Oka symposium, Dec 2, 2018

Deep Learning and AdS/CFTKoji Hashimoto (Osaka u)
ArXiv:1802.08313, 1809.10536
w/ S. Sugishita (Osaka),
A. Tanaka (RIKEN AIP),
A. Tomiya (RIKEN BNL)Deep Learning“cat”

AdS/CFT[Maldacena ‘97]

CFTAdSBlack
hole

Cf. Talk by Yi-Zhuang You

1. Formulation ofAdS/DL correspondence

2. Deeply learning QCD

1. Formulation of
AdS/DL correspondence
1-1

Three motivations

review

AdS/CFT: quantum response from geometry

review

Deep learning: optimized sequential map

1-2

From AdS to DL1-3

Dictionary of AdS/DL correspondence

1-1(1) Brief history of quantum gravity

1974 Yoneya, Scherk-Schwarz: String = quantum gravity.
Yoneya, Prog.Theor.Phys. 51 (1974) 1907.
Scherk, Schwarz, Nucl.Phys. B81 (1974) 118.

1976 Hawking: Information loss problem of black holes.

Hawking, Phys.Rev.D14(1976)2460.

1997 Maldacena: Discovery of AdS/CFT.
A quantum gravity is nonperturbatively defined.
2002 Holographic QCD.

Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231.
Karch, Katz, JHEP 0206:043.
Kruczenski,Mateos,Myers,Winters JHEP 0405:041.
Sakai, Sugimoto, PTP 113 (2004) 843.

2008 Holographic superconductor.
Hartnoll, Herzog, Horowitz, PRL 101(2008)031601.

2009 Bulk reconstruction.

Heemskerk,Penedones,Polchinski,Sully, JHEP 0910:079.

1-1

Emergent geometry?

Emergence of AdS radial direction?
Bulk reconstruction and locality.
[Heemskerk, Penedones, Polchinski, Sully 09]

Entanglement entropy reconstruction.

[Balasubramanian, Chowdhury, Czech, de Boer, Heller 13][Myers, Rao, Sugishita 14]

Optimization of boundary path integral.

[Caputa,Kundu,Miyaji,Takayanagi,Watanabe 17]Renormalization and effective LG theory.
[Ki-Seok Kim, Chanyong Park 16]

AdS/MERA. [Swingle 12]

Emergence of smooth neural network space?
Statistical neural network.

[Amari et al.]1-1Discretized QG spacetime?

Quantum gravity, discretizedCausal dynamical
triangulation [Ambjorn, Loll 1998]
AdS/MERA[Swingle 2009]

HaPPY code

[Pastawski, Yoshida, Harlow, Preskill 2015]

1-1(2) Solving inverse problem

AdS/CFT(No proof, no derivation)
Classical gravity
in d+1 dim. spacetime||Quantum field theory
in d dim. spacetime(Strong coupling limit,large DoF limit)

Conventional
holographic modeling
Model
MetricPredictiongµ
Prediction

Comparison
Experiment
data

Experiment
data

1-1

(2) Solving inverse problem

Our deep learning
holographic modeling

Conventional
holographic modeling

Model

Model

Metricgµ
Prediction

MetricPredictiongµ
Prediction

Comparison
Experiment
data

Experiment
data

Experiment
data

Experiment
data

1-1

(3) Most fundamental principle?
Brane(Superstring theory)

Brain(Neuroscience)review

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

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

S=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, g

const.

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

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

Deep learning : optimized sequential map
Layer N

Layer 2

Layer 1(1)xi

(2)
xi =(1) (1)(Wij xj )

(N )

xi

(N )

F = fi xi

(1)

Wij

(x)

“Activation function” (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 function” Em ))))
dataF

1-2

From AdS to DL

Bulk EoM

2+ h( )

metricDiscretization, Hamilton form
( +( +)= ( )+)= ( )+V[ ]h( )=0log

f ( )g( )d 1

( )V ( ( ))
( )h( ) ( )Neural-Network representation=0==0=0

1-3Dictionary 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 condition=0=0

Interaction
V( )

(a)

Wij

Input data(1)

xi

Output data
F
Activation function(x)

1. Formulation ofAdS/DL correspondence
1-1

Solving inverse problem

review

Deep learning : optimized sequential map

review

AdS/CFT: quantum response from geometry

1-2

From AdS to DL

1-3

Dictionary of AdS/DL correspondence1. Formulation ofAdS/DL correspondence

2. Deeply learning QCD

2. Deeply learning QCD
2-1

Demonstration of holographic modeling

2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

2-1

Demonstration of holographic modeling
ModelMetricgµ
Q Qbar potential

Lattice QCD data:chiral condensate
VS quark mass

Prediction
DeepLearning[RBC-Bielefeld collaboration, 2008](Courtesy of W.Unger)

Experiment
data

Experiment
data[T.Ishikawa et al., 2008,
CPPACS + JLQCD collaboration]

2-2Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

Chiral condensate VS quark mass.
Pick up
β=3.33data
β=3.30 ó T=196[MeV][RBC-Bielefeld collaboration, 2008](Courtesy of W.Unger)

Positive data
Negative data

2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

Map it to asymptotic scalar configuration. [Klebanov, Witten][DaRold,Pomarol][Karch,Katz,Son,Stephanov] [Cherman,Cohen,Werbos]

Nc
=mq e
4+2 Nc

q̄q e

32Nc
mq
43

• Conformal dimension of q̄q is 3.
• Sub-leading contribution, present.
• Everything measured in unit of AdS radius.

e 3

2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

asymptoticAdS
input

horizonmqq̄q
input

Klebanov-Wittendecomposition( = 0)

Unspecified metric h( ),coupling(to be trained)2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.

input

mqq̄q
input

QCD lattice data( = 0)

2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.
h( )

log

f ( )g( )d 1

Learned value of (AdS radius)-1 : 1/L = 237(3)[MeV]bulk coupling : λ/L = 0.0127(6)2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.
Q Qbar potential

Learned metricf ( )g( )3Procedures
based on
[Maldacena][Rey,Theisen,Yee]2-2

Deeply learning QCD
1) Use a QCD data.
2) Let the network learn the metric.
3) Calculate other physical quantities.
Q Qbar potential[T.Ishikawa et al., 2008,
CPPACS + JLQCD collaboration][Petreczky, 2010]

2-1

Demonstration of holographic modeling
ModelMetricgµ
Q Qbar potential

Lattice QCD data:chiral condensate
VS quark mass

Prediction
DeepLearning

β=3.30 ó T=196[MeV][RBC-Bielefeld collaboration, 2008](Courtesy of W.Unger)

Experiment
data

Experiment
data[T.Ishikawa et al., 2008,
CPPACS + JLQCD collaboration]

1. Formulation ofAdS/DL correspondence

2. Deeply learning QCD