ipi seminar, univ. Tokyo, online, 10 Nov 2022

Deep Learning and Emergent Spacetime
Koji Hashimoto (Kyoto U.)

Based on:- “Deriving dilaton potential in improved holographic QCD from chiral condensate” 2209.04638“Deriving dilaton potential in improved holographic QCD from meson spectrum” 2108.08091
w/ K.Ohashi (Keio), T.Sumimoto (Osaka u)- “Neural ODE and Holographic QCD” 2006.00712 w/ H.Y.Hu, Y.Z.You (UCSD)- “Deep Learning and AdS/QCD” 2005.02636 w/ T. Akutagawa, T. Sumimoto (Osaka u)- “Deep Boltzmann Machine and AdS/CFT” 1903.04951- “Deep Learning and Holographic QCD” 1809.10536
w/ S. Sugishita (Kentucky), A. Tanaka, A. Tomiya (RIKEN)- “Deep Learning and AdS/CFT” 1802.08313 w/ S. Sugishita (Kentucky), A. Tanaka, A. Tomiya (RIKEN)

3

Similarity!?

Wormholes in Penrose diagramof maximally extended eternal
AdS Schwarzschild black hole
[Iizuka, Sugishita, KH `17]

Deep Autoencoder

5Roadmap

Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

1.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime?

Quantummechanics
in d-dim.

Anti de Sitter
spacetime2.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

3.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Roadmap

4.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Deep Learning and Quantum Gravity
1. Quantum gravity

4 pages

2. Neural network quantum states 6 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

7 pages

Discussion: Quantum gravity ⊂ ML ?

Roadmap

1.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network1. Quantum Gravity

1/4

Brief History of quantum gravity
1974 ‘tHooft, Veltman:
Perturbation fails in Einstein gravity.
1970 Nambu, Susskind, Nielsen:
String theory of hadrons.

1971 Bekenstein:
Black hole entropy.

1974 Yoneya, Scherk, Schwarz:
String is quantum gravity.

1993 ‘tHooft, Susskind:
Holographic principle.

1997 Maldacena:
AdS/CFT correspondence.
13

1. Quantum Gravity

2/4

AdS/CFT correspondence, no proof[Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231]

“AdS”

“CFT”
“Large N”
Quantum mechanics
in d-dim. spacetime＝

Classical
Quantum gravityin (d+1)-dim. spacetime

・ Vast amount of examples known
・ No proof!
・ How does it work?
・ Given Left, how can one get Right?
14

1. Quantum Gravity

3/4

Dictionary : equating partition functions[Gubser, Klebanov, Polyakov, Phys.Lett.B428(1998)105][Witten, Adv.Theor.Math.Phys. 2 (1998) 253]

Partition function of
Quantum mechanics

Z[ 0 ]

Z[ 0 ][Dq(t)]eR

dt(L[q,q̇]+ 0 (t)O[q])

e

t

t

R

dtdz

p

＝

＝

Z

＝

Partition function ofClassical gravity?

g(R[g]+L[ ]+··· )(t, z) Classical sol.

?

gµ⌫ (t, z)

source0 (t)00 (t) =(t, z = 0)z

1. Quantum Gravity

4/4

Quantum geometry is a network
Regge calculus[Regge 1961]

Causal dynamical
triangulation[Ambjorn, Loll 1998]

AdS/MERA(Tensor Network)[Swingle `09]

Quantum codes
for holography[Pastawski, Yoshida,
Harlow, Preskill `15]

Deep Learning and Quantum Gravity
1. Quantum gravity

4 pages

2. Neural network quantum states 6 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

7 pages

Discussion: Quantum gravity ⊂ ML ?

Roadmap

Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime?

Quantummechanics
in d-dim.

