0-1 Solving inverse problem Model Metric g Q Qbar potencal - - PowerPoint PPT Presentation

Deep Learning and Holographic QCD

Koji Hashimoto (Osaka u)

ArXiv:1802.08313, 1809.10536 w/ S. Sugishita (Kentucky), A. Tanaka (RIKEN), A. Tomiya (RIKEN)

“Machine learning landscape” workshop, ICTP Trieste, 10 Dec 2018 East Asian string workshop, KIAS, Seoul, 7 Nov 2018 KMI colloquium, Nagoya U, 25 Oct 2018 “QG meets la\ce QCD workshop”, ECT*, Trento, 3 Sep 2018 APCTP focus program, Hanyang U, Seoul, 15 Aug 2018 QFT workshop, YITP, Kyoto, 31 July 2018

Solving inverse problem

0-1

Metric Model

gµν

Prediccon Experiment data Experiment data Deep Learning

[RBC-Bielefeld collaboracon, 2008] (Courtesy of W.Unger)

La\ce QCD data: chiral condensate VS quark mass Q Qbar potencal

[Petreczky, 2010]

Discre1zed QG space1me?

Quantum gravity, discreczed Causal dynamical triangulacon AdS/MERA HaPPY code

[Swingle 2009] [Pastawski, Yoshida, Harlow, Preskill 2015] [Ambjorn, Loll 1998]

0-2

Black hole CFT AdS (Emergent spacecme)

AdS/CFT

[Maldacena ‘97]

Neural network as a space1me

0-3

Deep neural network

• 1. Formulacon of

AdS/DL correspondence

• 2. Deeply learning QCD

AdS/CFT: quantum response from geometry

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

S =

• dd+1x
• − det g
• (∂ηφ)2 − V (φ)
• f ∼ η2, g ∼ const.

f ∼ g ∼ exp[2η/L] AdS boundary ( ) :

η ∼ ∞

Black hole horizon ( ) :

η ∼ 0 ds2 = −f(η)dt2 + dη2 + g(η)(dx2

1 + · · · + dx2 d−1)

Solve EoM, get response . Boundary condicons: ∂ηφ

• η=0= 0

AdS boundary ( ) :

η ∼ ∞

Black hole horizon ( ) :

η ∼ 0 φ = Je−∆−η + 1 ∆+ ∆− Oe−∆+η OJ

1-1

[Klebanov, Wioen]

Deep learning : op1mized sequen1al map

F = fix(N)

i

Layer 1 Layer 2 Layer N

“Weights” (variable linear map)

ϕ(x)

“Accvacon funccon” (fixed nonlinear fn.) 1) Prepare many sets : input + output 2) Train the network (adjust ) by lowering “Loss funccon”

{x(1)

i , F}

Wij W (1)

ij

x(1)

i

x(2)

i = ϕ(W (1) ij x(1) j )

x(N)

i

1-2

E ≡

• data
• fi(ϕ(W (N−1)

ij

ϕ(· · · ϕ(W (1)

lm x(1) m )))) − F

Neural network of AdS scalar

Discreczacon, Hamilton form

π(η + ∆η) = π(η) + ∆η

• h(η)π(η) − δV (φ(η))

δφ(η)

• φ(η + ∆η) = φ(η) + ∆η π(η)

Neural-Network representacon Bulk EoM

∂2

ηφ + h(η)∂ηφ − δV [φ]

δφ = 0

h(η) ≡ ∂η

• log
• f(η)g(η)d−1
• metric

1-3

φ π

η

η = 0

η = ∞

π

• η=0= 0

Dic1onary of AdS/DL correspondence AdS/CFT Deep learning

Emergent space Depth of layers Bulk gravity metric Network weights Nonlinear response Input data Horizon condicon Output data Interaccon Accvacon funccon

OJ

∂ηφ

• η=0= 0

h(η)

W (a)

ij

1-4

x(1)

i

F

ϕ(x)

V (φ)

∞ > η ≥ 0

i = 1, 2, · · · , N

• 1. Formulacon of

AdS/DL correspondence

• 2. Deeply learning QCD

Solving inverse problem

0-3

Metric Model

gµν

Prediccon Experiment data Experiment data Deep Learning

[RBC-Bielefeld collaboracon, 2008] (Courtesy of W.Unger)

La\ce QCD data: chiral condensate VS quark mass Q Qbar potencal

[Petreczky, 2010]

• Deeply learning QCD

2-1

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

• Deeply learning QCD

2-1

1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Chiral condensate VS quark mass.

Pick up β=3.33 data

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

Posicve data Negacve data

• Deeply learning QCD

2-1

1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Map it to asymptocc scalar configuracon. [Klebanov, Wioen]

[DaRold,Pomarol][Karch,Katz,Son,Stephanov] [Cherman,Cohen,Werbos]

φ = Nc 4π mqe−η + π 2Nc ¯ qqe−3η λ 2 Nc 4π mq 3 ηe−3η

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

¯ qq

• Deeply learning QCD

2-2

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

π(η = 0)

Unspecified metric , coupling (to be trained)

φinput

πinput Klebanov-Wioen decomposicon

mq

¯ qq

h(η)

λ

horizon asymptocc AdS

• Deeply learning QCD

2-2

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

π(η = 0)

φinput

πinput

mq

¯ qq

QCD la\ce data

• Deeply learning QCD

2-2

1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Learned value of (AdS radius)-1 : 1/L = 237(3)[MeV] bulk coupling : λ/L = 0.0127(6)

h(η) ≡ ∂η

• log
• f(η)g(η)d−1

Deeply learning QCD

2-3

1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Q Qbar potencal Learned metric

Procedures based on [Maldacena] [Rey,Theisen,Yee]

f(η)g(η)3

η

Bump Quantum gravity effect?Cf [Hyakutake 2014]

Deeply learning QCD

2-3

1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Q Qbar potencal

[T.Ishikawa et al., 2008, CPPACS + JLQCD collaboracon] [Petreczky, 2010]