Do Do NO NOT m measure co correlat ated observables, , but tr - PowerPoint PPT Presentation

Do Do NO NOT m measure co correlat ated observables, , but tr train ain an an Artif tific icial ial In Intellig elligenc ence e to pr predic edict them them Boram Yoon Los Alamos National Laboratory Lattice 2018, East Lansing, Michigan, USA, July 22-28, 2018 arXiv:1807.05971

Pr Prediction of ! "#$ fr from ! %#$ 1.236 1.06 1.044 1.046 1.045 1.04 1.040 1.232 1.044 1.02 1.036 1.043 1.228 1.00 1.032 1.042 1.224 1.041 0.98 1.028 1.040 1.220 0.96 1.024 g S g A g T 1.039 g V 1.216 0.94 1.020 1.038 Systematic error due to ML prediction included in errorbars • Genuine • ML Prediction Directly measured on 2263 confs Directly measured on 400 confs + ML prediction on 1863 confs

La Lattice ce QCD D Observables are Corr rrelated {M π (7) , F π (7) , C 3pt:A (7) , C 3pt:V (7) , …} {M π (1) , F π (1) , C 3pt:A (1) , C 3pt:V (1) , …} ExpectaQon value N O X ≈ 1 ∑ ( n ) O X N n = 1 Markov Chain Monte Carlo Trajectory U (1) U (2) U (3) U (4) U (5) U (6) U (7) U (8) U (9) of Gibbs Samples

Cor Correlation on Ma Map of of Nucleon on Observables 1 T,D C 3pt • Correlation between proton(uud) 0.9 | Correlation coefficient | 3-pt and 2-pt correlation functions A,D C 3pt 0.8 V,D C 3pt • Clover-on HISQ S,D 0.7 C 3pt ! = 0.089 fm, ' ( = 313 MeV T,U + = 10! , , = +/2 C 3pt 0.6 A,U C 3pt 0.5 • Using these correlations, V,U C 3pt / 012 can be estimated from / 312 0.4 S,U on each configuration C 3pt 0.3 C 2pt 0.2 S,U C 3pt V,U C 3pt A,U C 3pt T,U C 3pt S,D C 3pt V,D C 3pt A,D C 3pt T,D C 2pt C 3pt

Machine Learning Ma • One can consider the machine ) , ( # + , ( # , , … ) Input: " # = (( # learning (ML) process as a data fitting • The machine ! has very general fitting functional form with huge number of free parameters Machine • The free parameters are determined ! from large number of training data: ! " # ≈ % # • For example, " # : pixels of a picture Output: % # % # : “ cat ” or “ dog ”

Ma Machine Learning on on Lattice QCD QCD Observables • Assume N+M indep. measurements ( , ' " * , ' " + , … ) Input: ! " = (' " • Common observables ! " on all N+M Target observable # " on first N [Training Data] [Test Data] Machine N M $ (! " , # " ) (! " ) 1) Train machine F to yield # " from ! " on the Training Data Output: # " 2) Predict # " of the Test data from ! " / ≈ # " .(! " ) = # "

Pr Prediction Bias ( ≈ ' $ • !(# $ ) = ' $ • Simple average ./1 Low Variance High Variance ' = 1 ( + , ' $ Low Bias $-./0 is not correct due to prediction bias • Prediction = TrueAnswer + Noise + Bias • ML prediction may have bias ( ≠ ' $ ' $ High Bias ( − ' $ Bias = ' $

Bi Bias Cor Correction on [Bias Correction Data] [Test Data] [Training Data] N t N b M (4 & , ! & ) (4 & , ! & ) (4 & ) • Average of predictions on test data with bias correction ()+ ( 0 )( 1 ! = 1 , + 1 , $ % ! & % ! & − ! & . / &'()* &'( 0 )* , + ! & − ! & , = ! & • Expectation value, ! = ! & • Training data should not overlap with bias correction data • Not efficient: small training/bias correction data

Bi Bias Cor Correction on – Cr Cros oss Validation on N-m m /,( = > ( ? + → > ( , ! + 9 = 1 /,@ = > @ ? + → > @ , ! + 9 = 2 /,A = > A ? + → > A , ! + 9 = 3 … … … • Average of predictions on test data with bias correction ) ,-. 4 ! = 1 1 /,& + 1 & − ! 3 /,& $ % * % ! + 2 % ! 3 &'( +',-( 3'( $ = 6/2 , 2 ≪ 6 • Full training data & precise bias estimation • Systematic error of ML prediction naturally included in error estimation

Pr Prediction of ! "#$ fr from ! %#$ Input: & ' = {* +,- 0 ≤ 0/2 ≤ 3 456 } * +,- 0 Boosted Decision Tree Regression ! A,S,T,V " * 8,- 0, > 9,;,<,= 0, > Output: * 8,-

De Deci cision T Tree R Regression Input: {! 0#$ 0 ≤ &/( ≤ 20 } % Output: ! "#$ 10, 5 % ! "#$ &/( = 10, -/( = 5

Boos Boosted Decision on Tree (BD BDT) • Iterative boosting ! " = [Simple DT ℎ " ] ! % = ! " + [Simple DT ℎ % that corrects residual error of ! " ] ! & = ! % + [Simple DT ℎ & that corrects residual error of ! % ] ! ' = ! & + [Simple DT ℎ ' that corrects residual error of ! & ] … ! ( = ! ()% + ℎ ( ! + = ! , -../0 (+) • In this study, 3 45567 = 200 − 500