Anti de Sitter
spacetime2.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network2. Neural Network Quantum States 1/6
Various neural networks were invented

http://www.asimovinstitute.org/neural-network-zoo/2. Neural Network Quantum States 2/6
Machine learning = function approximator
Input: a vector (x1 , x2 , x3 , · · · )Output: a value f (x1 , x2 , x3 , · · · )<latexit sha1_base64="HwZg49KJQh0hpE2jTY+baB01xeg=">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</latexit><latexit sha1_base64="Me5UJD/BeRjwm06OfJziZ+a8yjo=">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</latexit>

Network architecture is the function ansatz

Perceptron model[Rosenblatt 1958][Rumelhart, McClelland 1986]

f = Wi(2)φ (Wij(1) xj)

Boltzmann machine[Ackley, Hinton, Sejnowski 1985]

2. Neural Network Quantum States 3/6
Neural network for classification
Perceptron model“Unit” (circle) : Vector component“Weight” (line) : Linear transformation
to be optimized

f = Wi(2)φ (Wij(1) xj)- Training protocol :“Activation function” (hidden line-end) :Nonlinear component-wise transf.
1
φ(x) ≡
1 + e −x1) Prepare many sets {(xj, f )} : (input, output)
2) Train the network (adjust W ) by lowering“Loss function” E ≡f − Wi(2)φ (Wij(1) xj)∑
data2. Neural Network Quantum States 4/6
Find ground state wave function ψ(s1, s2, ⋯, sN )Q : Minimize its energy E for a given Hamiltonian H,

A : Use ansatz and optimize parameters!
- Matrix product states
ψ(s1, s2, ⋯) = tr[A (s1) A (s2)⋯]- Tensor network states
ψ(s1, s2, ⋯) =∑m,n

Bmn Ams1s2 Ans3s4

2. Neural Network Quantum States 5/6
Neural network can be wave functions- Boltzmann machine states[Carleo, Troyer `17],[Nomura, Darmawan, Yamaji, Imada `17], ..

- Deep Boltzmann machine states[Carleo, Nomura, Imada `18], ..

- Feedfoward network states
ψ(s1, ⋯, sN ) =[Saito `18], ..

f σ
W s + bi∑ i
∑ ij jij2. Neural Network Quantum States 6/6
Better? and Why?
Neural states may beat conventional ones.
Ex) 2-dimensional
antiferromagnetic
Heisenberg model

Energy with
RBM states[Carleo, Troyer `17]
# of hidden units

Discovered intimate relations are there.
1) Boltzmann machine states are tensor network states[Chen, Cheng, Xie, Wang, Xiang `18]2) Tensor states are deep Boltzmann [Gao, Duan `17] [Huang, Moore `17]3) Tensor states are feedforward with “product pooling”[Cohen, Shashua `18]

Ex) Unified approach: MPO-Net [Gao, Cheng, He, Xie, Zhao, Lu, Xiang `19]

Deep Learning and Quantum Gravity
1. Quantum gravity

4 pages

2. Neural network quantum states 6 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

7 pages

Discussion: Quantum gravity ⊂ ML ?

Roadmap

3.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network3. When is NN a spacetime?
General NN is not a space
No notion of which unit is close to which

Perceptron model[Rosenblatt 1958][Rumelhart, McClelland 1986]

Boltzmann machine[Ackley, Hinton, Sejnowski 1985]

1/5

3. When is NN a spacetime?
Sparsity + weight sharing, for NN to be a space
No locality

Locality imposed＝
Convolutional
layer

Fully
connected

Parallellytranslated[Fukushima `80]

Input: (n x)
Output:<latexit sha1_base64="nwzfFCLRwUCM0jUUcWfpi10n7t8=">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</latexit>a (n x) + b@x (n x) + c@x2 (n x) + · · ·<latexit sha1_base64="OAM1XRpoTDK6qJPFgGfv9fB7zD0=">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</latexit>

2/5

3. When is NN a spacetime?
NN depth as time
Dynamical system
x·i = fi(x(t))

xi(tn+1) = xi(tn) + Δt ⋅ fi(x(tn))

tn+1 = tn + Δt

Discretized timeResNET (Residual network) : easily trained deep model[K.He et al.,1512.03385]

xi(n)

W

f

xi(n+1)xi(n+1) = f(Wij xj(n)) + xi(n)Skip connection

3/5

3. When is NN a spacetime?

4/5

Hamilton dynamics is a NN
·q = ∂H ,∂p

1802.08313·p = − ∂H∂qTime-dependent Hamiltonian = weights/activation
00v
0 1 + Δtw11
Δtw12
W=
0
Δtw21
1 + Δtw12
0
u
0

q

p
Wij

φi

W̃ij

Δtf(x)1
φi =1
Δtg(x)

