Exploration of Efficient Neural Network forPath Optimization Method
arXiv:2109.11710, 220X.XXXXX

Yusuke Namekawa(Kyoto Univ. → Hiroshima Univ.)in collaboration withK.Kashiwa, H.Matsuda, A.Ohnishi, H.Takase

Contents
1

Main message

32Motivation

4

3

Application to 2-dim U (1) gauge theory

104

Summary

16
Yusuke Namekawa(Hiroshima U)– 1 / 16 –TQFT 2022(YITP)[In memory of Taniguchi-san]• I have 36 papers with Taniguchi-san• Taniguchi-san and I are core members of an open source code forlattice QCD ”Bridge++” : now I maintain Taniguchi-san’s codes
Bridge++ party at Irish pub(Feb. 2019)
From left:(Namekawa’s hand), Aoyama-san,Matsufuru-san, Taniguchi-san[photo by Kanaya-san]Yusuke Namekawa(Hiroshima U)– 2 / 16 –TQFT 2022(YITP)

1Main message• For path optimization in a gauge theory,”it is efficient to employ a neural network
which respects the gauge symmetry”
ex. gauge invariant input / gauge covariant
neural network
cf. similar idea is used as a part of gauge equivariant convolutional neural network T.Cohen et al.(2019); Favoni et al.(2020)♦ Gauge variant neural network works but costs a lot

Yusuke Namekawa(Hiroshima U)– 3 / 16 –TQFT 2022(YITP)

2Motivation• QCD at high density has been investigated only by Complex LangevinMethod(CLM) Sexty(2013),...,YN et al.(2021) due to sign problem← Action becomes complex at µ 6= 0, which prohibits Monte Carlosimulation using probability weight P = e−S• CLM can not cover the whole phase diagram of QCD due tovalidity condition of CLM→ Alternative approach is needed♦ See next pageFukushima,Hatsuda(2011)

Yusuke Namekawa(Hiroshima U)– 4 / 16 –TQFT 2022(YITP)[Several methods to overcome sign problem]• Complex
Langevin method Parisi(1983),Klauder(1984),...
Low cost, generally work only in limited parameter region

 Revived by clarifying correctness criterion Aarts et al.(2009), Nishimura,Shimasaki(2015)

Calculated equation of state of QCD at finite density Sexty(2019), Attanasio et al.(2022)→ cf. Itou-san’s talk on 9/22

• Lefschetz thimble method Witten(2010),Cristoforetti et al.(2012),Fujii et al.(2013),Alexandru et al.(2015),...
Middle cost, difficult to resolve sign and ergodicity problems simultaneously ♦Tempered Lefschetz thimble method Fukuma,Umeda(2017),...
[ Middle cost, resolve sign and ergodicity problems simultaneously ] ♦Worldvolume Hybrid Monte Carlo(WV-HMC) method Fukuma,Matsumoto(2020),...
[ Low cost, resolve sign and ergodicity problems simultaneously ]→ See Fukuma-san’s talk@Lattice2022(Aug. 8th, 2022)• Pathoptimization method Mori,Kashiwa,Ohnishi(2017),...
Middle cost, find optimized path by machine learning,deal with ergodicity problem by tempering• Tensor network Levin,Nave(2007),...
[ High cost → See talks on 9/21 ]Yusuke Namekawa(Hiroshima U)– 5 / 16 –TQFT 2022(YITP)

[Path optimization method(POM)] Mori et al.(2017),Alexandru et al.(2018),Bursa,Kroyter(2018),...

• POM is a method which complexifies dynamical variables and deforms theintegration path using machine learning to minimize sign problem• POM has been successful in models with small redundant degrees offreedom, but is not efficient with large gauge degrees of freedom♦ One solution is gauge fixing but costs a lot Mori et al.(2019),...
♦ We found gauge invariant input / gauge covariant neural network
works well
→ This talk✓✏NB. Cauchy’s integral theorem ensures the following equality
1 Z
1 Z−S[U ]−S[U ]
hOi :=DU Oe=DU OeZ R
Z C
O : observable, Z : partition func, S : action, U : link variable−gImAµ (x) igReAµ (x)igAµ (x)eUx,µ := e→ Ux,µ = e
Aµ (x) ∈ R → Aµ (x) ∈ Cdetermined by machine learning✒

