186
12 Application to Superstring Theory
the data of (H, M) of matter can be converted to (φ, π). However, in the present
case, the fact that the AdS radius R of the matter is unknown, and the normalization
of the operators that must usually be determined from the two-point function of M
is unknown, we shall reluctantly put it as follows:
φ(η ini ) = αH + βM ,
π(η ini ) = − − αH − + βM .
(12.15)
Here α and β are unknown normalization constants, and
± = (d/2)
1 ±
1 + 4m 2 /h(∞) 2
(12.16)
determines the conformal dimension of the operator (d/ h(∞) gives the AdS radius).
On the numerical code, everything is measured in the unit R unit = 1. We consider
this (12.15) as the 0th layer and add it to the original neural network made of h(η).
The input data and output data are handled in the same way as in the previous
AdS Schwarzschild spacetime reproduction experiment, and the weight of the neural
network is trained. 11 In the 10-layer neural network, the parameters to be trained in
this case are the bulk mass m, λ in the interaction potential V = (λ/4)φ 4 , the
normalization constant α and β, and the metric h(η).
Figure 12.6 shows the training result. The left and middle figures show the fitting
of the positive data after the training and the emergent metric h(η), respectively. You
can see that a smooth emergent metric is obtained. 12 The figure on the right shows
statistical data from 13 trials. We can see how it converges to a certain function form.
At the same time, the value of the bulk mass and the strength of the interaction are
learned. The result is m 2 R 2 = 5.6 ± 2.5 and λ/R = 0.61 ± 0.22.
In this way, by using a deep neural network, the inverse problem for AdS/CFT
can be solved. From the data of a given material, interesting questions await, such
as what properties the emergent spacetime has and how generally it can be used for
AdS/CFT correspondence.
11 In this numerical experiment, we perform the training with two regularization terms: L reg =
L
(1)
reg + L
(2)
reg . The first one L
(1)
reg is the same as the previous (12.14). For the second term, we
introduced the regularization
L
(2)
reg ≡ c
(2)
reg (h(η
(N) ) − 1/η
(N) )
2 .
(12.17)
This makes h(η (N) ) behave like the horizon condition h(η) ∼ 1/η near the black hole horizon.
Examining the size of the regularization term so as not to hinder the training, we adopted the value
c
(2)
reg = 10 −4 as its coefficient.
12 We stopped the training when the total error function went below 0.02. This is based on the
judgment that the training result is sufficiently close to the training data.
12 Application to Superstring Theory
the data of (H, M) of matter can be converted to (φ, π). However, in the present
case, the fact that the AdS radius R of the matter is unknown, and the normalization
of the operators that must usually be determined from the two-point function of M
is unknown, we shall reluctantly put it as follows:
φ(η ini ) = αH + βM ,
π(η ini ) = − − αH − + βM .
(12.15)
Here α and β are unknown normalization constants, and
± = (d/2)
1 ±
1 + 4m 2 /h(∞) 2
(12.16)
determines the conformal dimension of the operator (d/ h(∞) gives the AdS radius).
On the numerical code, everything is measured in the unit R unit = 1. We consider
this (12.15) as the 0th layer and add it to the original neural network made of h(η).
The input data and output data are handled in the same way as in the previous
AdS Schwarzschild spacetime reproduction experiment, and the weight of the neural
network is trained. 11 In the 10-layer neural network, the parameters to be trained in
this case are the bulk mass m, λ in the interaction potential V = (λ/4)φ 4 , the
normalization constant α and β, and the metric h(η).
Figure 12.6 shows the training result. The left and middle figures show the fitting
of the positive data after the training and the emergent metric h(η), respectively. You
can see that a smooth emergent metric is obtained. 12 The figure on the right shows
statistical data from 13 trials. We can see how it converges to a certain function form.
At the same time, the value of the bulk mass and the strength of the interaction are
learned. The result is m 2 R 2 = 5.6 ± 2.5 and λ/R = 0.61 ± 0.22.
In this way, by using a deep neural network, the inverse problem for AdS/CFT
can be solved. From the data of a given material, interesting questions await, such
as what properties the emergent spacetime has and how generally it can be used for
AdS/CFT correspondence.
11 In this numerical experiment, we perform the training with two regularization terms: L reg =
L
(1)
reg + L
(2)
reg . The first one L
(1)
reg is the same as the previous (12.14). For the second term, we
introduced the regularization
L
(2)
reg ≡ c
(2)
reg (h(η
(N) ) − 1/η
(N) )
2 .
(12.17)
This makes h(η (N) ) behave like the horizon condition h(η) ∼ 1/η near the black hole horizon.
Examining the size of the regularization term so as not to hinder the training, we adopted the value
c
(2)
reg = 10 −4 as its coefficient.
12 We stopped the training when the total error function went below 0.02. This is based on the
judgment that the training result is sufficiently close to the training data.