' De Deci cision T Tree ℎ " fo for # $%& 10, 5

' De Deci cision T Tree ℎ " fo for # $%& 10, 5

' De Deci cision T Tree ℎ "# fo for $ "%& 10, 5

Pr Prediction of ! "#$ fr from ! %#$ 140 Genu − <C 3pt Genu > • Training and Test performed for Axial C 3pt Vector 120 Frequency Genu − C 3pt Pred 100 C 3pt - Clover-on-HISQ 80 - & = 0.089 fm, , - = 313 MeV 60 40 τ =10 - Measurements: 2263 confs ⨉ 64 srcs 20 t=5 0 • # of Training data: 400 confs -10 -5 0 5 10 -10 -5 0 5 10 A [ × 10 -16 ] V [ × 10 -16 ] # of Test data: 1864 confs C 3pt C 3pt 5 • Predictions of 1 234 10,5 / 1 934 10 140 Genu − <C 3pt Genu > Scalar C 3pt Tensor 120 Frequency Genu − C 3pt Pred 100 C 3pt 80 60 40 τ =10 20 t=5 0 -30 -15 0 15 30 -10 -5 0 5 10 S [ × 10 -16 ] T [ × 10 -16 ] C 3pt C 3pt

Pr Prediction of ! "#$ fr from ! %#$ (a) Train (b) Genuine (c) Pred.[2pt] (d) Pred.[2pt+3pt(12)] 1.260 1.240 1.220 u-d g A 1.200 1.180 τ = ∞ 1.055 1.050 1.045 u-d g V 1.040 1.035 1.030 τ =14 τ =10 1.025 τ =12 τ =8 1.020 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 t - τ /2 t - τ /2 t - τ /2 t - τ /2

Pr Prediction of ! "#$ fr from ! %#$ (a) Train (b) Genuine (c) Pred.[2pt] (d) Pred.[2pt+3pt(12)] 1.10 1.05 1.00 u-d 0.95 g S 0.90 0.85 0.80 τ =14 τ =10 τ =12 τ =8 1.15 u-d 1.10 g T 1.05 τ = ∞ 1.00 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 t - τ /2 t - τ /2 t - τ /2 t - τ /2

Pr Prediction of ! "#$ fr from ! %#$ • Results extrapolated to & → ∞ ! 1 Genuine Pred.[ C 2pt ] Pred.[ C 2pt + C 3pt (12)] g S 0.985(22) 1.013(30) 1.008(21) g A 1.2304(48) 1.2243(67) 1.2268(54) g T 1.0312(52) 1.0342(61) 1.0304(54) g V 1.0432(20) 1.0412(23) 1.0413(21) 2263 DM 400 DM 400 DM (Direct Meas.) + 1863 Pred. + 1863 Pred.

Qu Quark Ch Chromo omo EDM M (cE cEDM DM) • Simulation in presence of CPV cEDM interaction S = S QCD + S cEDM S cEDM = − i d 4 x ! seq" ∫ P" d q g s q ( σ ⋅ G ) γ 5 q P" 2 u" u" • Schwinger source method P ε" d" d" Include cEDM term in valence quark propagators P ε" d" d" by modifying Dirac operator D clov → D clov + i εσ µ ν γ 5 G µ ν seq" P )ε" P ε " • cEDM contribution to nEDM can be obtained u" u" P" by calculating vector form-factor F 3 with d" d" P" propagators including cEDM & O " # = %& ' % d" d"

%&' fr Pr Prediction of ! "#$ from ! "#$ • Predict 5 678 for cEDM and 9 : insertions from 5 678 without CPV ( /01234/ • CPV interactions è phase in neutron mass ;< = 9 = + ?@ A6BC- . D E = 0 • At leading order, H can be obtained from ≡ Tr 9 : M N I 5 678 M ( )*+, ( - .

%&' fr Pr Prediction of ! "#$ from ! "#$ • Training and Test performed for Input: 6,7 0 ≤ :/< ≤ 16 ]} - Clover-on-HISQ ( ) = {Re, Im[2 345 - < = 0.12 fm, K L = 305 MeV - Measurements: 400 confs ⨉ 64 srcs • # of Training data: 100 confs # of Test data: 300 confs Boosted Decision Tree 50 Genu − <C 2pt Genu > cEDM C 2pt γ 5 Regression 40 Frequency Genu − C 2pt Pred C 2pt 30 20 t=10 10 7 (BCDE, F G ) : Output: Im 2 345 0 -2 -1 0 1 2 -2 -1 0 1 2 P, cEDM [ × 10 -11 ] P, γ 5 [ × 10 -11 ] C 2pt C 2pt

%&' fr Pr Prediction of ! "#$ from ! "#$ • ( )*+, 0.065 cEDM 0.060 Genuine: 0.0527(16) α cEDM 0.055 Prediction: 0.0523(16) 0.050 0.045 • ( - . Genuine ML Prediction Genuine: -0.1462(14) -0.135 -0.140 Prediction: -0.1462(16) α γ 5 -0.145 -0.150 γ 5 Ø Genuine: DM on 400 confs -0.155 2 4 6 8 10 12 Ø Prediction: DM on 100 confs t + ML prediction on 300 confs

Su Summary • Machine learning is used to predict unmeasured observables from measured observables • Unbiased estimator using cross-validation is presented • Demonstrated for two lattice QCD calculations: 1) Prediction of ! "#$ from ! %#$ &'( from ! %#$ 2) Prediction of ! %#$ • The approach can be applied to various lattice calculations and reduce measurement cost