Machine Learning for QCD Matter:from Inverse Problems to Generative Models
Lingxiao Wang(王凌霄) (RIKEN)
May 18, 2024
The 1st edition of Spicy Gluons Workshop for Young Scientists
Prog.Part.Nucl.Phys. 104084(2023);
Phys. Rev. D 103, 116023, Phys. Rev. C 106, L051901, Phys. Rev. D 107, 083028, Phys. Rev. D 106, L051502;
Chin. Phys. Lett. 39, 120502, Phys. Rev. D 107, 056001, JHEP05(2024)060.

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Facilitators:
Lingxiao Wang (RIKEN iTHEMS) *Contact at lingxiao.wang@riken.jp
Catherine Beauchemin (RIKEN iTHEMS)
Akinori Tanaka (RIKEN AIP/RIKEN iTHEMS)
Enrico Rinaldi (Quantinuum K.K./RIKEN iTHEMS)
Akira Harada (NITIC/RIKEN iTHEMS)

https://sites.google.com/view/deep-in-wg/homepage

L. Wang2

Spicy Gluons 2024

Outline• Machie Learning for QCD Matter• Inverse Problems• Data-Driven Learning• Physics-Driven Learning• Generative Models• Probabilistic Models• Di usion Models

ff• Outlooks

Generated by ChatGPT-4 + DALL·E

Why Machine Learning?

…
…

Tempetrature
Tempetrature

Prog.Part.Nucl.Phys. 104084(2023) (Invited Review)

Quark-Gluon Plasma

Lattice
LatticeQCD
QCD
Property
PropertyExtraction,
Extraction,
Computational
ComputationalEffciency
Effciency

Heavy-Ion
Heavy-IonCollisions
Collisions
Data
DataAnalysis,
Analysis,
SignalSignalDetection,
Detection,
Fast
FastSimulations
Simulations

HadronicMatter
MatterHadronic

Neutron
NeutronStars
Stars
Data
DataAnalysis,
Analysis,
EoS
EoS Reconstruction
Reconstruction

Nuclei
Nuclei

VacuumVacuumBaryon Chemical
Chemical Potential
BaryonPotential• Heavy-Ion Collisions : Large number of data! Complicated simulations!
• Neutron Star : Accumulating observations! Poor signal-noise ratio!
• Lattice QCD : Computationally consuming! Physics extraction!
L. Wang4

Spicy Gluons 2024

What is ML?
Big Data + Deep Models
GPU

Successful Deep Learning!

Machine Learning (ML) is a subset of arti cial intelligence that involves thecreation of algorithms that allow computers to learn from and makedecisions or predictions based on data. It's essentially a way for computers to"learn" from data without being explicitly programmed to do so.
— ChatGPT-4

L. Wangfi

5DALL-E 3

Spicy Gluons 2024

Machine Learning and Physics
An inverse problem in science isthe process of inferring from a setof observations the causal factorsthat produced them.

Galileo Galilei (1564-1642)

Phys.Rev. Lett. 124, 010508 (2020)

Prediction
Estimation
Data, X

L. Wang

Machine, {θ}6Spicy Gluons 2024

Inverse Problems
Heavy-Ion Collisions

Phys. Rev. C 106, L051901; Phys. Rev. D 103, 116023

QCD matter properties

Physics

M-R

EoS

Neutron Star

Phys. Rev. D 107, 083028; JCAP08 (2022) 071(b) TOV Solver

32 x 2"!
!!

32 x 32
32 x 132 x 132 x 64

32 x 64

32 x 32

32 x 32

32 x 132 x 64("! , $! )(a) NN EoS
∇" ℒ

Data

Spectral Function

Correlation Function

Lattice QCD

-

#!"#

ℒ=

2

(""#$,! , $"#$,! )&!

(∆"! , ∆$! )Phys. Rev. D 106, L051502; Comput. Phys. Commun. 282, 108547 (2023);" !!

$(&, !! )
…

…

D(&)
"!

×

) Δ!

=

…

!

…
!
(a)

" !

(b)
2
∇# ℒ

…

""

=ℒ

-

)

…

!

D(&! )…

7…

…

L. Wang

Spicy Gluons 2024

Machine Learning and Inference

Maximum Likelihood Estimation(MLE)

Bayesian
Inference

max
θ

N

∏
i=1

p(xi ∣ θ)

Maximum A Posterior(MAP)p(X ∣ θ)π(θ)
p(θ ∣ X) =
p(X)

{θ}

Posterior p(θ ∣ X), Prior π(θ) , Evidence p(X)

L. Wang8

Spicy Gluons 2024

Data-Driven Learningfθ : X → Y

Forward process

Physics

Data

Model Parameters/
Properties/States

Observations

Inverse Mapping, fθ
L. Wang9

Spicy Gluons 2024

fi

Data-Driven Learningfθ : X → Y
Universal Approximation Theorem (1989,1991)
A feed-forward network with a single hidden layer containing anite number of neurons can approximate arbitrary continuousfunctions.

