Deep Learning and AdS/CFT

Transcription

1 KIAS, 26 March, 2018 Cquest, Sogang u., 29 March, 2018 MIT, CTP, 4 Apr, 2018 MPI, AEI, 13 Apr, 2018 HET group, Osaka, 30 May, 2018 DLAP2018 workshop, 1 June, 2018 Deep Learning and AdS/CFT Koji Hashimoto (Osaka u) ArXiv: w/ S. Sugishita (Osaka), A. Tanaka (RIKEN AIP), A. Tomiya (CCNU)

2 Brane (Superstring theory) Brain (Neuroscience) 2

3 Deep Learning cat AdS/CFT [Maldacena 97] CFT AdS Black hole

4 1. Formula`on of AdS/DL correspondence 2. Implementa`on of AdS/DL and emerging space 4

5 1. Formula`on of AdS/DL correspondence 1-1 review review Solving inverse problem AdS/CFT: quantum response from geometry Deep learning: op=mized sequen=al map From AdS to DL Dic=onary of AdS/DL correspondence

6 1-1 Solving inverse problem AdS/CFT (No proof, no deriva`on) Classical gravity in d+1 dim. space`me Conven`onal holographic modeling Model Metric g µ Quantum field theory in d dim. space`me (Strong coupling limit) Predic`on Experiment data Comparison Predic`on Experiment data

7 1-1 Solving inverse problem Our deep learning holographic modeling Model Metric g µ Conven`onal holographic modeling Model Metric g µ Predic`on Predic`on Predic`on Comparison Experiment data Experiment data Experiment data Experiment data

8 review AdS/CFT: quantum response from geometry Classical scalar field theory in (d+1) dim. geometry S = d d+1 x det g ( ) 2 V ( ) [Klebanov, Wiken] ds 2 = f( )dt 2 + d 2 + g( )(dx dx 2 d 1) AdS boundary ( ) : f g exp[2 /L] Black hole horizon ( 0 ) : f 2,g const. Solve EoM, get response O J AdS boundary ( ) : = Je + Black hole horizon ( 0 ) :. Boundary condi`ons: + 1 O e + =0 =0 8

9 review Deep learning : op=mized sequen=al map Layer 1 Layer 2 Layer N x (1) i x (2) i = (W (1) ij x(1) j ) x (N) i F = f i x (N) i W (1) ij (x) Ac`va`on func`on (fixed nonlinear fn.) Weights (variable linear map) 1) Prepare many sets i,f} : input + output 2) Train the network (adjust W ij ) by lowering (N 1) Loss func`on E f i ( (W {x (1) data ij ( (W (1) lm x(1) m )))) F

10 1-2 From AdS to DL Bulk EoM 2 + h( ) V [ ] =0 Discre`za`on, Hamilton form ( + )= ( )+ ( ) ( + )= ( )+ h( ) ( ) Neural-Network representa`on metric h( ) log f( )g( ) d 1 V ( ( )) ( ) ( = 0) = =0 10

11 1-2 From AdS to DL Bulk EoM 2 + h( ) V [ ] =0 Discre`za`on, Hamilton form ( + )= ( )+ ( ) ( + )= ( )+ h( ) ( ) Neural-Network representa`on metric h( ) log f( )g( ) d 1 V ( ( )) ( ) = =0 =0 =0 11

12 1-3 Dic=onary of AdS/DL correspondence AdS/CFT Emergent space Deep learning Depth of layers > 0 i =1, 2,,N Bulk gravity metric Network weights h( ) W (a) ij Nonlinear response O J Horizon condi`on =0 =0 Interac`on V ( ) Input data x (1) i Output data F Ac`va`on func`on (x) 12

13 1. Formula`on of AdS/DL correspondence 1-1 review review Solving inverse problem Deep learning : op=mized sequen=al map AdS/CFT: quantum response from geometry From AdS to DL Dic=onary of AdS/DL correspondence

14 1. Formula`on of AdS/DL correspondence 2. Implementa`on of AdS/DL and emerging space 14

15 2. Implementa`on of AdS/DL and emerging space Emergent geometry in deep learning Can AdS Schwarzschild be learned? Emergent space from real material? Numerical experiment summary Machines learn, what do we learn?

16 2-1 Emergent geometry in deep learning Experiment 1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. Experiment 2: Emergent space from real material? 1) Use material experimental data. Ex) Magne`za`on curve of strongly correlated material 2) 3) (same as above.) 4) Watch how space emerges! 16

17 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. 2 + h( ) V [ ] =0 h( ) = 3 coth(3 ) V [ ]= AdS Schwarzschild metric in the unit of AdS radius L =

18 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. ( + )= ( )+ ( ) ( + )= ( )+ h( ) ( ) V ( ( )) ( ) = =0 =0 =0 18

19 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. input Horizon condi`on : true : false input 19

20 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. input input Unspecified metric (10 layers, to be trained) =0 =0 Generated data from AdS Schwarzschild (10000 data points) 20

21 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. 21

22 2-2 Exp1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. With a regulariza`on 22

23 2-1 Emergent geometry in deep learning Experiment 1: Can AdS Schwarzschild be learned? 1) Use AdS Schwarzschild and generate input data. 2) Prepare network with unspecified metric. 3) Let the network learn it by the data. 4) Check if AdS Schwarzschild is reproduced. Experiment 2: Emergent space from real material? 1) Use material experimental data. Ex) Magne`za`on curve of strongly correlated material 2) 3) (same as above.) 4) Watch how space emerges! 23

24 2-3 Exp2: Emergent space from real material? 1) Use material experimental data. Ex) Magne`za`on curve of strongly correlated material 2) 3) (same as above.) 4) Watch how space emerges! 24

25 2-4 Numerical experiment summary Experiment 1 AdS Schwarzschild is successfully learned. Experiment 2 Experimental data is explained by emergent space. 25

26 2-5 Machines learn, what do we learn? Conven&onal holographic modeling Model Our deep learning holographic modeling Model Metric g µ Metric g µ Predic&on Predic&on Predic&on Comparison Experiment data Experiment data Experiment data Experiment data 26

27 1. Formula`on of AdS/DL correspondence 2. Implementa`on of AdS/DL and emerging space 27