182
12 Application to Superstring Theory
Here η = η fin ∼ 0 is the cutoff near the horizon. In η → 0, only the first term of F
is important, so simply the horizon condition is as follows:
π = 0 .
(12.12)
That is, whether the input data is correct or not is determined by whether the value
of π is zero at the last layer of the neural network.
Of course, since this is a numerical experiment, training will not progress at all
if you request that the output π be completely zero. Therefore, a trick is needed to
judge that the data is correct when the value is close to 0. This can be done by setting
a threshold: for example, if the value of the magnitude of the output is less than 0.1,
it is considered to be 0. Also, the value of π in the final layer may be very large,
while considering that neural networks are good at binary classification problems,
it is also effective to restrict the value range by converting the value of π in the final
layer to tanh(π). With that modification, if “incorrect data” that does not reproduce
the one-point function is input, the training is made such that the output should be 1
instead of 0.
In this way, we have a method to solve the inverse problem of determining
the gravity metric from the quantum data of the boundary quantum field theory.
The important thing in this “transition” of the problem was not only to rewrite the
differential equation into a neural network form, but also to divide the differential
equation and the boundary value problem into unknown and given, so that the
unknown is the network weights, and the known data is the input and the output. If
such separation is possible, the inverse problem can be solved using the techniques
described in this chapter.
12.3 Emergent Spacetime on Neural Networks
Now, let us actually implement the above neural network on a computer and perform
a numerical experiment to see if spacetime metrics can be obtained by learning.
Here, we introduce two independent numerical experiments [118]. The first one is a
numerical experiment to see if a known metric can be reproduced. And the second
one is to answer the question: when a neural network is fed with experimental data
of a real material whose gravity side is not known at all, will a spacetime emerge? 8
Time and space emerging from QCD data [130] will be described in more detail
in the next section.
8 In physics, the term “emergent” refers to a phenomenon in which properties and equations that
are not expected from the degrees of freedom that define a physical system appear dynamically.
12 Application to Superstring Theory
Here η = η fin ∼ 0 is the cutoff near the horizon. In η → 0, only the first term of F
is important, so simply the horizon condition is as follows:
π = 0 .
(12.12)
That is, whether the input data is correct or not is determined by whether the value
of π is zero at the last layer of the neural network.
Of course, since this is a numerical experiment, training will not progress at all
if you request that the output π be completely zero. Therefore, a trick is needed to
judge that the data is correct when the value is close to 0. This can be done by setting
a threshold: for example, if the value of the magnitude of the output is less than 0.1,
it is considered to be 0. Also, the value of π in the final layer may be very large,
while considering that neural networks are good at binary classification problems,
it is also effective to restrict the value range by converting the value of π in the final
layer to tanh(π). With that modification, if “incorrect data” that does not reproduce
the one-point function is input, the training is made such that the output should be 1
instead of 0.
In this way, we have a method to solve the inverse problem of determining
the gravity metric from the quantum data of the boundary quantum field theory.
The important thing in this “transition” of the problem was not only to rewrite the
differential equation into a neural network form, but also to divide the differential
equation and the boundary value problem into unknown and given, so that the
unknown is the network weights, and the known data is the input and the output. If
such separation is possible, the inverse problem can be solved using the techniques
described in this chapter.
12.3 Emergent Spacetime on Neural Networks
Now, let us actually implement the above neural network on a computer and perform
a numerical experiment to see if spacetime metrics can be obtained by learning.
Here, we introduce two independent numerical experiments [118]. The first one is a
numerical experiment to see if a known metric can be reproduced. And the second
one is to answer the question: when a neural network is fed with experimental data
of a real material whose gravity side is not known at all, will a spacetime emerge? 8
Time and space emerging from QCD data [130] will be described in more detail
in the next section.
8 In physics, the term “emergent” refers to a phenomenon in which properties and equations that
are not expected from the degrees of freedom that define a physical system appear dynamically.