Post on 22-Apr-2023
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
Solvinginverseproblem1-1
AdS/CFT:quantumresponsefromgeometry
Deeplearning:op=mizedsequen=almap
FromAdStoDL
Dic=onaryofAdS/DLcorrespondence
review
review
1-2
1-3
1. Formula\onofAdS/DLcorrespondence
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)
||
Conven\onalholographicmodeling
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
Dic=onaryofAdS/DLcorrespondence
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
Dic=onaryofAdS/DLcorrespondence
review
review
1-2
1-3
1. Formula\onofAdS/DLcorrespondence
Emergentgeometryindeeplearning2-1
CanAdSSchwarzschildbelearned?
Emergentspacefromrealmaterial?
Numericalexperimentsummary
Machineslearn…,whatdowelearn?
2-2
2-3
2-4
2-5
2.Implementa\onofAdS/DLandemergingspace
18
Experiment1:“CanAdSSchwarzschildbelearned?”
Experiment2:“Emergentspacefromrealmaterial?”
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
1) Usematerialexperimentaldata.Ex)Magne\za\oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
Emergentgeometryindeeplearning2-1
19
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
AdSSchwarzschildmetricintheunitofAdSradius
�2�� + h(�)���� �V [�]
��= 0
V [�] = ��2 +14�4h(�) = 3 coth(3�)
L = 1
20
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
��
� � = 0� =�
�(� + ��) = �(�) + ��
�h(�)�(�)� �V (�(�))
��(�)
��(� + ��) = �(�) + �� �(�)
����=0
= 0
21
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
�input
�input
Horizoncondi\on:true:false
22
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
Unspecifiedmetric(10layers,tobetrained)
GenerateddatafromAdSSchwarzschild(10000datapoints)
�input
�input
����=0
= 0
23
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
Witharegulariza\on
24
Experiment1:“CanAdSSchwarzschildbelearned?”
Experiment2:“Emergentspacefromrealmaterial?”
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
1) Usematerialexperimentaldata.Ex)Magne\za\oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
Emergentgeometryindeeplearning2-1
25
Exp2:Emergentspacefromrealmaterial?2-3
1) Usematerialexperimentaldata.Ex)Magne\za\oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
26
Numericalexperimentsummary2-4
Experiment1
AdSSchwarzschildissuccessfullylearned.
Experiment2
Experimentaldataisexplainedbyemergentspace.
Conven\onalholographicmodeling
Metric
Experimentdata
Model
gµ�
Predic\onPredic\on
Comparison
Experimentdata
Ourdeeplearningholographicmodeling
Metric
Model
gµ�
Predic\on
Experimentdata
Experimentdata
Machineslearn…,whatdowelearn?2-5
28
Machineslearn…,whatdowelearn?2-5- Unreproducedregionnearthehorizon?àIssueonregulariza\onanddatathickness- Canmorecomponentsofmetricbelearned?àBulkgaugefields,inhomogeneousdata- IsEinsteinequa\onsa\sfied?àNo,ingeneral.- HolographicQCD?Predic\on?- Einsteingravity?Emergentspace?Generalnetwork?