Deep Learning
and AdS/CFT
Sotaro Sugishita(Osaka U. → U. of Kentucky)
with Koji Hashimoto (Osaka U.),
Akinori Tanaka (RIKEN),
Akio Tomiya (CCNU→RIKEN)
Based on arXiv:1802.08313 (PRD 98, 046019)& 1809.?????
Discrete Approaches to the Dynamics of Fields and Space-Time 2018
Tohoku univ., Sep. 10, 2018

Title was changed by PRD.

l

Deep Learning
and AdS/CFTthecorrespondenceOutline Introduction

 Deep learning
 AdS/CFT Deep learning and AdS/CFT Summary

What is deep learning?
Deep learning is part of machine learning.

Input

OutputMachineあWhat is deep learning?
Deep learning is part of machine learning.

OutputInput
MachineあIn the machine,: non-linear transformation w/ tunable parameters••“Learning” is tuning of the parameters.
“Deep” means large 𝑁.

Related to coarse-graining?

あ Output captures some essential characteristics of input.

Related to RG?
It reminds us of the renormalizaton group.

Related to RG?
It reminds us of the renormalizaton group.
Or theoretical physicists are tuned so.

Related to RG?
It reminds us of the renormalizaton group.
Or theoretical physicists are tuned so.
There are papers discussing the relation between
machine learning and RG.
[Beny (2013), Mehta & Schwab (2014),
Koch-Janusz & Ringel (2017), Iso, Shiba & Yokoo (2018), …]

Inspired by
Dr. Amari’s talk

AdS/CFT and RG
AdS space

AdS boundary

scale transformation =-translation

AdS/CFT and RG
AdS space

AdS boundary
UV

scale transformation =IR-translation

holographic RG

AdS space as neural network
Can we regard the evolution in the bulk direction asthe propagation in a deep neural network?

AdS boundary

Outline Introduction Deep learning
 AdS/CFT Deep learning and AdS/CFT SummaryMachine Learning•classification

Hey, is this a bear?

ThebearmachineYes.

•regression

dataLearnNeural network
input

outputMachineLayer 0Layer 1

Layer 2

Layer 3

Neural network
input

outputLayer 0Layer 1

Layer 2

Layer 3

Neural network
input

outputLayer 0Layer 1#(Layers) is large

Layer 2

Layer 3

Deep LearningNeural network
 Proceed to next layer by linear trsf and non-linear trsf.

Neural network
 Proceed to next layer by linear trsf and non-linear trsf.

•parameters of linear transformation is tunable.

: weight,: biasupdated in thelearning process

Neural network
 Proceed to next layer by linear trsf and non-linear trsf.

•

parameters of linear transformation is tunable.

: weight,•: biasupdated in thelearning process

non-linear transformation
activation function
is not changed.
e.g., sigmoid function

Supervised learning Teach answers to the machine.
This is a bear.

ThebearmachineThis is not a bear.

ThebearmachineSupervised learning Prepare a data setyes,…

…
 Feed input data
The
bearmachine

no , …

to the machine.

Supervised learning Prepare a data set
yes,…

…
 Feed input data
The
bearmachine

no , …

to the machine.

Compare andadjust parameters

Supervised learning Adjust parameters
so that a loss function(error function, cost function) decreases.

optimize it.
••
•
•

•

mean squared error
L1 normcross entropy…

gradient descentRegularizatione.g.
•

to avoid the overfitting•

to get the expected properties

Unsupervised learning
For example, [Google Brain (2012)]
10 million images

Google’s AIThe concept of cats

The concept of faces

Machine learning for physics• Quantum many body system[Carleo & Troyer (2016), Fujita, Nakagawa, Sugiura &Oshikawa (2017),…]• Detection of phase transition in statistical systems[Wang (2016), Tanaka & Tomiya (2016), Ohtsuki^2 (2016),…]• String landscape[He (2017), Ruehle (2017), Carifio, Halverson, Krioukov &Nelson (2017),…]•LHC, Monte Calro, cosmology, neutron star, …,AdS/CFT

Outline Introduction

 Deep learning
 AdS/CFT Deep learning and AdS/CFT SummaryAdS/CFT correspondence[Maldacena (1997)]

Quantum Gravity on AdS

CFT on boundary of AdS

bulk

Many works about this conjecture.

boundary

Emergence of geometry AdS/CFT states that the bulk direction emerges from thedynamics of boundary QFT.

bulk

boundary Holography
Gravity can be defined in terms of low-dimension theories.
A definition of quantum gravity

AdS

CFT

Holographic approach
AdS

CFT

In some parameter regions, a QFT can be described bya classical Einstein gravity with matter fields.

We can analyze the QFT with strong couplings byusing classical gravity.
hardeasy•

holographic QCD,…

Note
 CFTs which have gravity dual descriptions are very specific.
e.g. 4d 𝒩 = 4 SYM

string theory on AdS5×S5

In order to have a “good gravity description”, CFT should[Heemskerk, Penedones, Polchinski, Sully (2009),
satisfy some criteria:
El-Showk & Papadodimas (2011)]
e.g.
• large DOF (large 𝑁, large 𝑐,…)• gapped spectrum (strong coupling)
We assume QFTS we will consider have semiclassical gravity
descriptions.

