are held constant at the base values, and eqn 18 is reduced to
C p ¼ C p F
, t
, T
19a where
F ¼ {a ,
K
s
; q}
19b T ¼ {q
, e
, l
1
, k
1
, n}
19c where F ¼ subset of flow parameters; and T ¼ subset of
transport parameters. It should be noted that when eqn 18 is simplified to eqn 19, a number of flow and transport
parameters, L; b; v
s
; v
r
, r
b
, l
2
, C , and t
, are kept constant. This simplification is made to simplify studying the applic-
ability of ANN while considering the most significant flow and transport parameters such as a or K
s
. Also, the simpli- fication helps to make conclusive remarks on the overall
applicability by considering the effects of the few most significant parameters instead of all parameters. In the
next section, the applicability of ANN in simulating eqn 19 is assessed.
5.3 Training and testing subset
The flow and transport parameters are allowed to vary over the domain given by a [ [0.025, 0.100]; K
s
[ [15, 60]; q [ [5, 20]; e [ [0.8, 3.2]; l
1
[ [0.0005, 0.0020]; k
1
[ [0.15, 0.50]; and n [ [0.5, 1.0], and the training and testing subset,
S, is sampled from this domain. A subset of 100 realizations of x is sampled at random; C
for each realization is deter- mined using HYDRUS, and the 100 patterns are placed in
the training and testing subset, S.
5.4 Allocation method
In allocating S to a training subset, S
1
, and a testing subset, S
2
, a simple allocation method is applied pattern-by-pattern on S and this method is based on an user-defined expected
fraction, ˜f, where ˜f [ ,
1 . In allocating a pattern to S
1
or S
2
, a random number r [ 0, 1 is generated. If r ˜f, the pattern is allocated to S
1
; otherwise, it is allocated to S
2
. The sequence of r is simulated using a random number
generator initiated by a seed, r, and the allocation method becomes a function of ˜f and r. For a given S, the allocation
method is expected to allocate ˜f and 1 ¹ ˜f fractions of S to S
1
and S
2
, respectively, and this approach is expected to generate S
1
and S
2
of different or similar sizes for different values of ˜f and r.
5.5 Artificial neural network development
In this manuscript, ANNs are developed using the Neural Works Professional IIPlus, a commercially available soft-
ware package.
12
The internal parameters of the ANN include initial
weight distribution, transfer function, input–output scaling, training rate m, momentum factor
y, training rule, and the number of weight updates ¯ M.
ANN training is performed using the default values in the package, except the seed, r. A common r is used for
the allocation method and training for consistency. As such, the initial weights are randomly distributed over
[ ¹ 0.1, þ 0.1]. The transfer function used was sgm· or tanh·. The input–output scaling is performed using the
minimum and maximum values of each input and output component contained in S
1
; the input components are line- arly scaled over [ ¹ 1, þ 1], and the output components are
linearly scaled over [ þ 0.2, þ 0.8] or [ ¹ 0.8, þ 0.8] depending on the use of sgm· or tanh·, respectively.
12
The default values of m and y are shown in Table 1. The generalized delta training rule for BPA and ¯
M ¼ 50 000 are used. As such, the performance of BPA may be
expressed as h ¼ h r
, ˜f
, J
ÿ 20
where h ¼ performance of BPA.
5.6 Performance criteria
ANN is trained to approximate a vector function, G·, and the performances of ANN in approximating each ith
component, G
i
·, during training and testing phases need to be assessed. In the ANN training phase, the objective is to
match the desired response, d
i
, with the ANN response y
i
, at each ith output neuron for all the patterns in S
1
. As such, the performance of training in approximating G
i
· is assessed using the correlation coefficient defined as
R
i
¼ jd
i
y
i
j
d
i
j
y
i
j
d
i
. 0; j
y
i
. 0 21a
j
2 d
i
þ y
i
¼ j
2 d
2 i
þ 2j
d
i
y
i
þ j
2 y
2 i
21b where R
i
¼ correlation coefficient between d
i
and y
i
; j
d
i
¼ standard deviation of d
i
; j
y
i
¼ standard deviation of y
i
; and j
d
i
þ y
i
¼ standard deviation of d
i
þ y
i
. Finally, an average performance of the training phase in approximating G· is
assessed using the average correlation coefficient, R,
Table 1. Default values for training rate m and momentum factor y
Layer Parameter
Weight update 0–10 000
10 001–30 000 30 001–50 000
Hidden Training rate, m
0.30 0.150
0.03750 Momentum factor, y
0.40 0.200
0.05000 Output
Training rate, m 0.15
0.075 0.01875
Momentum factor, y 0.40
0.200 0.05000
ANN and algorithms in flow and transport simulations 151
defined as R ¼
1 K
X
K i ¼ 1
R
i
22 In addition, the performance of training is assessed using
scatter plots of y
i
versus d
i
, and the scatter of ðd
i
, y
i
from the 458 line is assessed using two error bounds defined as
y
a
¼ 1 ¹ e
d 23a
y
b
¼ 1 þ e
d 23b
where e ¼ specified error in decimal values; y
a
¼ lower
error bound corresponding to e; and y
b
¼ upper error bound
corresponding to e. A scatter plot helps to assess ANN performance more effectively compared to R. In the
ANN testing phase, the objective is to match d
i
with y
i
at each ith output neuron for all the patterns in S
2
. As such, the performance of testing in predicting G· is assessed by
repeating the above procedure for S
2
. 6 ARTIFICIAL NEURAL NETWORK ASSESSMENT
Several example scenarios related to the GFCT model are solved to assess the ANN applicability. In general, the accu-
racy of ANN training is observed to be approximately 100, and the generalization of ANN testing is observed
to limit the applicability of ANN. In the next section, the applicability of ANN with respect to only ANN testing is
discussed.
6.1 Example 1: simulation of the BTC with linear adsorption
