9.2 Representation of Hamiltonian Dynamical System
153
yields the following formula:
˙
q = w 11 q + w 12 p + g 1 (q) ,
(9.18)
˙
p = w 21 q + w 22 p + g 2 (p) .
(9.19)
In order for them to be Hamilton equations (9.12), their right-hand sides must satisfy
the symplectic relation:
∂
∂q
(w 11 q + w 12 p + g 1 (q)) +
∂
∂p
(w 21 q + w 22 p + g 2 (p)) = 0 .
(9.20)
However, this equation does not allow any nonlinear g i (x). Therefore, simply
equating the unit vector with (p, q) yields only a linear Hamilton equation.
So, let us use some ingenuity. We extend the relationship between units and
canonical variables, and also extend the relationship between time evolution and
transformation between layers. We shall assume the following relationship:
x i (t + t) =
J ij σ j (J jk x k (t)) .
(9.21)
This differs from (9.13) in two ways: First, x 1 = q and x 2 = p are the
same as before, but we add x 0 and x 3 , and set i, j, k = 0, 1, 2, 3. Second, it
introduces
J . This second point does not change the neural network. We simply
split J between the layers into two linear transformations and rewrote the first
linear transformation as a transformation working at the previous layer. Thus, the
following combination, the linear translation J → nonlinear local transformation σ
→ the linear transformation
J , is regarded as a t time translation.
In the neural network expanded like this, we choose sparse weights and local
activation functions as follows:
J =
⎛
⎜
⎜
⎝
0
0
v
0
0 1 + w 11 t w 12 0
0 w 21 1 + w 22 0
0
u
0
0
⎞
⎟
⎟
⎠ ,
J =
⎛
⎜
⎜
⎝
0 0 0 0
λ 1 1 0 0
0 0 1 λ 2
0 0 0 0
⎞
⎟
⎟
⎠ ,
(9.22)
⎛
⎜
⎜
⎝
σ 0 (x 0 )
σ 1 (x 1 )
σ 2 (x 2 )
σ 3 (x 3 )
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
f (x 0 ))t
1
1
g(x 3 ))t
⎞
⎟
⎟
⎠ .
(9.23)
Here (u, v, w ij ) (i, j = 1, 2) is a weight constant. The conceptual picture of this
neural network is shown in the right panel of Fig. 9.3.
153
yields the following formula:
˙
q = w 11 q + w 12 p + g 1 (q) ,
(9.18)
˙
p = w 21 q + w 22 p + g 2 (p) .
(9.19)
In order for them to be Hamilton equations (9.12), their right-hand sides must satisfy
the symplectic relation:
∂
∂q
(w 11 q + w 12 p + g 1 (q)) +
∂
∂p
(w 21 q + w 22 p + g 2 (p)) = 0 .
(9.20)
However, this equation does not allow any nonlinear g i (x). Therefore, simply
equating the unit vector with (p, q) yields only a linear Hamilton equation.
So, let us use some ingenuity. We extend the relationship between units and
canonical variables, and also extend the relationship between time evolution and
transformation between layers. We shall assume the following relationship:
x i (t + t) =
J ij σ j (J jk x k (t)) .
(9.21)
This differs from (9.13) in two ways: First, x 1 = q and x 2 = p are the
same as before, but we add x 0 and x 3 , and set i, j, k = 0, 1, 2, 3. Second, it
introduces
J . This second point does not change the neural network. We simply
split J between the layers into two linear transformations and rewrote the first
linear transformation as a transformation working at the previous layer. Thus, the
following combination, the linear translation J → nonlinear local transformation σ
→ the linear transformation
J , is regarded as a t time translation.
In the neural network expanded like this, we choose sparse weights and local
activation functions as follows:
J =
⎛
⎜
⎜
⎝
0
0
v
0
0 1 + w 11 t w 12 0
0 w 21 1 + w 22 0
0
u
0
0
⎞
⎟
⎟
⎠ ,
J =
⎛
⎜
⎜
⎝
0 0 0 0
λ 1 1 0 0
0 0 1 λ 2
0 0 0 0
⎞
⎟
⎟
⎠ ,
(9.22)
⎛
⎜
⎜
⎝
σ 0 (x 0 )
σ 1 (x 1 )
σ 2 (x 2 )
σ 3 (x 3 )
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
f (x 0 ))t
1
1
g(x 3 ))t
⎞
⎟
⎟
⎠ .
(9.23)
Here (u, v, w ij ) (i, j = 1, 2) is a weight constant. The conceptual picture of this
neural network is shown in the right panel of Fig. 9.3.