w
m ijk
¼ w
m ¹ 1 ijk
¼ m
]E
m p
]w
m ¹ 1 ijk
þ y w
m ¹ 1 jk
¹ w
m ¹ 2 jk
ÿ m [
, 1
, y [
[ ,
1 ]
ð 7bÞ
where m ¼ training rate and y ¼ momentum factor. Note y ¼ 0 for m ¼ 1. BPA continues updating ¯
w and w until
m ¼ ¯ M, where ¯
M is a user-defined number.
3.3 Genetic algorithm
Holland
20
presented the GA as a heuristic, probabilistic, combinatorial, search-based optimization technique pat-
terned after the biological process of natural evolution. Goldberg
21
discussed the mechanism and robustness of GA in solving nonlinear, optimization problems, and
Montana and Davis
22
applied GA quite robustly in ANN
training. In general, GA determines ¯ w and w in three
steps. First, GA samples an initial population of N random configurations of ¯
w and w from the weight space
defined as ¯
w
ij
[ [
¹ q
, þ
q ]
q . 0 8a
w
ijk
[ [
¹ q
, þ
q ]
q . 0 8b
where q ¼ weight bound. GA encodes each ¯ w and w con-
figuration in binary strings with L
s
bits L
s
¼ string
length and associates each string with a fitness value defined as
F
g s
¼ 1
E
g s
s ¼ 1 ,
2 , …,
N 9a
E
g s
¼ 1
2P X
P p ¼ 1
X
K i ¼ 1
y
g spi
¹ d
pi 2
y
g spi
¼ G ¯
w
g s
,
w
g s
,
x
p
ÿ 9b
where s ¼ string index; F
g s
¼ fitness for sth string after gth
generation; E
g s
¼ mean squared error for sth string after gth
generation; ¯ w
g s
¼ ¯
w in sth string after gth generation; w
g s
¼ w in sth string after gth generation; and y
g spi
¼ y
i
for
¯ w
g s
, w
g s
, and x
p
. As such, GA associates a higher fitness to strings with smaller errors. Second, GA starts updating this
initial population using S
1
for G generations. During an update from g ¹ 1 to g g ¼ generation index, GA per-
forms four operations: scaling, selection, crossover, and mutation: 1 GA scales linearly the fitnesses in the g ¹
1th population within an appropriate range using a scaling coefficient, C, where C is defined as the number of best
strings expected in the scaled population;
21
2 GA updates this population by selecting strings with a higher fitness
with a higher probability; 3 GA perturbs the resulting population by performing crossover with a probability of
p
c
; and 4 GA further perturbs the resulting population by performing mutation with a probability of p
m
. As such, GA evaluates NG þ 1 configurations, and the optimal config-
uration is searched from these configurations.
4 FLOW AND CONTAMINANT TRANSPORT IN THE UNSATURATED ZONE
4.1 Unsaturated flow