L. Wang10

Spicy Gluons 2024

Data-Driven Learning
Determine Impact Parameter

QMC data
1 hidden layer
with 20 neurons
Input 5X5

S. A. Bass, A. Bischo , J. A. Maruhn, H. Stöcker, and W. Greiner, Phys. Rev. C 53, 2358 (1996)

L. Wangff

11

Spicy Gluons 2024

Data-Driven Learning
Determine Impact Parameter
Recognizing QCD
Phase Transitions
Hydro dataρ(pT , Φ)

CNNs+DNNs
Cross-Overor
1st order PT

Input 15X48

S. A. Bass, A. Bischo , J. A. Maruhn, H. Stöcker, and W. Greiner, Phys. Rev. C 53, 2358 (1996)
L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker, and X.-N. Wang, Nature Commun. 9, 210 (2018)

L. Wangff

12

Spicy Gluons 2024

Data-Driven Learning
Determine Impact Parameter
Recognizing QCD
Phase Transitions

Point Net for Jet Tomography

ρ(pT , Φ)

Cross-Overor
1st order PT

S. A. Bass, A. Bischo , J. A. Maruhn, H. Stöcker, and W. Greiner, Phys. Rev. C 53, 2358 (1996)
L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker, and X.-N. Wang, Nature Commun. 9, 210 (2018)

Z. Yang, Y. He, W. Chen, W.-Y. Ke, L.-G. Pang, and X.-N. Wang, Eur. Phys. J. C 83, 652 (2023).

L. Wangff

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Our Current Works
with Lijia and Kai

Phys. Rev. D 103, 116023

Phys. Rev. C 106, L051901(Letter)with Yuan-Sheng, Xu-Guang and Kai
p.b.c.

Configurations
40×30 Input

Pion Spectra
20×24 Inputlll

32 Features

3×3

ReLu

BN
AvgPool

3×3
64 Features, 10×12 convBN
AvgPool

32 Features
3×3

64 Features, 5×6 conv
3×3

Flatten

ReLu

•
•
•
Pooling

Activation

Neuron•••Conv. kernel

SoftMax01&‘0’: Cross-over‘1’: 1st order PTlll

Flatten

Fully connected

LegendDamping coefficientBN
AvgPool

Leaky ReLu
Leaky ReLu

Dense 100, BN
Dropout 0.5

Fully connected

LegendLeaky ReLu

l
l
l

ReLu

Dense 128, BN
Dropout

3×3

l
l
l

BN
AvgPool

•
•
•

3×3

l
l
l

64 Features, 20×24 conv

Conv. kernel

Pooling

Activation

Neuron•••32 Features, Dropoutlll

SoftMax01‘0’: no CS‘1’: CS

Learning phase transition orders, CME signals
from nal-stage distributionsfi

L. Wang

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Spicy Gluons 2024

Our Current Worksin Preparation

with Jiaxing Zhao, Liang Zhang

Correlations?
Potentials

Ra aele Del Grande | XQCD 2023

Whether this inverse mapping exists?
ff

L. Wang

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Spicy Gluons 2024

Our Current Works
V(r) = b1e−b2r 2+ b3(1 − e−b4r 2)(e(−mπr)

rin Preparation)nπwith Jiaxing Zhao, Liang Zhang

Schrödinger eq.
CATS Framework: D. Mihaylov et al.,Eur. Phys. J. C78 (2018) 394

2

− r2

S(r) = (4πr02)−3/2e 4r0

60 k-points, 10000 correlations

r0 = 1.3 fm
L. Wang16

Spicy Gluons 2024

Our Current Worksin Preparation

with Jiaxing Zhao, Liang Zhang

60 points

b1, b3

L. Wang

R-squaredb1

b3

Training

0.96

0.99

Testing

0.96

0.99

Neural networks for Femtoscopy
17

b2 = 73.9, b4 = 0.78, nπ = 2
Spicy Gluons 2024

Physics-Driven Learninĝθ = arg max {p(X ∣ θ)}θForward process

Physics

Data

Model Parameters/
Properties/States

Observations

Inference
L. Wang18

Spicy Gluons 2024

Physics-Driven Learning
Inferring Dynamical Information

J. E. Bernhard, etc.(Duke Group), arXiv:1804.06469 and Nat Phy 15, 1113 (2019).

L. Wang19

Spicy Gluons 2024

Physics-Driven Learning
Inferring Dynamical Information
Determing Nulcear Struture