0
λ1
W̃ =
0
0

0
1
0
0

0
0
1
0

0
0
0
0
0
0
λ2
0

λ1
λ2
1
1
2
2
H = w11 pq + w12 p − w21q + F(vp) − G(uq)22vu(F′ = f, G′ = g)

3. When is NN a spacetime?

5/5

Q. Find a Hamiltonian
Consider a particle motion x(t) in a given potential
V (x) in 1 dimension, with unknown time-dependent
friction force h(t)ẋ .
One tried many initial conditions (x(t = 0), ẋ(t = 0))
and collected those which stop at t = 10 .
Q. From given data of the initial conditions, find h(t) .
V (x)@V (x)
mẍ = h(t)ẋ +@xx

Deep Learning and Quantum Gravity
1. Quantum gravity

4 pages

2. Neural network quantum states 6 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

7 pages

Discussion: Quantum gravity ⊂ ML ?

Roadmap

4.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

AdS/CFT(No proof, no derivation)

Conventional modeling
Gravity model
ActionClassical gravity theory
in d+1 dim. spacetime||
Quantum field theory
in d dim. spacetime(Strong coupling limit,
large DoF limit)

MetricPredictiongµ
Prediction

Comparison
Exp/Lattice
data

Exp/Latticedata

Quantum field theory

Bulk reconstruction

Conventional modeling

Gravity model

Gravity model

ActionActionMetricgµ
Prediction

MetricPredictiongµ
Prediction

Comparison
Exp/Lattice
data

Exp/Latticedata

Quantum field theory

Exp/Lattice
data

Exp/Latticedata

Quantum field theory

4. Spacetime emergent from data 1/7
Gravity side
Classical scalar field theory in unknown 5-dim. spacetime
S=<latexit sha1_base64="TFyvaO4nUBjzbL0k2HMZOKHqRSU=">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</latexit>

Z

p
⇥
4
d⌘d x det g (@⌘ )2

V( )

ds2 =

1802.08313
1809.10536⇤

f ( )dt2 + d 2 + g( )(dx21 + · · · + dx2d 1 )
3 2
4
+
V[ ]=
L2
4

Data: ( 0 , Z[ 0 ])

t

0
Black
Hole

(⌘)
( |⌘=1 , @⌘ |⌘=1 , @⌘ |⌘=0 )

1

⌘

0

4. Spacetime emergent from data 2/7
Equation of motion as a feedforward NN
Eq. of motion2+ h( )

metricDiscretization
Hamilton form

V[ ]

( +
( +

=0

h( )

)= ( )+)= ( )+log

f ( )g( )d 1

h( ) ( )V ( ( ))
( )( )Feedfoward neural network for classification

=0

=

=0

=0

4. Spacetime emergent from data 3/7
Training with data of quark condensate
Data of quantum chromodynamics
simulated by supercomputer<latexit sha1_base64="l+XJxifsHha+KUEDRdBoVx0fszc=">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</latexit><latexit sha1_base64="Y1lb4mzByJPqNHPD1TVSTPNFncc=">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</latexit>

0 : Quark mass@
Z[ 0 ] : Quark condensate@ 0

4. Spacetime emergent from data 4/7
Spacetime metric emergent as NNh(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>

Trained values of potential :1/L = 237(3)[MeV] , λ/L = 0.0127(6)4. Spacetime emergent from data 5/7
Reconstructing gravity model dual to QCD
Gravity model
ActionMetric h(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>

Prediction

QCD

Chiralcondensate

Quarkpotential
44

4. Spacetime emergent from data 6/7
Deriving the dilaton potential
Gravity model
Action2209.04638

Metric h(⌘)

