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Deep Learning and AdS/CFT Koji Hashimoto (Osaka u) ArXiv:1802.08313 w/ S. Sugishita (Osaka), A. Tanaka (RIKEN AIP), A. Tomiya (CCNU) KIAS, 26 March, 2018 Cquest, Sogang u., 29 March, 2018 MIT, CTP, 4 Apr, 2018 MPI, AEI, 13 Apr, 2018 ET group, Osaka, 30 May, 2018 DLAP2018 workshop, Osaka, 1 June, 2018 Paris QCD workshop, 11 June, 2018 Yau Center, China, 15 June, 2018

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Page 1: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

Page 2: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

����AdS boundary

WatchhowmachinelearnsAdSblackhole

AdSradialdirec^on

Page 3: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Brane Brain(Superstringtheory) (Neuroscience)

Page 4: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

DeepLearning

BlackholeCFT AdS

AdS/CFT

“cat”

[Maldacena‘97]

Page 5: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

1.  Formula^onofAdS/DLcorrespondence

2.Implementa^onofAdS/DLandemergingspace

Page 6: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Solvinginverseproblem1-1

AdS/CFT:quantumresponsefromgeometry

Deeplearning:op=mizedsequen=almap

FromAdStoDL

Dic=onaryofAdS/DLcorrespondence

review

review

1-2

1-3

1.  Formula^onofAdS/DLcorrespondence

Page 7: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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)

||

Page 8: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Conven^onalholographicmodeling

Metric

Experimentdata

Model

gµ�

Predic^onPredic^on

Comparison

Experimentdata

Solvinginverseproblem1-1Ourdeeplearning

holographicmodeling

Metric

Model

gµ�

Predic^on

Experimentdata

Experimentdata

Page 9: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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]

Page 10: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

����

Page 11: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

Page 12: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

Page 13: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

Page 14: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Solvinginverseproblem1-1

Deeplearning:op=mizedsequen=almap

AdS/CFT:quantumresponsefromgeometry

FromAdStoDL

Dic=onaryofAdS/DLcorrespondence

review

review

1-2

1-3

1.  Formula^onofAdS/DLcorrespondence

Page 15: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

1.  Formula^onofAdS/DLcorrespondence

2.Implementa^onofAdS/DLandemergingspace

Page 16: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Emergentgeometryindeeplearning2-1

CanAdSSchwarzschildbelearned?

Emergentspacefromrealmaterial?

Numericalexperimentsummary

Machineslearn…,whatdowelearn?

2-2

2-3

2-4

2-5

2.Implementa^onofAdS/DLandemergingspace

Page 17: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^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

Page 18: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

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

Page 19: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp1:CanAdSSchwarzschildbelearned?2-2

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

��

� � = 0� =�

�(� + ��) = �(�) + ��

�h(�)�(�)� �V (�(�))

��(�)

��(� + ��) = �(�) + �� �(�)

����=0

= 0

Page 20: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp1:CanAdSSchwarzschildbelearned?2-2

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

�input

�input

Horizoncondi^on:true:false

Page 21: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp1:CanAdSSchwarzschildbelearned?2-2

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

Unspecifiedmetric(10layers,tobetrained)

GenerateddatafromAdSSchwarzschild(10000datapoints)

�input

�input

����=0

= 0

Page 22: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp1:CanAdSSchwarzschildbelearned?2-2

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

Page 23: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp1:CanAdSSchwarzschildbelearned?2-2

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

Witharegulariza^on

Page 24: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^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

Page 25: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Exp2:Emergentspacefromrealmaterial?2-3

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

Page 26: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Numericalexperimentsummary2-4

Experiment1

AdSSchwarzschildissuccessfullylearned.

Experiment2

Experimentaldataisexplainedbyemergentspace.

Page 27: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

Machineslearn…,whatdowelearn?2-5Conven&onal

holographicmodeling�Ourdeeplearning

holographicmodeling�

Metric�

Experimentdata�

Model�

gµ�

Experimentdata�

Predic&onPredic&on

Comparison�

Metric�

Experimentdata�

Model�

gµ�

Experimentdata�

Predic&on

Page 28: Deep Learning and AdS/CFT - Brown University...Deep learning : op=mized sequen=al map F = f ix (N ) i Layer 1 Layer 2 Layer N “Weights” (variable linear map) (x) “Ac^va^on func^on”

1.  Formula^onofAdS/DLcorrespondence

2.Implementa^onofAdS/DLandemergingspace


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