J. E. Bernhard, etc.(Duke Group), arXiv:1804.06469 and Nat Phy 15, 1113 (2019).
Y.-L. Cheng, S. Shi, Y.-G. Ma, H. Stöcker, and K. Zhou, Phys. Rev. C 107, 064909 (2023)

L. Wang20

Spicy Gluons 2024

Physics-Driven Learning
Inferring Dynamical Information
Determing Nulcear Struture
Building Nuclear Matter EoS

J. E. Bernhard, etc.(Duke Group), arXiv:1804.06469 and Nat Phy 15, 1113 (2019).
Y.-L. Cheng, S. Shi, Y.-G. Ma, H. Stöcker, and K. Zhou, Phys. Rev. C 107, 064909 (2023)

F. Özel and P. Freire, Annu. Rev. Astron. Astrophys. 54, 401 (2016)

L. Wang21

Spicy Gluons 2024

Physics-Driven Learning
Building Nuclear Matter EoS

Physics Parameters are Finite
EoS, Wave-Function, Potential, …
Inference is Easy-To-Compute
ODEs, PDEs, Simulations, …

F. Özel and P. Freire, Annu. Rev. Astron. Astrophys. 54, 401 (2016)

L. Wang

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Spicy Gluons 2024

Physics-Driven Deep Learninĝθ = arg max {p(X ∣ θ)}θ

L. Wang

Deep Neural Network represented Physics, fθBack-Propagation

Flexible Representation

Easy-To-Compute on GPUs

23

Spicy Gluons 2024

Physics-Driven Deep Learning
1. Building Neutron Star EoS

Tolman–Oppenheimer–Volko equations{dPm(r)ε(r)P(r)
4πr P(r)2Gm(r)
=−G
1
+
1
+
1
−
(
dr
r2
ε(r) ) (m(r) ) (
r )

−1

3

dm(r)
= 4πr 2ϵ(r)
dr
EoS

Core
SurfaceNat. Rev. Phys. 4, 237–246 (2022)

Hydrostatic conditionin each shell (dr)P(ε) = 0

r = 0, ε(r = 0) = εc, P(r = 0) = P(εc)

r = R, ε(r = R) ≃ 0, M = 4πr 2ε(r)dr∫

M, R

L. Lindblom, A.J., 398, 569 (1992).
If the whole M(R) is known,it’s well-de ned problem.

Pressure
L. WangGravity

ff

fi

24

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Physics-Driven Deep Learning
1. Building Neutron Star EoS
Tolman–Oppenheimer–Volko equations?

L. Wangff

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Physics-Driven Deep Learning
1. Building Neutron Star EoS

Phys. Rev. D 107, 083028; JCAP08 (2022) 071(b) TOV Solver

32 x 2"!

(Mi − Mobs,i)(Ri − Robs,i)
ℒ=χ =
+
2
2
∑
ΔM
ΔR
i
i
i=1

!!

2

32 x 32
32 x 132 x 132 x 64

32 x 64

32 x 32

2

2

Loss function

32 x 32

32 x 132 x 64("! , $! )ℒ=(a) NN EoS

Pθ(ρ)

∇" ℒ

-

#!"#

L. Wang

Nobs(""#$,! , $"#$,! )&!

(∆"! , ∆$! )26

2• (Mi, Ri) : predicted Mass-Radius• (Mobs,i, Robs,i) : observed Mass-Radius• (ΔMi, ΔRi) : uncertainties

Spicy Gluons 2024

Physics-Driven Deep Learning
1. Building Neutron Star EoS

Phys. Rev. D 107, 083028
18 (Mi, Ri) , sample size = 10k

Blue dots: NN results, Fujimoto-Fukushima-Murase
Yellow and Green dashed lines: Bayesian Approaches

causality (dϵ/dp < 1)
Maximum mass ≥ 1.9 M⊙

Our results: R1.4 = 11.6 ± 0.43 km (68% CI)

L. Wang27

Spicy Gluons 2024

Physics-Driven Deep Learning
2. Reconstructing Spectral FunctionD̄ηη′ = Tr(Tτ[Jη(τ, x)Jη′†(0,0)]e −H/T )/Z