V( )<latexit sha1_base64="/TkiKyiMuAimf1zKvgHhRhSPEVI=">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</latexit><latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>h(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit><latexit sha1_base64="N6O5MPFx52853e29CytQLLZ93NA=">AAACZ3ichVHLSsNAFD2Nr1ofrQoiuFGL4qrc+kBxVXTjsj5aBS2SxLEdTJOQpAUt/oALtxVcKYiIn+HGH3DRTxCXFdy48CYNiBb1DjNz5sw9d87MaLYhXY+oEVE6Oru6e6K9sb7+gcF4Ymg471oVRxc53TIsZ1dTXWFIU+Q86Rli13aEWtYMsaMdr/n7O1XhuNIyt70TWxTKatGUR1JXPZ/az5bkQSJJKQpioh2kQ5BEGFkrcYd9HMKCjgrKEDDhMTagwuW2hzQINnMF1JhzGMlgX+AMMdZWOEtwhsrsMY9FXu2FrMlrv6YbqHU+xeDusHIC0/RM99SkJ3qgF/r4tVYtqOF7OeFZa2mFfRA/H9t6/1dV5tlD6Uv1p2cPR1gOvEr2bgeMfwu9pa+e1ptbK5vTtRm6oVf2f00NeuQbmNU3/XZDbF4hxh+Q/vnc7SA/l0rPpxY3FpKZ1fArohjHFGb5vZeQwTqyyPG5JVygjsvIixJXRpWxVqoSCTUj+BbK5Cd0/4si</latexit>

Chiralcondensate
45

4. Spacetime emergent from data 7/7
Prediction of QCD string breaking
Gravity model
ActionMetric h(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">AAACanichVG7SgNBFD1Z3/EVY6OkCUYlNmHWB4qVaGOZhzGCSthdJzq4L3YnAQ3+gJ2VoJWCiPgZNv6ART5BtFOwsfBmsyAq6h1m5syZe+6cmdFdU/iSsUZEaWvv6Ozq7on29vUPDMaG4uu+U/UMXjQc0/E2dM3nprB5UQpp8g3X45qlm7yk768090s17vnCsdfkgcu3LW3XFhVhaJKo0l56i0ttqhxLsQwLIvkTqCFIIYysE7vGFnbgwEAVFjhsSMImNPjUNqGCwSVuG3XiPEIi2Oc4QpS0VcrilKERu0/jLq02Q9amdbOmH6gNOsWk7pEyiQn2wG7YC7tnt+yRvf9aqx7UaHo5oFlvablbHjweKbz9q7Joltj7VP3pWaKChcCrIO9uwDRvYbT0tcPTl8JifqI+yS7ZE/m/YA12Rzewa6/GVY7nzxGlD1C/P/dPsD6dUWcyc7nZ1NJy+BXdSGAMaXrveSxhFVkUA3cnOMN55FmJK6NKopWqRELNML6EMv4BjlmMEg==</latexit>

Predictionh(⌘)<latexit sha1_base64="vTlG9iOIrhww+0CnFBDonAQYKbI=">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</latexit>

Chiralcondensate

Quarkpotential
46

Roadmap

4.
Quantum
gravityin (d+1)-dim.
‘tHooft `93
Susskind `94
Maldacena `97

Generalspacetime

Anti de Sitter
spacetime?

Quantummechanics
in d-dim.

Swingle `10

Carleo,Troyer `17

Neural network

Tensor network

Deep Learning and Quantum Gravity
1. Quantum gravity

4 pages

2. Neural network quantum states 6 pages
3. When is NN a spacetime?

5 pages

4. Spacetime emergent from data

7 pages

Discussion: Quantum gravity ⊂ ML ?

Discussion: Quantum gravity ⊂ ML ?
3 steps for quantum gravity
Quantum Mechanics
side

Gravity side
metric gµ⌫ field

Architecture

Large DoF limit
Large DoF expansion
Finite DoF

Classical
Classical

Classical
Quantum

Feedforward NN
Deep Boltzmann

Quantum

Quantum?

- Neural ODE : free from discretization- Quantum AdS/CFT ⊂ Deep Boltzmann machine- Which part of geometry is the neurons?

Discussion: Quantum gravity ⊂ ML ?
Neural ODE : free from discretization
2006.00712

d (⌘)= f ( (⌘), ⌘, h(⌘))
d⌘Emergent metric

Q Qbar potential
Neural ODE [R.T.Q.Chen, Y.Rubanova,
J.Bettencourt, D.Duvenaud 1806.07366]

Discussion: Quantum gravity ⊂ ML ?
Quantum AdS/CFT ⊂ Deep Boltzmann
AdS/CFT[Maldacena 1997]

Deep Boltzman machine[Salakhutdinov, Hinton 2009][KH `19] [You,Yang,Qi `18] (See also [Gan,Shu `17][Howard `18])

