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
ETgroup,Osaka,30May,2018DLAP2018workshop,Osaka,1June,2018
ParisQCDworkshop,11June,2018YauCenter,China,15June,2018
����AdS boundary
WatchhowmachinelearnsAdSblackhole
AdSradialdirec^on
Brane Brain(Superstringtheory) (Neuroscience)
DeepLearning
BlackholeCFT AdS
AdS/CFT
“cat”
[Maldacena‘97]
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
1. Formula^onofAdS/DLcorrespondence
Conven^onalholographicmodeling
Metric
Experimentdata
Model
gµ�
Predic^onPredic^on
Comparison
Experimentdata
Solvinginverseproblem1-1
AdS/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
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,Wimen]
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
����
FromAdStoDL
Discre^za^on,Hamiltonform
�(� + ��) = �(�) + ��
�h(�)�(�)� �V (�(�))
��(�)
��(� + ��) = �(�) + �� �(�)
��
� � = 0
Neural-Networkrepresenta^on
�(� = 0)
� =�
BulkEoM �2�� + h(�)���� �V [�]
��= 0
h(�) � ��
�log
�f(�)g(�)d�1
�metric
1-2
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
Dic=onaryofAdS/DLcorrespondence
AdS/CFT Deeplearning
Emergentspace 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
1. Formula^onofAdS/DLcorrespondence
2.Implementa^onofAdS/DLandemergingspace
Emergentgeometryindeeplearning2-1
CanAdSSchwarzschildbelearned?
Emergentspacefromrealmaterial?
Numericalexperimentsummary
Machineslearn…,whatdowelearn?
2-2
2-3
2-4
2-5
2.Implementa^onofAdS/DLandemergingspace
Experiment1:“CanAdSSchwarzschildbelearned?”
Experiment2:“Emergentspacefromrealmaterial?”
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
1) Usematerialexperimentaldata.Ex)Magne^za^oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
Emergentgeometryindeeplearning2-1
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
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
��
� � = 0� =�
�(� + ��) = �(�) + ��
�h(�)�(�)� �V (�(�))
��(�)
��(� + ��) = �(�) + �� �(�)
����=0
= 0
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
�input
�input
Horizoncondi^on:true:false
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
Unspecifiedmetric(10layers,tobetrained)
GenerateddatafromAdSSchwarzschild(10000datapoints)
�input
�input
����=0
= 0
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
Exp1:CanAdSSchwarzschildbelearned?2-2
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
Witharegulariza^on
Experiment1:“CanAdSSchwarzschildbelearned?”
Experiment2:“Emergentspacefromrealmaterial?”
1) UseAdSSchwarzschildandgenerateinputdata.2) Preparenetworkwithunspecifiedmetric.3) Letthenetworklearnitbythedata.4) CheckifAdSSchwarzschildisreproduced.
1) Usematerialexperimentaldata.Ex)Magne^za^oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
Emergentgeometryindeeplearning2-1
Exp2:Emergentspacefromrealmaterial?2-3
1) Usematerialexperimentaldata.Ex)Magne^za^oncurveofstronglycorrelatedmaterial2)3)(sameasabove.)4)Watchhowspaceemerges!
Numericalexperimentsummary2-4
Experiment1
AdSSchwarzschildissuccessfullylearned.
Experiment2
Experimentaldataisexplainedbyemergentspace.
Machineslearn…,whatdowelearn?2-5Conven&onal
holographicmodeling�Ourdeeplearning
holographicmodeling�
Metric�
Experimentdata�
Model�
gµ�
Experimentdata�
Predic&onPredic&on
Comparison�
Metric�
Experimentdata�
Model�
gµ�
Experimentdata�
Predic&on
1. Formula^onofAdS/DLcorrespondence
2.Implementa^onofAdS/DLandemergingspace