154
9 Dynamical Systems and Neural Networks
Using this definition of time translation, the transformation between layers is
˙
q = w 11 q + w 12 p + λ 1 f (vp) ,
(9.24)
˙
p = w 21 q + w 22 p + λ 2 g(uq) .
(9.25)
The condition of the symplectic structure of the Hamilton equation is
w 11 + w 22 = 0
(9.26)
which can be easily satisfied. The corresponding Hamiltonian is
H = w 11 pq +
1
2
w 12 p
2
−
1
2
w 21 q
2
+
λ 1
v
F (vp) −
λ 2
u
G(uq) ,
(9.27)
and we chose here
F
(x 0 ) = f (x 0 ) , G
(x 3 ) = g(x 3 ) .
(9.28)
This is a nonlinear Hamiltonian with a deep neural network representation.
As an example, if we choose
w 11 = w 21 = 0 , w 12 = 1/m , λ 1 = 0 , λ 2 = 1 , u = 1 ,
(9.29)
the corresponding Hamiltonian is the Hamiltonian of a nonrelativistic particle
moving in an arbitrary potential:
H =
1
2m
p
2
− G(q) .
(9.30)
It is easy to see that if you devise F (p), you can also create a relativistic particle
Hamiltonian.
In order to construct, using a neural network, a general nonlinear Hamiltonian
in which both p and q are included as nonlinear functions, a network configuration
using a basis of linear combinations for the whole nonlinearity is required. Although
not described in detail here, various types of Hamiltonians can be built by
generalizing the network as described above. 7
In this chapter, we have seen that the time evolution of differential equations and
Hamilton equations can be reconstructed as data propagation over neural networks.
By using the method introduced here, the time evolution of physical systems can be
directly applied to the deep learning scheme. The main issue of machine learning is
generalization and solving inverse problems. If the problem of time evolution of a
7 Nonlinear Schrödinger equations can be constructed in a similar way.
9 Dynamical Systems and Neural Networks
Using this definition of time translation, the transformation between layers is
˙
q = w 11 q + w 12 p + λ 1 f (vp) ,
(9.24)
˙
p = w 21 q + w 22 p + λ 2 g(uq) .
(9.25)
The condition of the symplectic structure of the Hamilton equation is
w 11 + w 22 = 0
(9.26)
which can be easily satisfied. The corresponding Hamiltonian is
H = w 11 pq +
1
2
w 12 p
2
−
1
2
w 21 q
2
+
λ 1
v
F (vp) −
λ 2
u
G(uq) ,
(9.27)
and we chose here
F
(x 0 ) = f (x 0 ) , G
(x 3 ) = g(x 3 ) .
(9.28)
This is a nonlinear Hamiltonian with a deep neural network representation.
As an example, if we choose
w 11 = w 21 = 0 , w 12 = 1/m , λ 1 = 0 , λ 2 = 1 , u = 1 ,
(9.29)
the corresponding Hamiltonian is the Hamiltonian of a nonrelativistic particle
moving in an arbitrary potential:
H =
1
2m
p
2
− G(q) .
(9.30)
It is easy to see that if you devise F (p), you can also create a relativistic particle
Hamiltonian.
In order to construct, using a neural network, a general nonlinear Hamiltonian
in which both p and q are included as nonlinear functions, a network configuration
using a basis of linear combinations for the whole nonlinearity is required. Although
not described in detail here, various types of Hamiltonians can be built by
generalizing the network as described above. 7
In this chapter, we have seen that the time evolution of differential equations and
Hamilton equations can be reconstructed as data propagation over neural networks.
By using the method introduced here, the time evolution of physical systems can be
directly applied to the deep learning scheme. The main issue of machine learning is
generalization and solving inverse problems. If the problem of time evolution of a
7 Nonlinear Schrödinger equations can be constructed in a similar way.