“Machine learning landscape” workshop, ICTP Trieste, 10 Dec 2018 East Asian string workshop, KIAS, Seoul, 7 Nov 2018 KMI colloquium, Nagoya U, 25 Oct 2018 “QG meets la\ce QCD workshop”, ECT*, Trento, 3 Sep 2018 APCTP focus program, Hanyang U, Seoul, 15 Aug 2018 QFT workshop, YITP, Kyoto, 31 July 2018 Deep Learning and Holographic QCD � Koji Hashimoto (Osaka u) ArXiv:1802.08313, 1809.10536 w/ S. Sugishita (Kentucky), A. Tanaka (RIKEN), A. Tomiya (RIKEN)
0-1 � Solving inverse problem � Model � Metric � g µ ν Q Qbar potencal � La\ce QCD data: chiral condensate Prediccon VS quark mass � Deep Learning � Experiment Experiment data � data � [RBC-Bielefeld collaboracon, 2008] (Courtesy of W.Unger) [Petreczky, 2010]
0-2 � Discre1zed QG space1me? � Quantum gravity, discreczed � Causal dynamical triangulacon � [Ambjorn, Loll 1998] AdS/MERA � [Swingle 2009] HaPPY code � [Pastawski, Yoshida, Harlow, Preskill 2015]
0-3 � Neural network as a space1me � AdS/CFT [Maldacena ‘97] � Deep neural network � Black hole AdS (Emergent spacecme) CFT
1. Formulacon of AdS/DL correspondence � 2. Deeply learning QCD
1-1 � AdS/CFT: quantum response from geometry � [Klebanov, Wioen] � Classical scalar field theory in (d+1) dim. geometry � � � d d +1 x ( ∂ η φ ) 2 − V ( φ ) � � S = − det g ds 2 = − f ( η ) dt 2 + d η 2 + g ( η )( dx 2 1 + · · · + dx 2 d − 1 ) AdS boundary ( ) : � f ∼ g ∼ exp[2 η /L ] η ∼ ∞ Black hole horizon ( ) : � f ∼ η 2 , g ∼ const. η ∼ 0 Solve EoM, get response . Boundary condicons: � � O � J AdS boundary ( ) : � η ∼ ∞ 1 φ = Je − ∆ − η + � O � e − ∆ + η ∆ + � ∆ − Black hole horizon ( ) : � � ∂ η φ η =0 = 0 η ∼ 0 �
1-2 � Deep learning : op1mized sequen1al map � Layer 2 � Layer N � Layer 1 � x ( N ) x (1) x (2) i = ϕ ( W (1) ij x (1) j ) i i F = f i x ( N ) i W (1) ϕ ( x ) ij “Accvacon funccon” (fixed nonlinear fn.) � “Weights” (variable linear map) � { x (1) 1) Prepare many sets : input + output i , F } 2) Train the network (adjust ) by lowering W ij � � “Loss funccon” � � f i ( ϕ ( W ( N − 1) ϕ ( · · · ϕ ( W (1) � lm x (1) � � m )))) − F E ≡ � � ij � data
1-3 � Neural network of AdS scalar � η φ + h ( η ) ∂ η φ − δ V [ φ ] Bulk EoM � ∂ 2 = 0 δφ � � � metric � h ( η ) ≡ ∂ η log f ( η ) g ( η ) d − 1 Discreczacon, Hamilton form � φ ( η + ∆ η ) = φ ( η ) + ∆ η π ( η ) � � h ( η ) π ( η ) − δ V ( φ ( η )) π ( η + ∆ η ) = π ( η ) + ∆ η δφ ( η ) Neural-Network representacon � φ π � η =0 = 0 π � η η = 0 η = ∞
1-4 � Dic1onary of AdS/DL correspondence � AdS/CFT � Deep learning � Emergent space � Depth of layers � ∞ > η ≥ 0 i = 1 , 2 , · · · , N Bulk gravity metric � Network weights � W ( a ) h ( η ) ij Nonlinear response � Input data � x (1) � O � J i Horizon condicon � Output data � � ∂ η φ η =0 = 0 F � Interaccon � Accvacon funccon � ϕ ( x ) V ( φ )
1. Formulacon of AdS/DL correspondence � 2. Deeply learning QCD
0-3 � Solving inverse problem � Model � Metric � g µ ν Q Qbar potencal � La\ce QCD data: chiral condensate Prediccon VS quark mass � Deep Learning � Experiment Experiment data � data � [RBC-Bielefeld collaboracon, 2008] (Courtesy of W.Unger) [Petreczky, 2010]
2-1 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. ��
2-1 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Chiral condensate VS quark mass. � Pick up β=3.33 data � Posicve data β=3.30 ó T=196[MeV] [RBC-Bielefeld collaboracon, 2008] Negacve data � (Courtesy of W.Unger) ��
2-1 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Map it to asymptocc scalar configuracon. � [Klebanov, Wioen] [DaRold,Pomarol][Karch,Katz,Son,Stephanov] [Cherman,Cohen,Werbos] � � N c � � N c � 3 π qq � e − 3 η � λ 4 π m q e − η + η e − 3 η 2 � N c � ¯ φ = 4 π m q 2 • Conformal dimension of is 3. � ¯ qq � • Sub-leading contribucon, present. • Everything measured in unit of AdS radius. ��
2-2 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. asymptocc horizon � AdS � φ input m q � ¯ qq � π ( η = 0) π input Klebanov-Wioen Unspecified metric , h ( η ) decomposicon coupling (to be trained) � λ ��
2-2 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. φ input m q � ¯ qq � π ( η = 0) π input QCD la\ce data � ��
2-2 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. � � � f ( η ) g ( η ) d − 1 h ( η ) ≡ ∂ η log Learned value of (AdS radius) -1 : 1/L = 237(3)[MeV] bulk coupling : λ/L = 0.0127(6) � ��
2-3 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Q Qbar potencal � Learned metric � f ( η ) g ( η ) 3 Procedures η based on [Maldacena] [Rey,Theisen,Yee] � Bump � Quantum gravity effect? � Cf [Hyakutake 2014] �
2-3 � Deeply learning QCD � 1) Use a QCD data. 2) Let the network learn the metric. 3) Calculate other physical quancces. Q Qbar potencal � [Petreczky, 2010] [T.Ishikawa et al., 2008, CPPACS + JLQCD collaboracon]
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