Correlation Function

k = 0,η = η′d 3k−i(ωnτ−k⋅x)
≡T
D
(iω
,
k)e
∑ ∫ (2π)3 ηη′ nnSpectrum representation
G(τ, T ) =∞∫0

M. Asakawa, Y. Nakahara, and T. Hatsuda, Prog. Part. Nucl. Phys. 46, 459 (2001)

Energy eigenspaceC(τ) =

K(ω, τ, T )ρ(ω, T )dω1cosh ω(τ − 2T )K(ω, τ, T ) =ωsinh 2T∑e −Enτ ∣ < Ω0 ∣ J(0) ∣ Ωn > ∣2∑

| cn |2 e −Enτ

n=0

≡

T→0

n=0

G(τ) =

Thermal Kernel

ρ(ω),

En

Discretization∞∫0e −τωρ(ω)dω

Laplace Kernel

￼

￼

￼

L. Wang￼￼Spectral Function

k = 0,η = η′, T → 0

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Spicy Gluons 2024

Physics-Driven Deep Learning
2. Reconstructing Spectral Function
Kallen—Lehmann(KL) representation
D(p) ≡

L. Wang29∞∫0K(p, ω)ρ(ω)dωK(p, ω) =ωπ(ω 2 + p 2)

Spicy Gluons 2024

Physics-Driven Deep Learning
2. Reconstructing Spectral Function

Phys. Rev. D 106, L051502∞∫0K(p, ω)ρθ(ω)dω = D(p)Forwardχ 2 Loss function
ℒ=

Np

∑
i

wi(Di − D(pi))2

Backward∇θ ℒ =

L. Wang∑j,kK( pj, ωk )

30

∂ℒ
∇ ρ
∂D( pj ) θ k

Spicy Gluons 2024

Physics-Driven Deep Learning
2. Reconstructing Spectral Function

Phys. Rev. D 106, L051502

"!
(a)

!

Regularization

" !

(b)

Softplus
Weights
Nodes

L2 : λ ∣ θ ∣22
Smoothness: λs

Nω

∑
i

(ρi − ρi−1)2

Gradient-based Optimization
Adam : θt+1 = θt −

ηvt̂ + ϵm̂ tPhysical Prior

fi

ff

ff

L. Wang…

Di erentiable variables : Network weights {θ}…

…
NN : (ρ1, ρ2, ⋯, ρNω)

Positive-de ned condition(for hadrons):
Softplus log(1 + e x)

NN-P2P : ρ(ω)
Di erentiable variables : Network weights {θ}Adam, L2 ( λ = 10−3 → 10−8 ), Smoothness ( λs = 10−4 → 0 )Adam, L2 ( λ = 10−6 → 0 )width = 64 and depth = 3 with bias

width = 64 and depth = 3 with bias

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Spicy Gluons 2024

Physics-Driven Deep Learning
2. Reconstructing Spectral Function

Phys. Rev. D 106, L051502

noise level ϵ
Np = 25 points

Mock Test I.

ρ (BW)(ω) =4AΓω(M 2 + Γ2 − ω 2) + 4Γ2ω 2
2

A = 1.0, Γ = 0.5, M = 2.0

A1 = 0.8, A2 = 1.0, Γ1 = Γ2 = 0.5
M1 = 2.0, M2 = 5.0

L. Wang

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Physics-Driven Deep Learning
2. Reconstructing Spectral Function

Phys. Rev. D 106, L051502
noise level ϵ = 10−4 with Np = 25 points

Mock Test II.
1.

1. Single-peak functions
2. Non-positive-de nited SPs
3. Lattice QCD mock data
Thermal (details see arXiv:2110.13521)

G(τ, T ) =

∞

∫0

dω
K(ω, τ, T )ρ(ω, T )2πK(ω, τ, T ) =1cosh ω(τ − 2T)ωsinh 2T

3.
2.

L. Wangfi

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Physics-Driven Deep Learning
3. Extracting Nuclear Force

N. Ishii, S. Aoki, and T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007)

Local Approx.
Gradient Expansion

Nambu-Bethe-Salpeter (NBS) wave function

Nulcear ForceHAL QCD method

L. Wang

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Spicy Gluons 2024

Physics-Driven Deep Learning
3. Extracting Nuclear Forcein preparation ( with HAL QCD)

Two identical-particle
interactions(Ek − H0)ϕk(r) = d 3 r′U(r, r′)ϕk(r′)
∫

NN-2

k2
Ek =
,
2m

∇2
H0 = −
,
2m

mN
m=
2

Phys. Lett. B 712, 437 (2012)

