entrapment of BPA at different local optimal solutions. For J ¼ {3, 5, 10}, ANN performance increases with increasing
J. As J increases, ANN acquires more freedom for approximating C
. Also, this observation suggests that J
opt
. 10 . J
HN
ð J
opt
¼ J at optimal performance and
J
HN
¼ J recommended by Hecht-Nielsen
3
for the present example.
In investigating this problem, an interesting phenomenon is noted; i.e. t is changing more rapidly than n. In these
scenarios, t is the primary, independent variable and is sup- posed to change more rapidly compared to other parameters
such as n, and the weights may fail adjusting to these inputs with disproportionate variations. Although this problem
may be handled to some extent by increasing J, this approach will lead to larger ANN, a larger optimization
problem, and a greater difficulty in training. Alternatively, an innovative ANN architecture may be considered. Instead
of simulating C
as a continuos function of t, ANN may be used to simulate C
as a discrete function of time. Notation- ally, the continuous function, C
¼ C
t, n, may be replaced by the discrete function, C
¼ {C
t
i
, n, i ¼ 1, 2,.., T}, and an 1-3-T ANN may be used to simulate this discrete func-
tion. As such, t is distributed across the output layer, and this approach resembles the concept of distributed input–output
representation.
6.3 Example 3: use of ANN to simulate BTC parameters
In the previous example, an innovative ANN architecture based on the concept of distributed input–output represen-
tation is introduced to simulate C ¼
C t, n. In the present
example, this innovative architecture is implemented. In typical groundwater remediation problems, the four key
parameters of the BTC are 1 breakthrough time, t
b
; 2 time to reach MCL, t
MCL
; 3 time to maximum concentra- tion, t
max
; and 4 maximum concentration, C
max
. If one can predict these four values, the prediction of the remaining
point values of C can be avoided.
In the present example, the applicability of the innovative ANN in simulating the four key parameters of the BTC is
assessed. These values are considered to be functions of flow parameters, F, and transport parameters, T, defined
by eqns 19b and 19c, respectively. The ANN application is divided into three parts: 1 effect of F; 2 effect of T; and
3 combined effect of F and T. In general, as there are seven inputs I ¼ 7 and four outputs J ¼ 4, a 7-J-4 ANN is used,
and J is determined, depending on the number of active input components. In studying the effect of F or T, the
respective input components in T or F are held constant at base values. However, this condition should not change the
Ð5
100 80
20
Predicted t
b
days
20 40
60 80
100 40
60
Desired t
b
days a
Ð5
100 80
20
Predicted t
MCL
days
20 40
60 80
100 40
60
Desired t
MCL
days b
+5
+5 Ð5
100 80
20
Predicted t
max
days
20 40
60 80
100 40
60
Desired t
max
days c
Ð5
16 14
2
Predicted C
max
µ gL
2 8
12 14
16 4
12
Desired C
max
µ gL
d +5
+5
6 8
10
4 6
10
Fig. 7. Performance of 7-7-4 ANN in Example 3: a breakthrough time; b time to reach MCL; c time to reach maximum concentration;
and d maximum concentration
154 J. Morshed, J. J. Kaluarachchi
results since ANN considers a constant input component to be a threshold weight, and ANN training adjusts other
weights to take this into account. In the next section, a subset of 100 realizations is sampled at random and C
for each realization is determined using HYDRUS. Thereafter,
the 100 patterns are placed in the corresponding training and testing subset, S. The allocation method with ˜f ¼ 0.5 and
r ¼ 1 is then used for allocating S to S
1
and S
2
. ANN used tanh· as the transfer function and trained using
BPA. At this stage, it may be noted that S of the present example is much smaller than those of the previous two
examples.
6.4 Effect of flow parameters
