Geology Reference
In-Depth Information
E
Τ
G G
G H G
Τ
W W
=
−
0
G
(10)
W W
=
−
0 0
G
(15)
0
0
0
0
G G
Τ
0 0
0 0 0
Eq.(10) can be used to implement an iterative
process which, however, may not converge. The
process can be stopped when the new calculated
error is greater tan the previous one. To improve
convergence, it is sometimes useful to advance
only a fraction of λ.
Eq.(15) can also be used to implement an
iterative process, and convergence is improved
if only a fraction of the step λ is applied. As a
simplification, the full Hessian matrix can be
replaced by using only its diagonal terms.
C. Combination of optimization approaches
B. Search for weights W trying to zero out the
gradient G
The gradient-free search OPT described in
4.2 can be utilized to obtain a first solution for
the optimum weights
W
. This approach can be
subsequently combined with either the strategy
described in a) or b), in an attempt to improve the
optimization results. This can be done after each
of the
NREP
repetitions of OPT.
The components of the gradient or of the Hes-
sian matrix are calculated numerically.
A zero gradient is associated with a minimum
value of the error
E
. Since, in general, the network
predictions will not all agree exactly with the input
data, searching for a minimum error is better than
searching for
E
= 0. In this case, the gradient
G
is
expressed as a linear function around the gradient
G
0
evaluated for the weights
W
0
,
G G
=
+
H W W
(
−
)
(11)
0
0
0
4. SEARCH-BASED OPTIMIZATION
FOR THE DESIGN PARAMETERS
in which
H
0
is now the Hessian matrix
The optimization process OPT described in 4.2
for the training of neural networks can also be
adapted to the problem of determining optimum
design parameters, minimizing the total cost and
satisfying the required minimum reliability con-
straints. Thus, the numerical procedure follows
the steps represented in Figure 2.
The starting point or anchor,
x
d
0
, is the result
of a preliminary design for the structure. This first
step normally utilizes deterministic, simpler
methods and may follow capacity design guide-
lines specified in the Codes. For this preliminary
choice of the design parameters, the neural net-
works for the reliability indices are used to estimate
the achieved levels
β
j
(
x
d
0
) and, using Eq.(1), the
total cost
C
(
x
d
0
), Figure 2(a). This Figure shows,
as a schematic illustration, the case of just two
design parameters, x
d
(1) and x
d
(2), which could
E
W W
∂
2
H
=
(12)
0
∂
∂
i j
i
j
W
0
The search is now for a new vector
W
-
W
0
,
also in the gradient direction, with the objective
of achieving
G
= 0. Thus,
(
W W
)
G
−
= −
λ
(13)
0
0
The step magnitude λ is now obtained from
G G
G H G
Τ
λ
=
0 0
(14)
Τ
0 0 0
giving a new
W
as
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