NN-1

￼

￼

￼
￼

￼
￼

L. Wang

￼

NN-1

35

1 ∂2
∂
2
−
−
H
R(t,
r)
=
4πr′
dr′ U(r, r′)R(t, r′)
02∫∂t{ 4mN ∂t}

Spicy Gluons 2024

Physics-Driven Deep Learningin preparation ( with HAL QCD)

3. Extracting Nuclear Force

Neural Network Hadron Force
Separable Potential

UNN(r, r′) = ωfθ(r, r′)U(r, r′) ≡ ων(r)ν(r′),
eϕk0(r) =iδ0(k)

sin{kr + δ0(k)} − sin δ0(k)e

kr

−μr

ν(r) ≡ e−μrr(μ 2 + k 2)
1+
2μ
(
)

1
3μ 2 + k 2 2(μ 2 + k 2)42
2
2 2
k cot δ0(k) = − 2 2μ(μ − k ) −(μ + k ) +
4μ [
4μ 3
8π mω ]

Nambu-Bethe-Salpeter (NBS) wave function
No Approx.

ϕk(r)e −Wkt ≡ ⟨0 | N(x + r, t)N(x, t) | NN, Wk⟩(Ek − H0)ϕk(r) = d 3 r′U(r, r′)ϕk(r′)
∫

￼

￼

￼

￼
￼

￼

￼

￼

￼

L. Wang

￼

Ek =

k2
,
2m

H0 = −

∇2
,
2m

m =

mN
2

ℒ=

∑∫
k

d 3 r (Ek − H0) ϕk(r) − d 3 r′Uθ(r, r′)ϕk(r′)
[
]
∫

2

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Spicy Gluons 2024

Physics-Driven Deep Learning
3. Extracting Nuclear Forcein preparation ( with HAL QCD)

Neural Network Hadron ForceΩcccΩccc Interaction

1 ∂2
∂
2
−
−
H
R(t,
r)
=
4πr′
dr′
U(r,
r′
)R(t,
r′
)
0
∫
{ 4mN ∂t 2 ∂t
}

s
t
l
u
s
e
R
ry

a
n
i
m
Preli

Nambu-Bethe-Salpeter (NBS) wave function
R2 = Rt+1 − 2Rt + Rt−1, R1 = (Rt+1 − Rt−1)/2,Rr = ∇2 R(t, r)

mN = 2.073, a −1 = 2333.0MeV
t = 26No Approx.

Phys. Rev. Lett. 127, 072003 (2021)

￼

￼
￼

￼
￼

￼

L. Wang

￼

￼

1
1
ℒ=
R2(t, r) − R1(t, r) +Rr(t, r) − 4πr′2dr′ Uθ(r, r′)R(t, r′)∑ { 4mN
∫
mN
}
t

37

Spicy Gluons 2024

Physics-Driven Deep Learning
3. Extracting Nuclear Forcein preparation ( with HAL QCD)

Neural Network Hadron Force

Pr
el

im

in
ar

y

R
es

ul
ts

ΩcccΩccc Interaction

L. Wang38

Spicy Gluons 2024

Summary I• Inverse Problems• Data-driven learning• Physics-driven learning• Physics-driven deep learning• Neural network representations• Gradient-based optimization• Future workŝθ = arg max {p(X ∣ θ)}• Nuclear Matter EoSθ• Spectroscopy [github1, github2]• NN-Nulcear Force
L. Wang39

Spicy Gluons 2024

Generative Models
as Inverse ModelingGenerative Models@blogs of OpenAI

Generative models→ Underlying Distributions in Data

L. Wang

max
θ

41

N

∏
i=1

pθ(xi)
Spicy Gluons 2024

High-Dimensional Distribution−S(ϕ)p(ϕ) = e
/Z
1
⟨O⟩ ≈
Oi
∑
N i

Lattice QCD © Derek Leinweber/CSSM/University of Adelaide→ Physical Distribution, Sampling
via Generative Models
Global Sampling
Fast and Independent Sampler
Heavy-Ion Collisions © 2010 CERN

Prog.Part.Nucl.Phys. 104084(2023) (Invited Review)

L. Wang42

Spicy Gluons 2024

Probabilistic Models
x ∼ pdata(x)
max
θ

∏

pθ(xi)

Maximum likelihood estimation(MLE)log pθ(xi)Maximum log-likelihood estimation

i=1

max
θ

N

N

∑
i=1

Bishop, C. M. & Bishop, H. Deep Learning: Foundations and Concepts. doi:10.1007/978-3-031-45468-4.
I. Goodfellow, arXiv:1701.00160 (2017)