Yusuke Namekawa(Hiroshima U)✑– 6 / 16 –Ohnishi(2017)

TQFT 2022(YITP)[Gauge variant neural network]
Ux,µ|{z}→ hidden layer →input layer

Ux,µ|{z}output layer• Machine learning chooses best path which enhances phase factoreiθ := Je−S /|Je−S |,J := det(∂U/∂U )♦ Averaged phase factor | hexp(iθ)i | is an indicator of sign problem:| hexp(iθ)i | = 1 for mild, | hexp(iθ)i | = 0 for severe✓Input layer : ti = Ux,µHidden layer : hj = F (wh5✏(1)
t + bj )ji i

U1,2U1,2

Output layer : zn = ωn F (wh4U1+2̂,1U1+2̂,1w, b, ω := parameters of neural network

h3F (x) := tanh(x), activation func
iθ(t)−1Fcost (t):=|Z| |heipq |−1 ,U1+1̂,2

U1+1̂,2h2

U1,1U1,1h1(2)h + bn )nj j✒Yusuke Namekawa(Hiroshima U)

hOipq :=– 7 / 16 –1 ZZDUpq : phase quenchedhi−SO JeU ∈CTQFT 2022(YITP)✑[Gauge invariant neural network] YN et al.(2021)
Ux,µ → Px,µν → hidden layer →|{z}input layer

Ux,µ|{z}output layer• We adopt gauge invariant plaquette in the input layer
Px,12 := Ux,1 Ux+1̂,2 U −1x+2̂,1−1Ux,2♦ Similar idea is used as a part of gauge equivariant convolutional neural network
T.Cohen et al.(2019); Favoni et al.(2020)

h5h5
U1,2U1,2U1,2h4h4
U1+2̂,1U1+2̂,1h3
U1+1̂,2U1+1̂,2U1+1̂,2h2h2U1,1U1,1U1+2̂,1U1+2̂,1h3
U1+1̂,2U1,2P1,12U1,1U1,1h1h1(old)
Yusuke Namekawa(Hiroshima U)(new)– 8 / 16 –TQFT 2022(YITP)[Gauge covariant neural network] Tomiya,Nagai(2021)
Ux,µ|{z}input layer(1)→ Ũx,µ→ ··· →| {z }

(l+1)
(l)
(l)
Ũx,µ= exp[iWx,µ] Ũx,µ,hidden layer(l)Wx,µ:=Ux,µ|{z}output layerX  (l) (l)(l) (l) −1
ρ+ Px,µν + ρ− Px,µνν6=µ(l)ρ± : parameters in neural network, (l) : number of smearing• The hidden layer is constructed by Stout-like smearing, which is gaugecovariant• We use Nstout = 2 in this work

Yusuke Namekawa(Hiroshima U)– 9 / 16 –TQFT 2022(YITP)

3

Application to 2-dim U (1) gauge theory• Sign problem is originated from the complex coupling β = 1/(ga)2 ∈ R → C• Analytic result has been obtained→ Good testbed for new approach Kashiwa,Mori(2020),Pawlowski et al.(2021)
cf. 2-dim U (1) + θ-term, another type of sign problem, is investigated by tensor renormalization
Kuramashi and Yoshimura(2019) and complex Langevin Hirasawa et al.(2020)

β X
−1Px,12 + Px,12S=−2 xβ = 1/(ga)2 ∈ R → CPx,12 := Ux,1 Ux+1̂,2 U −1x+2̂,1−1Ux,2[Analytic result] Wiese(1988),...
Z :=ZdU e−S =1In (β) :=2πYusuke Namekawa(Hiroshima U)+∞
XIn (β)Vn=−∞Z πdφ eβ cos φ−inφ−π– 10 / 16 –TQFT 2022(YITP)

[Neural network iteration dependence of average phase factor]• Gauge invariant input and gauge covariant neural network successfullyenhances averaged phase factor | hexp(iθ)i |♦ The peak structure in histogram of the averaged phase factor ismade clear• (See next page for gauge variant link-variable input)
1.00

12000

POM(plaq input)
4 × 4, β = 0.0+0.5 i
Naive reweighting
10000 NN input = plaq
4 × 4, β = 0.0 + 0.5 i

