deep learning and ads/cft - brown university...deep learning : op=mized sequen=al map f = f ix (n )...

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