Discussion: Quantum gravity ⊂ ML ?
Physical picture of Deep Boltzmann
Ground state wave function for given Hamiltonian
is identified as a deep Boltzmann machine[Carleo, Nomura, Imada `18], ..

| ψ⟩ = lim e −τH | any⟩ = e −ΔτH e −ΔτH⋯ | any⟩τ→∞H<latexit sha1_base64="XagScydK2LYSdBnKEe0o0+PseEo=">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</latexit>

Euclidean time⌧<latexit sha1_base64="/eEqvXEnTni57AUUEIhrBngQ2w0=">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</latexit>(xi ) =

X(n)<latexit sha1_base64="AvgKTR8IQdExyctru1rwnMXbrXc=">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</latexit>hj

2{0,1}

2

exp 4

X
ij

(0)
wij xi hj

XX
n

ij

3

(n) (n) (n+1) 5

wij hi hj

Discussion: Quantum gravity ⊂ ML ?
AdS/CFT discretized the bulk, but fixed
AdS/MERA

Quantum codes for holography[Swingle `09]

[Pastawski, Yoshida, Harlow, Preskill `15]

Discussion: Quantum gravity ⊂ ML ?
Quantum spacetime? Regge vs Matrix

Regge calculus

Dynamical triangulation

Fixed lattice architecture,
variable lengths

Randomly generated
lattice architecture,
fixed lengths

Suits conventional NN

Novel “QG NN”[Regge `61]

[Ambjorn, Loll `98]

3. When is NN a spacetime?
Hamilton dynamics is a NN
·q = ∂H ,∂p·p = − ∂H∂qTrial NN representation:q(t + Δt) = φ1(W11q(t) + W12 p(t))

p(t + Δt) = φ2(W21q(t) + W22 p(t))Δt → 0Consistency requires

q· = w11q + w12 p + g1(q)p· = w21q + w22 p + g2(p)1802.08313

q

φ1

p

φ2

Wij = δij + Δtwij
φi(x) = x + Δtgi(x)Trivial linear Hamiltonian…

57

4. Spacetime emergent from data
Emergent metric can predict physics
Emergent metric
VolumefactorBump

Cf) Bottom-up modele cz /2
12
ds = 2(1 − z 4)dt 2 + dxi2 +
dz 2
4
)
z (
1−z
2

[Andreev, Zakharov, `06, `07]58

4. Spacetime emergent from data
Emergent metric can predict physics
Emergent metric

Q Qbar potential

Volumefactor

Bump

Procedures
based on
[Maldacena][Rey,Theisen,Yee][Petreczky, `10]59

3. Holography is a NN
Detailed mapping possible
Scalar field theory
in curved spacetime
Properdiscreti-zationEnergy function ofdeep Boltzmann60

3. Holography is a NN
Dictionary: Spacetime is a NN

61

5. Holographic spacetime is a NN
Algebraize PDE by Fourier transformation[Karch, Kaz, Son, Stephanov `06]

Classical gauge theory in 5-d dilaton gravity backgroundS = d xdz e−g (FMN)∫
Dilaton Φ(z) , metric ds 2 = e 2A(z) (dz 2 + ημνdx μdx ν)
4

−Φ

2

AdS boundary ( z ∼ 0 ) : B(z) ≡ Φ(z) − A(z) ∼ log z

Solve EoM for gauge field Aμ(z, x μ) = vn(z)ρμ(x μ)∂−B ∂evn + ω 2e −Bvn = 0
∂z (
∂z )

When frequency takes a proper discrete value ω 2 ∼ mn2,
gauge field is normalizable : vector meson spectra.
62

5. Holographic spacetime is a NN
Algebraize PDE by Fourier transformation
Bulk EoM ∂−B ∂evn + ω 2e −Bvn = 0
∂z (
∂z )
Discretization
Hamilton formvn(z + Δz) = vn(z) + Δz πn(z)πn(z + Δz) = πn(z) + Δz (B′(z)πn(z) − ω 2vn(z))

Neural-Network representationω2v(z = ∞) = 0

v
z=0

z

z=∞

63

5. Holographic spacetime is a NN
Training with QCD data: hadron spectra
PDG data for rho meson mass :mρ(1) = 0.77 GeV, mρ(2) = 1.45 GeV

Positive
NegativeB′(z) = Φ′(z) − A′(z)z<latexit sha1_base64="SJhDWOzNl6n2GkZuiMiTY+AO8D8=">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</latexit>64

