweights

Nonlinearresponse Inputdata

Horizoncondi\on Outputdata

Interac\on Ac\va\onfunc\on

�O�J

��=0

= 0

h(�) W (a)ij

1-3

x(1)i

F

�(x)V (�)

� > � � 0 i = 1, 2, · · · , N

Solvinginverseproblem1-1

Deeplearning:op=mizedsequen=almap

　AdS/CFT:quantumresponsefromgeometry

FromAdStoDL


review

review

1-2

1-3


16



Emergentgeometryindeeplearning2-1

CanAdSSchwarzschildbelearned?

Emergentspacefromrealmaterial?

Numericalexperimentsummary

Machineslearn…,whatdowelearn?

2-2

2-3

2-4

2-5


18

Experiment1:“CanAdSSchwarzschildbelearned?”

Experiment2:“Emergentspacefromrealmaterial?”

1)  UseAdSSchwarzschildandgenerateinputdata.2)  Preparenetworkwithunspecifiedmetric.3)  Letthenetworklearnitbythedata.4)  CheckifAdSSchwarzschildisreproduced.

1)  Usematerialexperimentaldata.Ex)Magne\za\oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!


19

Exp1:CanAdSSchwarzschildbelearned?2-2


AdSSchwarzschildmetricintheunitofAdSradius

�2�� + h(�)�� V [�]

��= 0

V [�] = ��2 +14�4h(�) = 3 coth(3�)

L = 1

20



��

� � = 0� =�

�(� + ��) = �(�) + ��

�h(�)�(�)� �V (�(�))

��(�)

��(� + ��) = �(�) + �� (�)

��=0

= 0

21



�input

�input

Horizoncondi\on:true:false

22



Unspecifiedmetric(10layers,tobetrained)

GenerateddatafromAdSSchwarzschild(10000datapoints)

�input

�input

��=0

= 0

23



Witharegulariza\on

24

Experiment1:“CanAdSSchwarzschildbelearned?”

Experiment2:“Emergentspacefromrealmaterial?”




25

Exp2:Emergentspacefromrealmaterial?2-3


26

Numericalexperimentsummary2-4

Experiment1

AdSSchwarzschildissuccessfullylearned.

Experiment2

Experimentaldataisexplainedbyemergentspace.


Metric

Experimentdata

Model

gµ�

Predic\onPredic\on

Comparison

Experimentdata

Ourdeeplearningholographicmodeling

Metric

Model

gµ�

Predic\on

Experimentdata

Experimentdata

Machineslearn…,whatdowelearn?2-5

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Machineslearn…,whatdowelearn?2-5-  Unreproducedregionnearthehorizon?àIssueonregulariza\onanddatathickness-  Canmorecomponentsofmetricbelearned?àBulkgaugefields,inhomogeneousdata-  IsEinsteinequa\onsa\sfied?àNo,ingeneral.-  HolographicQCD?Predic\on?-  Einsteingravity?Emergentspace?Generalnetwork?

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Deep Learning and AdS/CFT - CERN Indico

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