144
8 Detection of Phase Transition by Machines
Fig. 8.2 Heat map of weight J adjacent to the final layer of the neural network after the training.
(Excerpt from [108])
Table 8.1 The top three rows show the phase transition temperature values detected by the neural
network. The values of their inverse temperature β are given. The limit L → ∞ corresponds to
an infinite volume, where the exact inverse temperature of the phase transition is known: β Exact
c
=
1
2 log(
√
2 + 1). “CNN” is a convolutional neural network, and “FC” is a fully connected neural
network
System size
β c CNN
β c FC
8×8
0.478915
0.462494
16×16
0.448562
0.433915
32×32
0.451887
0.415596
L → ∞
β Exact
c
∼ 0.440686
of (8.9) that repeats linear and nonlinear transformations twice:
f θ,ϕ (x) = σ (θ z) , z = σ (ϕs) .
(8.9)
Here s is a vector-like arrangement of the spin configuration s. The activation
function σ used is a softmax function, and its definition is
[σ (z)] I =
e z I
J e z J
.
(8.10)