5. Holographic spacetime is a NN
Training with QCD data: hadron spectra
PDG data for rho meson mass :mρ(1) = 0.77 GeV, mρ(2) = 1.45 GeV

PDG data for a2 meson mass :

(2)
ma(1)
=
1.32
GeV,
m
a2 = 1.70 GeV
2B′(z) = Φ′(z) − A′(z)B̃′(z) = Φ′(z) − 3A′(z)z<latexit sha1_base64="SJhDWOzNl6n2GkZuiMiTY+AO8D8=">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</latexit>z<latexit sha1_base64="SJhDWOzNl6n2GkZuiMiTY+AO8D8=">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</latexit>65

5. Holographic spacetime is a NN
Neural geometry emergent in the bulk
Metric- consistent with pure AdS

Dilaton- grows at IR,
like QCD running couplingexp[2A(z)](z)<latexit sha1_base64="3Z+uiwYkdbPeEwNXnvxKOQsOTys=">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</latexit><latexit sha1_base64="BybCGHCvw1nykS9JflNfjIaoVE0=">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</latexit>z<latexit sha1_base64="SJhDWOzNl6n2GkZuiMiTY+AO8D8=">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</latexit>z<latexit sha1_base64="SJhDWOzNl6n2GkZuiMiTY+AO8D8=">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</latexit>66

4. Spacetime emergent from data
Simplest holographic model
Classical scalar field probing 5-dim. curved spacetime
S=<latexit sha1_base64="TFyvaO4nUBjzbL0k2HMZOKHqRSU=">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</latexit>

Z

ds2 =

4

d⌘d x

p

⇥

det g (@⌘ )

2

V( )

⇤

1802.08313
1809.10536

f ( )dt2 + d 2 + g( )(dx21 + · · · + dx2d 1 )AdS boundary (Black hole horizon (): f g0): f

exp[2 /L]2

, gconst.

Solve eq. of motion to get response ⟨ψ̄ ψ⟩mq. [Klebanov, Witten `98]
AdS boundary () : ϕ = mq e −η + ⟨ψ̄ ψ⟩ e −3ηBlack hole horizon (

0):

=0

=0
67

1. Quantum Gravity
Brief History of quantum gravity
1970 Nambu, Susskind, Nielsen: String theory of hadrons.
1971 Bekenstein: Black hole entropy is area.
Bekenstein, Lettere al Nouvo Cimento 4(1972)737.

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

1974 ‘tHooft, Veltman: Perturbation fails in gravity.

‘tHooft, Verltman, Ann.Inst.Henri Poincare A20 (1974) 69.

1993 ‘tHooft, Susskind: Holographic principle.

‘tHooft, Conf.Proc.C930308 (1993)284.
Susskind, J.Math.Phys.36(1995)6377.

1997 Maldacena: Discovery of AdS/CFT correspondence.
Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231.

6
9

1-1

Is deep learning useful? 3 reasons.

1. Holographic QCD is an inverse problem,
and the deep learning is good at it.

A = f(B)

Normal problem:
System “f” given, calculate A for given B.
Inverse problem:
For given many data (A,B), find “f”.

2. Discrete gravity is a network optimization,and so is the deep learning.
3. Deep learning architecture resembles holography.
701-2 Holographic QCD is an inverse problem
Gravity Model
Metricgµ

Prediction

Prediction

Experiment
data

Experimentdata

1-2 Holographic QCD is an inverse problem
Gravity Model
Metricgµ
Q Qbar potential

Lattice QCD data:chiral condensate
VS quark mass

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

Experiment
data

Experiment
data[Petreczky, 2010]

1-3

Discrete gravity = Network optimization?