L. Wang

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Probabilistic Models
x ∼ pdata(x)pθ(x) =e−fθ(x)

Curse of DimensionalityZθ 😵

from a statistical phsyics perspective∇x log p(x)Approaching Score Functionsθ(x) → ∇x log p(x)sθ(x) = ∇x log pθ(x)= − ∇x fθ(x) − ∇x log Zθ = − ∇x fθ(x)=0

L. Wang44

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L. Wang

45

Space Opera,
JasonGluons
Allen via Midjourney
Spicy
2024

Diffusion Models• Forward di usion process graduallyadds noise to input• Reverse denoising process learns togenerate data by denoising• Train Probabilistic Models

to learn how to convert a simple
distribution to a target distribution
Ho et al., Denoising Di usion Probabilistic Models, NeurIPS 2020

L. Wangff

ff

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Diffusion Models
Song et al., Score-Based Generative Modeling through Stochastic Di erential Equations, ICLR 2021

• Forward Di usion SDE

dϕ= f(ϕ, ξ) + g(ξ)η(ξ)
dξ• Drift term: pulls towards mode
• Di usion term: injects noise

Anderson, in Stochastic Processes and their Applications, 1982

• Reverse Generative Di usion SDE• Drift term is adjusted with a “Score Function”• Represent the score function with neural networks

ff

47
ff

ff

ff

L. Wangdϕ= [f(ϕ, t) − g 2(t) ∇ϕ log pt(ϕ)] + g(t)η̄(t)dt

Spicy Gluons 2024

Diffusion Models
Stochastic Quantization

Parisi G. and Wu Y. S., Sci. China, A 24, ASITP-80-004 (1980).

δSE[ϕ]∂ϕ(x, τ)=−+ η(x, τ)∂τδϕ(x, τ)⟨η(x, τ)⟩ = 0,

⟨η(x, τ)η(x′, τ′)⟩ = 2αδ(x − x′)δ(τ − τ′)τ: ctitious time, α: di usion constant• Fokker-Planck equation
δSE∂P[ϕ, τ]δδn=α d x+P[ϕ, τ]{ δϕ ( δϕ
∫
∂τ
δϕ )}

!~#[!, & = 0]!

Thermal equilibrium limit → Quantum distribution

Equilibrium solution (long-time limit),

Peq[ϕ] ∝ e− α1 SE[ϕ]1. No need gauge- xing!
2. Can handle fermionic elds naturally→(Complex Langevin method)….

• Set the di usion constant as α = ℏPeq[ϕ] ∼ e− ℏ1 SE[ϕ]!~#[!, & = *]= Pquantum[ϕ]

￼

￼
fi

￼
fi

ff

ff

ff

fi

L. Wang￼P. H. Damgaard and H. Hü el, Stochastic Quantization, Phys. Rept. 152, 227 (1987).
M. Namiki, Basic Ideas of Stochastic Quantization, PTPS 111, 1 (1993).
G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty, and I.-O. Stamatescu, Eur. Phys. J. A 49, 89 (2013).

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Diffusion Models
JHEP 05(2024)060

DMs as SQ• Di usion models(Reverse SDE):

peq(ϕ) ∝ edϕ= − g(t)2 ∇ϕ log pt(ϕ) + g(t)η̄dt• De ne: τ ≡ T − t(dτ ≡ − dt)

SDM
− ᾱ

pτ=T(ϕ) → P[ϕ, T]dϕ= gτ2 ∇ϕ log qτ(ϕ) + gτ η̄dτϕ(τn+1) = ϕ(τn) + gτ2 ∇ϕ log qτn[ϕ(τn)]Δτ + gτ Δτ η̄(τn)O(ᾱ) ∼ O(ℏ)

introducing Noise scale: ⟨η̄2⟩ ≡ 2ᾱ, time scale: gτ2Δτ

The reverse mode ofa well-trained di usion model at τ → T serves as
the stochastic quantization for the input• FP equation∂pτ(ϕ)
δδ1n2= d x gτ ᾱ+ ∇ϕ SDM
pτ(ϕ)
{
)}
∫
∂τ
δϕ ( δϕ ᾱ
∇ϕ SDM ≡ − ∇ϕ log qτ(ϕ)

49
ff

fi

ff

L. Wang

Spicy Gluons 2024

Diffusion Models
DM for Scalar Field• Euclidean action on latticed(ϕ0(x + aμ)̂ − ϕ0(x))2 m02 2 λ0 4
SE =
a [
+
ϕ0 + ϕ0 ]
2
∑
∑
a
2
4!
x
μ=1
d

• Dimensionless form
SE =

∑
x

[−2κ

d

∑

ϕ(x)ϕ(x + μ)̂ + (1 − 2λ)ϕ(x)2 + λϕ(x)4]

JHEP 05(2024)060• Di usion models• T = 1.0, σ = 25
• Data generation• 2-d 32×32 lattice; Hamiltonian Monte Carlo(HMC);
5120 con gurations for training.