0.60

8000
N|<exp(i θ)>|

0.80

6000

0.40
4000
0.20
0.000

1000

POM(gauge covariant)
POM(plaq input)
POM(link input)
2000
3000
4000
NN iter

2000
5000

Yusuke Namekawa(Hiroshima U)

0-1– 11 / 16 –-0.50
θ/π

TQFT 2022(YITP)

0.51

[Neural network iteration dependence of average phase factor(continued)]• Naive link-variable input does not enhance the averaged phase factor by5000 neural network iterations with Nunit = 16 hidden layer units• Naive link-variable input with much larger neural network iterations
and larger hidden layer units enhances the averaged phase factor♦ Neural network can learn the gauge symmetry, but the cost is high→ It is better to employ gauge invariant input or gauge covariant
neural network
1.00

1.000.95|<exp(i θ)>||<exp(i θ)>|0.80
4 × 4, β = 0.0 + 0.5 i

0.60
0.40

0.90
0.85
2 × 2, β = 0.0 + 0.5 i

0.80

Nunit = 1530
45

0.75
0.20
0.000

1000

POM(gauge covariant)
POM(plaq input)
POM(link input)
2000
3000
4000
NN iter

0.70
05000

10000

NN iter

5000

Calculated by H.Matsuda
Yusuke Namekawa(Hiroshima U)– 12 / 16 –TQFT 2022(YITP)

15000

[Comparison with the analytic solution]• At small Imβ, the POM successfully reproduces the exact solution withcontrolled error• At large Imβ, the POM fails due to no enhancement of the peakin histogram of the averaged phase factor→ Further improvement is required, such as tempering
1.0

4×4Re β = 0.0

6000

No clear peakin θ histogram

4×4
5000 NN input = plaq

β = 0.0 + 1.250 i
β = 0.0 + 1.375 i

4000
0.0N

Im <plaq>

0.5

3000
2000

-0.5

POM(plaq input)
Naive reweighting
Exact-1.00
0.511000Im β

1.52Yusuke Namekawa(Hiroshima U)

2.50-1– 13 / 16 –-0.50
θ/π

TQFT 2022(YITP)

0.51

[Volume dependence]• Enhancement of the averaged phase factor is confirmed♦ Gauge invariant input / gauge covariant neural network shows milder
volume dependence than that of naive reweighting
100|<exp(i θ)>|

10-1
10-2
10-3

β = 0.0 + 0.5 i

POM(plaq input)
POM(gauge covariant)
Naive reweighting-5
10 020 40 60 80 100 120 140 160
L210-4

Yusuke Namekawa(Hiroshima U)– 14 / 16 –TQFT 2022(YITP)[Test approximated Jacobian in neural network]
For a large scale simulation, cost reduction is necessary
3• The main bottleneck is cost of Jacobian O(Ndof)• We test J = 1 approximation in the learning process← We still need the exact Jacobian for final output and measurement
cf. WV-HMC needs no explicit form of Jacobian in Monte-Carlo update Fukuma,Matsumoto(2020)

1.00

0.266

0.900.264
4 × 4, β = 0.0 + 0.5 i0.80

Im <plaq>|<exp(i θ)>|♦ POM using J = 1 approximated neural network can enhance theaveraged phase factor with a slightly larger error by 1%

0.70
0.60
0.50
0.400

1000

POM(with Jacobian)
POM(without Jacobian)
2000
3000
4000
NN iter4×4
β = 0.0 + 0.5 i

Exact
Naive reweighting
POM(w/ Jacobian)
POM(w/o Jacobian)

0.262
0.260
0.258

5000

Yusuke Namekawa(Hiroshima U)

0.256– 15 / 16 –TQFT 2022(YITP)

4Summary

We explored efficient ways for the path optimization method, which reduces sign
problem by complexification of path using machine learning• Gauge invariant input / gauge covariant neural network successfullyenhances the average phase factor, i.e., reduces sign problem← Gauge variant neural network can also enhance the average phasefactor with much larger cost
• J = 1 approximated neural network still leads to large enhancement of
the average phase factor (at least in our setup)[Future direction]• Try non-Abelian case, such as SU (2), SU (3)• Test another type of sign problem, such as system with θ-term

Yusuke Namekawa(Hiroshima U)– 16 / 16 –TQFT 2022(YITP)