Regge calculus[Regge 1961]

Causal dynamical
triangulation[Ambjorn, Loll 1998]

AdS/MERA[Swingle `09]

Quantum codes
for holography[Pastawski, Yoshida,
Harlow, Preskill `15]

1-4

Machine learning = Optimizing networks

http://www.asimovinstitute.org/neural-network-zoo/

74

75

1-5 Machine learning = function approximator
Input: a vector (v1 ,v2 , v3, …)
Output: a value f (v1 , v2 , v3 ,…)Network architecture = Function ansatz

Perceptron model[Rosenblatt 1958][Rumelhart, McClelland 1986]

Boltzmann machine[Ackley, Hinton, Sejnowski 1985]

Universal approximation theorem :Any function can be approximated with more
hidden units [Cybenko 1989] [Roux, Bengio 2008]

76

1-6

Neural network for classification

Perceptron model“Unit” (circles) : Vector components“Weight” (lines) : Linear transformation
to be optimized

f = Wi(2)φ (Wij(1) xj)- Training protocol :“Activation function” (hidden line-end) :Nonlinear component-wise transf.
1
φ(x) ≡
1 + e −x1) Prepare many sets {(xj, f )} : input + output
2) Train the network (adjust W ) by lowering“Loss function” E ≡f − Wi(2)φ (Wij(1) xj)
∑
data

77

1-7

Holography as a classifier

AdS/CFT[Maldacena ‘97]

Deep neural network

CFT

AdS
(Emergent spacetime)

Black hole

1. Is deep learning useful?

7 pages

ArXiv:1903.04951

Deeply learning
2.
QCD chiral condensate

8 pages

ArXiv:1802.08313, 1809.10536 w/ S. Sugishita, A. Tanaka, A. Tomiya

Deeplylearning
3.
QCD hadron spectra

4 pages

ArXiv:1909.????? w/ T.Akutagawa, T. Sumimoto

2-1 AdS/CFT: quantum response from geometry[Klebanov, Witten]

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

dd+1 xds2 =det g (

)2

V( )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 conditions:
AdS boundary (
):
= Je

1

+
+

Black hole horizon (

0):

O e
=0

=0

+

2-2

Neural network of AdS scalar

Bulk EoM

2+ h( )

metricDiscretization, Hamilton form
( +

( +

)= ( )+)= ( )+

V[ ]
h( )

=0
log

f ( )g( )d 1

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

=0

=

=0

=0

2-3

Dictionary 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 J

Horizon condition=0=0Interaction
V( )

(a)

Wij

Input data(1)

xi

Output data
F
Activation function(x)Holographic QCD is an inverse problem
Gravity Model
Metricgµ
Q Qbar potential

Lattice QCD data:chiral condensate
VS quark mass

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

Experiment
data

Experiment
data[Petreczky, 2010]

2-4

QCD data as a classification

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

84

2-5

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

Nc
=
mq e
4

+

2 Nc

q̄q e

3

2

Nc
mq
4

3

e 3

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

q̄q
input

Klebanov-Witten
decomposition( = 0)

Unspecified metric h( ),coupling(to be trained) 85

2-5

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

Nc
=
mq e
4

+

2 Nc

q̄q e

3

2

Nc
mq
4

3

e 3

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

q̄q

QCD lattice data86

2-6 Machine learns QCD data, metric emerges

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

87

2-7

Machine learned a bottom-up model
Learned metric
Volumefactor

Bump
Quantum gravity effect?
Coexistence ofCf) Bottom-up modele cz /2
12
ds = 2(1 − z 4)dt 2 + dxi2 +
dz 2
4
(
)
z
1−z
2

[Andreev, Zakharov, `06, `07]

Cf [Hyakutake `14]

Confinement
Deconfinement

2-8Machine predicts a Q Qbar potential
Q Qbar potential

Learned metric
VolumefactorBump

Coexistence ofProcedures
based on
[Maldacena][Rey,Theisen,Yee]

Confinement à Linear potential
Deconfinement à Debye screening

2-8

Machine predicts a Q Qbar potential
Q Qbar potential[T.Ishikawa et al., `08,
CPPACS + JLQCD collaboration]

Coexistence of[Petreczky, `10]

Confinement à Linear potential
Deconfinement à Debye screening

1. Is deep learning useful?

7 pages

ArXiv:1903.04951

Deeply learning
2.
QCD chiral condensate

8 pages

ArXiv:1802.08313, 1809.10536 w/ S. Sugishita, A. Tanaka, A. Tomiya

Deeplylearning
3.
QCD hadron spectra

4 pages

ArXiv:1909.????? w/ T.Akutagawa, T. Sumimoto

3-1

Hadron spectra from geometry[Karch, Kaz, Son, Stephanov `06]

Classical gauge theory in 5-d dilaton gravity backgroundS = d xdz e−g (FMN)∫
Dilaton Φ(z) , metric ds 2 = e 2A(z) (dz 2 + ημνdx μdx ν)
4

−Φ

2

AdS boundary ( z ∼ 0 ) : B(z) ≡ Φ(z) − A(z) ∼ log z

Solve EoM for gauge field Aμ(z, x μ) = vn(z)ρμ(x μ)∂−B ∂evn + ω 2e −Bvn = 0
∂z (
∂z )

When frequency takes a proper discrete value ω 2 ∼ mn2,
gauge field is normalizable : vector meson spectra.