GKP-Witten[Gubser, Klebanov, Polyakov (1998), Witten (1998)]

 Correlation functions in CFT can be computed from theclassical solution of the dual field in gravitational theory.
boundary condition is fixed by

Dictionary in AdS/CFTscalar operator

in CFT𝑑 with dim(2-pt function in the g.s..
scalar fieldwith mass
on 𝑑 + 1 -AdS)

Free scalar on AdS

AdS boundary•

Asymptotic behavior of the classical solution near

1-pt func w/
source in CFT[Klebanov & Witten (1999)]

Asymptotically AdS
 Background does not need to be the exact AdS space.

Asympto AdS is OK.

aroundIt corresponds to an excited state in CFT.
•AdS Schwarzschildhorizonfinite temperatureOutline Introduction

 Deep learning
 AdS/CFT Deep learning and AdS/CFT SummaryHolographic modeling Two approaches••

Top-down
Bottom-up

embed into string theory
prepare a phenomenological model conventional holographic QCD models in the bottom up
approach• Hard wall, Soft wall,…
[Erlich, Katz, Son & Stephanov (2005), Da Rold & Pomarol (2005),…]

Guessmodel(ansatz of geometry)

results

DL holographic modeling Conventional approach

Guessmodel(ansatz of geometry)

results

DL holographic modeling Our approach[Hashimoto, SS, Tanaka, Tomiya (1802.08313)]

Data

model
(ansatz of geometry)
Deep Learning

results

Setup
We consider general backgrounds (not sol. of Einstein eq.).
•bulk scalar dual to boundary op.

general metric with horizon
bdry
data

Learn the metric
AdS bdry

bdry
data

horizonsmooth b.c.

Learn the metric
AdS bdryhorizonbdry
datasmooth b.c.

EOM in given background
input

outputLearn the metric
AdS bdryhorizonbdry
datasmooth b.c.

compareEOM in given backgroundinput

outputpropagation in neural network

Learn the metric
AdS bdryhorizon

bdrydatasmooth b.c.

compareEOM in given background
input

outputpropagation in neural network•

In the appropriate metric, output should satisfy thecorrect boundary condition.

background metric = parameters to be updatedNeural network rep. of EOM•

Suppose the field is homogeneous

to be learned

Neural network rep. of EOM•

Suppose the field is homogeneous

to be learned•

Discretize・・・

The metric function

plays the role of weights!

Boundary cond. at the horizon
Near the horizon

・・・

Boundary cond. at the horizon
Near the horizon

Suppose the field is finite.

・・・

Boundary cond. at the horizon
Near the horizon

Suppose the field is finite.

・・・Data for learning• Positive data
correct response• Negative data
wrong response

If the metric function is tuned appropriately,for positive data
for negative data

Test learning
Can the machine reproduce the black hole metric?
•4d AdS Schwarzschild•

Generate positive and negative data randomly

•

Learn

from the data.

?

Test learning
loss function

bdryhorizonloss function < 0.0002

Test learning Introduce a regularization

Smooth function is preferred.

bdryhorizon

Deep learning and holographic QCD[Hashimoto, SS, Tanaka, Tomiya (in preparation)] Use the result of lattice QCD.
• quark mass vs chiral condensate[Unger (2010)]

 Learn the metric, and compute other quantities

Data for learning・・・

Learned metriclayer

is also learned.

In this approach,
only a combination of metric compos. can be obtained.

ഥ potential
Holographic QQ
Ifandare obtained, we can compute quark
quark-antiquarkpotential holographically.
Evaluate a classical string action. [Maldacena (1998), Rey & Yee (1998)]

horizonഥ potential
Holographic QQ
Ifandare obtained, we can compute quark
quark-antiquark
potential holographically.
Evaluate a classical string action. [Maldacena (1998), Rey & Yee (1998)]

horizonഥ potential
Compute QQ
has been obtained.
can be determined if•

Ansatz(pure AdS)

is fixed.

ഥ potential
Compute QQ
has been obtained.
can be determined if•

is fixed.

Ansatz(pure AdS)

(another ansatz givesa similar shape)ഥ potential
Obtained QQ
string breaking

similar?
Coulomb + linear

lattice result
Fig from [Petreczky (2010)]

The first holographic model which shows both of
confinement and string breaking (as far as we know)

bump (effective IR wall)
confinement

horizonstring breaking

a regionbumpOutline Introduction

 Deep learning
 AdS/CFT Deep learning and AdS/CFT SummarySummary Presented a deep neural network representation of thepropagation to the bulk direction.

•metric is weight and updated in the process oflearning. Spacetime is a neural network?

 Applied it to a holographic QCD model.
••ഥ potential.
obtained a QQ
there are many points which should be improved.

 Machine learning techniques may be also useful in otherregions in physics.