• Broken phase: κ = 1.0, λ = 0.022• Symmetric phase: κ = 0.21, λ = 0.022

μ=1

d−2

a 2 ϕ0 = (2κ)1/2ϕ(am0)2 =1 − 2λ6λ− 2d, a −d+4 λ0 = 2κκ• Hopping parameter κ,
Coupling constant λfiff

L. Wang50

Spicy Gluons 2024

Summary II
Input(Configurations)

Masked Convolutional Layers

Chinese Phys. Lett. 39, 120502 (2022)• Generative Models• Probabilistic Models• Score Matching• Di usion Models• Stochastic Quantization scheme

SampleOutputHidden Layers(Parameters of Mixture Distributions)

qθ(s)

…

Phys. Rev. D 107, 056001• Future works• Di usion models as SQJHEP 05(2024)060• 2+1D Gauge Field• Complex Langevin Method(CLM) for
Fermions

ff

ff

L. Wang

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Spicy Gluons 2024

Thank You !
ML meets Physics, Opportunities and Challenges

Kouzou Sakai @Quanta MagazineRepresentation Learninggθ : X → Y
fθ : Y → X

Forward Mapping, fθPhysics

Data

Model Parameters/
Properties/States

Observations

Inverse Mapping, gθ
L. Wang53

Spicy Gluons 2024

Representation Learning
Recognizing nuclear liquid-gas phase transition

Anomaly Detection
Latent Variable Extraction

R. Wang, Y.-G. Ma, R. Wada, L.-W. Chen, W.-B. He, H.-L. Liu, and K.-J. Sun, Phys. Rev. Res. 2, 043202 (2020)

L. Wang54

Spicy Gluons 2024

Representation Learning
Recognizing nuclear liquid-gas phase transition

Predict relativistic hydrodynamics

R. Wang, Y.-G. Ma, R. Wada, L.-W. Chen, W.-B. He, H.-L. Liu, and K.-J. Sun, Phys. Rev. Res. 2, 043202 (2020)

H. Huang, B. Xiao, Z. Liu, Z. Wu, Y. Mu, and H. Song, Phys. Rev. Res. 3, 023256 (2021)

L. Wang55

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Backups I
1. Building Neutron Star EoS

Phys. Rev. D 107, 083028

Uncertainty estimations

Recall: Importance Sampling• x : random variables• f (x) : observables• n : number of samples• p(x) : original(true) distribution• q(x) : reference distribution•x : reconstructed EoSs given a sample•O(x) : observables, M, R, P
Variance

{

Mean

p(x)
E[ f(x)] = f(x)p(x)dx = f(x)q(x)dx∫∫q(x)σ(O)2 = ⟨Ô 2⟩ − Ō2Ō = ⟨O⟩̂ =N∑jw ( j)O ( j)Weights

Posterior(θ ( j) | data)
EoSw ( j) =Proposal(θ ( j) )EoS∝
p(data | θ ( j) ) ∝ exp(−χ 2(Mθ ( j) , Rθ ( j) ))EoS
EoS EoSp(θ ( j)EoS

2

L2 : exp(−λ | | θ | |2 ) ∼ 1,p(data | θ ( j) ) ⋅ Prior(θ ( j) )EoS
EoS
| samples( j)) ⋅ p(samples( j) | data) ⋅ Prior(data)w̃( j) = w ( j) /λ = 10−8∑w ( j)

jnormalizationp(θ ( j) | samples( j)) = 1
EoS

p(samples | data) =(Mobs, ΔM 2) (Robs, ΔR 2)

deterministic

56
𝒩

𝒩

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Spicy Gluons 2024

Backups I
1. Building Neutron Star EoS

Phys. Rev. D 107, 083028

GW170817+390
Λ1.4 = 190−120+115.9
Our results, Λ̃1.4 = 286.47−115.9

L. Wang57

Spicy Gluons 2024

Comput. Phys. Commun. 282, 108547

2. Reconstructing Spectral Function

Kij, i ∈ Nx, j ∈ Nω, Nx < Nω
D ⃗ ≡ K ρ⃗

• In practice, the Euclidean correlations

have nite number of points and withnite precision;

highly rectangular

vectorization• The ill-posedness of the spectral

reconstruction fundamentally exists
even for continuous correlation
functions(in nite observations);

D(x) ≡• It’s caused by the numerical inaccuracyof the correlation measurements(induced high degeneracy in solution
space).