3-2

Neural network of AdS scalar

Bulk EoM∂−B ∂evn + ω 2e −Bvn = 0
∂z (
∂z )

Discretization, Hamilton formvn(z + Δz) = vn(z) + Δz πn(z)πn(z + Δz) = πn(z) + Δz (B′(z)πn(z) − ω 2vn(z))

Neural-Network representationω2v(z = ∞) = 0

v
z=0

z

z=∞

3-3

QCD data as a classification

PDG data for rho meson mass :mρ(1) = 0.77 GeV, mρ(2) = 1.45 GeV

Positive
NegativeB′(z) = Φ′(z) − A′(z)

z
94

3-3

QCD data as a classification

PDG data for rho meson mass :mρ(1) = 0.77 GeV, mρ(2) = 1.45 GeV

PDG data for a2 meson mass :

(2)
ma(1)
=
1.32
GeV,
m
= 1.70 GeVa2
2

B̃′(z) = Φ′(z) − 3A′(z)B′(z) = Φ′(z) − A′(z)

z

z
95

3-4

Emergent spacetime is confining

Pure AdS

Bump

Confining geometry emerged…… consistent with Wilson loop96

1. Is deep learning useful?

7 pages

ArXiv:1903.04951

Deeply learning
2.
QCD chiral condensate

8 pages

ArXiv:1802.08313, 1809.10536 w/ S. Sugishita, A. Tanaka, A. Tomiya

Deeplylearning
3.
QCD hadron spectra

4 pages

ArXiv:1909.????? w/ T.Akutagawa, T. Sumimoto

A

What do we learn?
Strategy for a universal modeling?
- Consistency with more observables:
1-pt, 2-pt, 3-pt, … and spacetime dependence?
- More hadron spectroscopy.
- Rather, observable-oriented modeling?

Consistency with top-down approach?
- Supergravity as a regularization for learning

Quantum gravity path-integral?
98A-1 Neural network for probability distribution
Boltzmann machine“Unit” (circles) : Spins“Visible units” (yellow)“Hidden units” (green)“Weight” (lines) : Spin-spin coupling
to be optimized- Training protocol :1) Prepare many sets {(vi, Pex(vi))} : input + output
2) Train the network (adjust W ) by lowering“Loss function” E ≡ DKL (Pex(vi) | | P(vi))
99

A-2

QFT on AdS as a Boltzmann machine[KH `19] [You,Yang,Qi `18] (See also [Gan,Shu `17][Howard `18])

AdS/CFT[Maldacena 1997]

Deep Boltzmann machine[Salakhutdinov, Hinton 2009]

101

Summary: Neural network as a spacetime
Deep neural network

CFT

AdS
(Emergent spacetime)

Black hole

A

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.]
103

1-3

Quantum gravity = Network optimization

Regge calculus[Regge 1961]

Causal dynamical
triangulation[Ambjorn, Loll 1998]

AdS/MERA[Swingle `09]

Quantum codes
for holography[Pastawski, Yoshida,
Harlow, Preskill `15]

2-2

Deep learning : optimized sequential map
Layer 2

Layer 1
(1)
xi

(2)
xi =

Layer N(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 {x(1)i , F } : input + output
2) Train the network (adjust Wij ) by lowering
(N 1)
(1)
“Loss function” E
fi ( (Wij(· · · (Wlm x(1)m ))))
data

F

106

Is Spacetime a Neural Network?

107

Spacetime is a Neural Network
1. Space is a NN
2. Time is a NN

1802.08313

3. Holography is a NN

1903.04951

4. Holographic space is a NN

1802.08313
1809.10536

5. Holographic spacetime is a NN

2001.?????

6. Quantum spacetime is a NN?