KIAS, 26 March, 2018
Cquest, Sogang u., 29 March, 2018
MIT, CTP, 4 Apr, 2018
MPI, AEI, 13 Apr, 2018
HET group, Osaka, 30 May, 2018
DLAP2018 workshop, 1 June, 2018

Deep Learning and AdS/CFTKoji Hashimoto (Osaka u)
ArXiv:1802.08313 w/ S. Sugishita (Osaka),
A. Tanaka (RIKEN AIP),
A. Tomiya (CCNU)

Brane(Superstring theory)

Brain(Neuroscience)2

Deep Learning“cat”

AdS/CFT[Maldacena ‘97]

CFTAdSBlackhole1. Formula`on ofAdS/DL correspondence

2. Implementa`on of AdS/DLand emerging space41. Formula`on ofAdS/DL correspondence
1-1

Solving inverse problem

review

AdS/CFT: quantum response from geometry

review

Deep learning: op=mized sequen=al map

1-2

From AdS to DL

1-3

Dic=onary of AdS/DL correspondence

1-1

Solving inverse problem

AdS/CFT(No proof, no deriva`on)Classical gravity
in d+1 dim. space`me||Quantum ﬁeld theory
in d dim. space`me(Strong coupling limit)

Conven`onal
holographic modeling
ModelMetricPredic`ongµPredic`onComparison
Experiment
data

Experiment
data

1-1

Solving inverse problem

Our deep learning
holographic modeling

Conven`onal
holographic modeling

Model

Model

Metricgµ
Predic`on

MetricPredic`ongµPredic`onComparison
Experiment
data

Experiment
data

Experiment
data

Experiment
datareview

AdS/CFT: quantum response from geometry[Klebanov, Wiken]

Classical scalar ﬁeld theory in (d+1) dim. geometryS=

dd+1 xds2 =det g ()2V( )f ( )dt2 + d 2 + g( )(dx21 + · · · + dx2d 1 )AdS boundary (Black hole horizon (): f g0): f

exp[2 /L]

2, gconst.

Solve EoM, get response O J . Boundary condi`ons:
AdS boundary ():= Je

1++Black hole horizon (0):O e=0=0+8

review

Deep learning : op=mized sequen=al map
Layer N

Layer 2

Layer 1
(1)
xi

(2)
xi =

(1) (1)(Wij xj )(N )xi(N )

F = fi xi

(1)

Wij

(x)

“Ac`va`on func`on” (ﬁxed nonlinear fn.)
“Weights” (variable linear map)(1)
Prepare many sets {xi , F } : input + output

1)2) Train the network (adjust Wij ) by lowering
(N 1)
(1)
fi ( (Wij(· · · (Wlm x(1)“Loss func`on” Em ))))
dataF

1-2

From AdS to DL
2

Bulk EoM+ h( )metricDiscre`za`on, Hamilton form
( +( +)= ( )+)= ( )+V[ ]h( )=0log

f ( )g( )d 1( )h( ) ( )V ( ( ))( )Neural-Network representa`on( = 0)==010

1-2

From AdS to DL

Bulk EoM

2+ h( )

metricDiscre`za`on, Hamilton form
( +( +)= ( )+)= ( )+V[ ]h( )=0log

f ( )g( )d 1( )V ( ( ))
( )h( ) ( )Neural-Network representa`on=0==0=011

1-3

Dic=onary of AdS/DL correspondence
AdS/CFT

Deep learning

Emergent space

Depth of layers
i = 1, 2, · · · , N
Network weights

0>

Bulk gravity metrich( )Nonlinear response
O JHorizon condi`on=0=0

Interac`on
V( )

(a)

Wij

Input data(1)

xi

Output dataFAc`va`on func`on(x)121. Formula`on ofAdS/DL correspondence
1-1

Solving inverse problem

review

Deep learning : op=mized sequen=al map

review

AdS/CFT: quantum response from geometry

1-2From AdS to DL

1-3

Dic=onary of AdS/DL correspondence1. Formula`on ofAdS/DL correspondence

2. Implementa`on of AdS/DLand emerging space14

2. Implementa`on of AdS/DLand emerging space
2-1

Emergent geometry in deep learning

2-2

Can AdS Schwarzschild be learned?

2-3

Emergent space from real material?

2-4

Numerical experiment summary

2-5Machines learn…, what do we learn?

2-1Emergent geometry in deep learning

Experiment 1: “Can AdS Schwarzschild be learned?”
1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
Experiment 2: “Emergent space from real material?”
1) Use material experimental data.
Ex) Magne`za`on curve ofstrongly correlated material
2) 3) (same as above.)4) Watch how space emerges!
16

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
2+ h( )

V[ ]h( ) = 3 coth(3 )=0V[ ]=AdS Schwarzschild metricin the unit of AdS radius L = 1

21+4

417

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
( +( +)= ( )+)= ( )+( )h( ) ( )V ( ( ))
( )=0==0=018

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
Horizoncondi`on: true
: false

input

input

19

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
input
input=0=0

Unspeciﬁed metric(10 layers, to be trained)
Generated data from AdS Schwarzschild(10000 data points)
20

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.

21

2-2

Exp1: Can AdS Schwarzschild be learned?

1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.

With a regulariza`on22

2-1

Emergent geometry in deep learning

Experiment 1: “Can AdS Schwarzschild be learned?”
1) Use AdS Schwarzschild and generate input data.
2) Prepare network with unspeciﬁed metric.
3) Let the network learn it by the data.
4) Check if AdS Schwarzschild is reproduced.
Experiment 2: “Emergent space from real material?”
1) Use material experimental data.
Ex) Magne`za`on curve ofstrongly correlated material
2) 3) (same as above.)4) Watch how space emerges!
232-3 Exp2: Emergent space from real material?

1) Use material experimental data.
Ex) Magne`za`on curve ofstrongly correlated material
2) 3) (same as above.)4) Watch how space emerges!
24

2-4

Numerical experiment summary

Experiment 1

Experiment 2

AdS Schwarzschild is
successfully learned.

Experimental data is
explained by emergent
space.

25

2-5

Machines learn…, what do we learn?
Conven&onal
holographic modeling

Our deep learning
holographic modeling

ModelModel

Metric gµMetric gµ

Predic&on

Predic&on

Predic&on

Comparison
Experimentdata

Experimentdata

ExperimentdataExperimentdata26

1. Formula`on ofAdS/DL correspondence

2. Implementa`on of AdS/DL
and emerging space27