L. Wang∞∫0

K(x, ω)ρ(ω)dωeigenvalue problem

∞

∫0

58
fi

fi

fi

Backups I

ψs(ω) K(x, ω) dω = λsψs(x)Spicy Gluons 2024

Backups II: Diffusion Models
x ∼ pdata(x) ≡ p(x)
max
θ

N

∏
i=1

pθ(xi)∇x log p(x)Maximum Likelihood Estimation

Parameterizing probability density functions

Approaching Score Functionsθ(x) ≈ ∇x log p(x)sθ(x) = ∇x log pθ(x) = − ∇x fθ(x) − ∇x log Zθ = − ∇x fθ(x)=0

L. WangParameterizing score functions@Yang Song’s Blog

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Spicy Gluons 2024

𝔼
𝔼

Backups II: Diffusion Models
x ∼ pdata(x) ≡ p(x)Adding noise2
p(x)[∥∇x log p(x) − sθ(x)∥2]{
1-D

x̃ = x + σϵ

Minimizing the Fisher divergencepσ(x̃) ≡ pσ(x̃ | x)pdata(x)dx∫pσ(x̃ | x) ∼ N(x̃; x, σ 2 I ), ϵ ∼ N(0,I )∂ log(pσ(x̃ | x)pdata(x))∂ log(pσ(x̃))∂ log(pσ(x̃ | x))
≡
=
∂x̃
∂x̃
∂x̃
Score-matching a noise-perturbed distribution

2
[∥∇
log
p
(x)
−
s
(x)∥
pσ(x)
x
σ
θ
2]

L. Wang

Estimated scores are only accurate in high density regions

Matching a noise-perturbed distribution

60

No enough data in low-density region!
But we sample from the low-density region using
Langevin dynamics…

🤔

Spicy Gluons 2024

2
[∥∇
log
p
(x)
−
s
(x)∥
pσ(x)
x
σ
θ
2]

Matching a noise-perturbed distribution

Small noise → Good approximation to data, poor in low-density region!
Large noise → Poor approximation to data, good in low-density region!

Multiple noise perturbations!

L

λi pσ (x)[∥∇x log pσi(x) − sθ(x, i)∥22]∑i
i=1Matching multi-noise
perturbed distributions

🥳

• Choose the noise scheme as a geometric progression,

σ1 /σ2 = σi−1 /σi = … = σL−1 /σL > 1, with σ1 being su ciently small and σLcomparable to the maximum pairwise distance between all training data
points. L is typically on the order of hundreds or thousands.
• Parameterize the score-based model, sθ(x, i), with U-Net skip connections.

Ho et al., Denoising Diffusion Probabilistic Models, NeurIPS 2020

Multiple scales of Gaussian noise to perturb the data distribution ( rstrow), and jointly estimate the score functions for all of them (second row).

Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, ICLR 2021

𝔼

L. Wang
fi

61
ffi

𝔼

Backups II: Diffusion Models

Spicy Gluons 2024

Backups II: Diffusion ModelsL2
λi pσ (x)[∥∇x log pσi(x) − sθ(x, i)∥2]∑ii=1

xi+1 = xi + σiϵ

Matching multi-noise perturbed distributions

dμ
=0
dt

∇x log pσi(x) = ∇x log(pσi(xτ | x)p(x)) = ∇x log pσi(xτ | x)2

dσ
= σ 2tdt

pσi(xτ ∣ x) =t ∈ [0,τ]τσ≡
σ
Choosethenoiseschemeas,, where τ indicates “time•istep” for adding noise.

12τ
xτ; x,(σ − 1)I()
2 log σTraining is Matching𝒩

𝔼

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications

L. Wang62

Spicy Gluons 2024

Backups II: Diffusion Models
2

μ 2 g 4S(ϕ) = ϕ + ϕ ,24!
gf(ϕ) = − μ 2ϕ − ϕ 3
3!

JHEP 05(2024)060

0+0 dimensional FT(1 variable)

L. Wang63

Spicy Gluons 2024

Backups II: Diffusion Models
2

μ 2 g 4S(ϕ) = ϕ + ϕ ,24!
gf(ϕ) = − μ 2ϕ − ϕ 3
3!

JHEP 05(2024)060

0+0 dimensional FT(1 variable)

L. Wang64

Spicy Gluons 2024