16
K. Kakuda et al.
Fig. 1. Particle interaction models (3D).
and W (r) = 0 for q ≥ 3, where q = 3r/h, σ = 63/478π h 2 and 9/40π h 3 for 2D and
3D, respectively. The particle form at particle a for the pressure gradient term in Eq. (1)
can be written as
−
1
ρ
∇p
a
= −
b
m b
p a
ρ 2
a
+
p b
ρ 2
b
+ Π ab
∇ a W ab
(6)
where ∇ a stands for the gradient taken with respect to the coordinates of particle a, and
Π ab denotes the numerical viscosity. On the other hand, the second term of right-hand
side, namely, the viscous term in Eq. (1) is discretized as
ν∇
2 u a =
b
m b
ρ a ρ b
(μ a + μ b )u ab
r ab · ∇ a W ab
r 2
ab
(7)
4 CNN Architectures
In this section, we construct the latent space network [3] based on the deep CNN using
some datasets, namely velocities and the reciprocal sum of the distance between particles,
obtained from the SPH simulations. Following to Reference [3], the learning procedure
in this study is given as
Step 1: As illustrated in Fig. 2(a), the input datasets such as velocity vectors and so on are
compressed by the encoder and converted to latent codes in the autoencoder. The CNN
architecture of the encoder consists of 17 convolutional layers with one fully-connected
tanh layer (see Fig. 2(b)).
Step 2: The latent codes as input are reconstructed by the decoder, and stored as output
data. For the decoder, we adopt the CNN architecture consisting of 13 convolutional
layers (see Fig. 2(c)). By repeating these two steps, the velocity fields etc. using the
so-called autoencoder are obtained as shown in Fig. 2.
K. Kakuda et al.
Fig. 1. Particle interaction models (3D).
and W (r) = 0 for q ≥ 3, where q = 3r/h, σ = 63/478π h 2 and 9/40π h 3 for 2D and
3D, respectively. The particle form at particle a for the pressure gradient term in Eq. (1)
can be written as
−
1
ρ
∇p
a
= −
b
m b
p a
ρ 2
a
+
p b
ρ 2
b
+ Π ab
∇ a W ab
(6)
where ∇ a stands for the gradient taken with respect to the coordinates of particle a, and
Π ab denotes the numerical viscosity. On the other hand, the second term of right-hand
side, namely, the viscous term in Eq. (1) is discretized as
ν∇
2 u a =
b
m b
ρ a ρ b
(μ a + μ b )u ab
r ab · ∇ a W ab
r 2
ab
(7)
4 CNN Architectures
In this section, we construct the latent space network [3] based on the deep CNN using
some datasets, namely velocities and the reciprocal sum of the distance between particles,
obtained from the SPH simulations. Following to Reference [3], the learning procedure
in this study is given as
Step 1: As illustrated in Fig. 2(a), the input datasets such as velocity vectors and so on are
compressed by the encoder and converted to latent codes in the autoencoder. The CNN
architecture of the encoder consists of 17 convolutional layers with one fully-connected
tanh layer (see Fig. 2(b)).
Step 2: The latent codes as input are reconstructed by the decoder, and stored as output
data. For the decoder, we adopt the CNN architecture consisting of 13 convolutional
layers (see Fig. 2(c)). By repeating these two steps, the velocity fields etc. using the
so-called autoencoder are obtained as shown in Fig. 2.