SU(3)非可換ゲージ理論のトポロジー分類への機械学習の適用
北沢正清, 松本拓也, 河野泰宏
(大阪大学)
T. Matsumoto, M. Kitazawa, Y. Kohno, arXiv:1909.06238

Deep learning and Physics 2020, 2020/6/11, online seminar理論物理学と機械学習
機械学習は理論物理学に応用可能か？
 MLはblack box. 「理解」にはならない？
 MLにできることは、より洗練した方法で実現可能？

 100%の正答率は得られない。

Deep learning and Physics 2020, 2020/6/11, online seminar

Topological Charge in YM Theory
q(x) in SU(3) YM: integerin continuum theory

 Interests / applications
 Instantons
 Axial U(1) anomaly
 Axion cosmology
 Topological freezingDeep learning and Physics 2020, 2020/6/11, online seminar

Machine Learning
Input: q(x)OutputQ4-dimensional fieldAnalysis of 4D data by 4D-CNN
Capture “instanton”-like structure?
Acceleration of the analysis of Q?
Deep learning and Physics 2020, 2020/6/11, online seminar

topologicalcharge

Gradient Flow and Topology
gradient flow ≃ Smoothing
flow time t = smearing radius

Deep learning and Physics 2020, 2020/6/11, online seminar

Trial 2: Topol. Density @ t>0

OutputFull connect

Full connect

GAP

Convolution

Convolution

Input

 Input: q(x,t) in 4-dim space at nonzero flow time Data reduction to 84 (average pooling)

Recalls of each topological sector (%)

Deep learning and Physics 2020, 2020/6/11, online seminar

QBenchmark

Distribution of Q(t)Simple estimator from Q(t)1) Naïve:2) Improved:c>1: optimization param.

3) zero:Deep learning and Physics 2020, 2020/6/11, online seminar

Comparison: NN vs Benchmark
accuracy at b=6.2
t/a2=0ML (Trial 2)
37.0

naïve
27.3

improved
27.3

t/a2=0.1
t/a2=0.2
t/a2=0.3

40.3
53.7
76.1

38.3
54.0
69.8

38.354.6
77.3

 Machine learning cannot exceed the benchmark value.
 NN would be trained to answer the “improved” value.
 No useful local structures found by the NN.
Deep learning and Physics 2020, 2020/6/11, online seminar

Trial 3: Multi-Channel Analysis
 Input: q(x,t) in four-dimensional space at t/a2=0.1, 0.2, 0.3

QResultmachine learning

b=6.2
b=6.5

93.8
94.1

benchmark @ t/a2=0.3
77.3
71.3

 non-trivial improvement from the benchmark!!
Deep learning and Physics 2020, 2020/6/11, online seminarIs this a non-trivial result?
inputanswer

We can estimate the answer from Q(t) by our eyes…
Deep learning and Physics 2020, 2020/6/11, online seminar

Trial 4: Feed Q(t) [0-dim]

Q(0.3)

Result
b=6.2
b=6.5

Q(t)
94.1
95.7

tuninginput t95.9
99.0

outputQ

input

OutputQ(0.2)

Full connect (5)Q(0.1)

Input (3) Input: Q(t) at t/a2=0.1, 0.2, 0.3

Trial 3 (4dim) benchmark
93.8
94.1

 Good accuracy is obtained only from Q(t)Deep learning and Physics 2020, 2020/6/11, online seminar

77.3
71.3

Reducing the Training Data
 Smaller training data will reduce numerical costfor the training.
Training data

10,000

5,000

1,000

500

100

b=6.2

95.9(2) 95.9(2) 95.9(2) 95.5(3)90.3(7)b=6.5

99.0(2) 99.0(2) 98.9(2) 98.9(1)90.2(8)

 1000 configurations are enough to train the NN
successfully!
 Numerical cost for the training is small.

Deep learning and Physics 2020, 2020/6/11, online seminarRobustness Test Analyze configurations with a different parameter set

training
data

analyzed data
b=6.2

b=6.5

b=6.2

95.9(2)98.6(2)b=6.5

95.6(2)99.0(2)both95.8(1)98.9(2)

 NNs trained for b=6.2 and 6.5 can be used foranother parameter successfully.
 Universal NN would be developed!
 Note: same physical volume
Deep learning and Physics 2020, 2020/6/11, online seminar

Untrained Data
standard training

training w/o |Q|=4,5

CNN can make accurate predictions even for
untrained values of Q.
Deep learning and Physics 2020, 2020/6/11, online seminar

Trial 5: Dimensional Reduction
 Optimal dimension between d=0 and 4?
 d-dimensional CNN
 Input: qd(x) after dimensional reduction
 3-channel analysis: t/a2=0.1, 0.2, 0.3
 No d dependence
 Failed in finding
features in multi-dim.
space.
 No instanton-like
local structure inQCD vacuum?

Deep learning and Physics 2020, 2020/6/11, online seminarSummary and Outlook 4次元データのCNN解析は、配位の局所的構造を特徴抽出するには至っていないようである。
 次元縮約も精度向上に寄与しない。
 ゼロ次元FNN解析は、トポロジカル電荷推定の効率化に有効。Gradient flowはt/a2=0.3までで良い。
 高い正答率／少ない学習データ／高い堅牢性／未知のデータへの適用将来展望
 連続極限 / 体積依存性
 DIGAが適用可能な高温配位の解析
 論文の出版
Deep learning and Physics 2020, 2020/6/11, online seminarbackup

Deep learning and Physics 2020, 2020/6/11, online seminar

Neural Network Setting
 convolutional neural network by CHAINER framework
 supervised learning convolutional layer: 4-dim., periodic BC regression analysis / round off to obtain integer activation: logistic
 answer of Q
 Q(t) @ t/a2=4.0
 round off

Deep learning and Physics 2020, 2020/6/11, online seminar

Using different flow times
t/a2

b=6.2

b=6.50.3, 0.25, 0.2

95.9(2) 99.0(2)0.3, 0.2, 0.1

95.5(2)95.7(2)

0.25, 0.2, 0.15

95.1(3)95.0(2)

0.2, 0.15, 0.1

86.9(3)83.1(4)0.2, 0.1, 075.6(5)68.2(4)

0.15, 0.1, 0.05

71.8(4)65.2(4)0.1, 0.05, 054.8(5)49.9(3)

 t/a2=0.3, 0.25, 0.2 gives the best accuracy.
 Better accuracy on the finer lattice.
 More than three t values do not improve accuracy.
 error: variance in 10 independent trainings
Deep learning and Physics 2020, 2020/6/11, online seminar

Topological Charge Density

No isolated instanton structure…

Deep learning and Physics 2020, 2020/6/11, online seminar

Lattice Setting

 Gradient flow for smoothing

N4164

Nconf
20,000

6.5

244

20,00020,000 confs. in total SU(3) Yang-Mills Wilson gauge action
 2 lattice spacings with same
physical volume
 LTc~0.63


b6.2

Training: 10,000
Validation: 5,000
Test: 5,000

distribution of Q

Deep learning and Physics 2020, 2020/6/11, online seminar

Topology on the Lattice Distinct topological sectors on sufficiently fine lattices
 Definitions of Q on the lattice: fermionic: Atiyah-Singer index theorem
 gluonic: q(x) after smoothing cooling, smearing
 gradient flow
Luscher, Weisz, 2011

 Good agreement b/w various
definitions
 Faster algorithm is desirable!

Deep learning and Physics 2020, 2020/6/11, online seminar

Luscher, 1981

Details of NN Trial 1~3

 Trial 4

 Trial 5

Deep learning and Physics 2020, 2020/6/11, online seminar