Transcript Document

Artificial Neural Networks:
An Alternative Approach to
Risk – Based Design
By George Mermiris
Introduction
Inspiration from the study of the human brain
and physical neurons
Response speed for physical neurons is 10-3 s
compared to electrical circuits with 10-9 s
Massive parallel structure: 1011 neurons with
104 connections per neuron
The efficiency of the brain is directly
dependent on the accumulated experience 
new connections are established which
determine our capabilities
The Biological Model
Dendrites
Synapses
Axon
Cell Body
Artificial Neural Networks (ANN):
Basic Forms, Feed-Forward Networks
General pattern:
n  w1,1p1  w1,2p2  ...  w1,R pR  b
a = f(n) = f(wp + b)
• p: input vector
• w: weight matrix
• b: bias vector
• n: net output of the
neuron
• : activation function
• a: output vector of the
network
a = f(n) = f(wp + b)
Multi-Neuron, Single-Layer ANN
a = f(n) = f(Wp + b)
Multi-Layer, Multi-Neuron Network
Abbreviated Form of a Network
Activation Functions
Linear
Function
f (x)  x
Log – Sigmoid
Function
1
f (x) 
1  e x
Hyperbolic
Tangent Sigmoid
Function
ex  ex
f (x)  x
e  e x
Training Neural Networks
The training of a network has the same concept as
for humans: the larger its experience the better its
response
For an ANN the learning is established with suitable
adjustment of its weights and biases
Requirements: training data and proper algorithm
The Backpropagation Algorithm
A three-fold concept
1. Performance Index: Approximate Square
Error: F(x) = (t - a)T(t – a) = eTe
The Steepest Descent Algorithm for function F
and modifications:
xk 1  xk  k gk g: gradient
F
w k 1  w k  
w
F
b k 1  b k  
b
The Backpropagation Algorithm
2. Chain Rule of Calculus:
F F n


w n w
F F n


b n b
3. Calculation of the first derivatives of the
performance index starting from the last layer
and backpropagating to the first (!)
Levenberg – Marquardt algorithm: Main variation
of the method based on the concept of Newton’s
method with small approximation
Example 1: Resistance Experiment
Case 1: 1 cm wave
amplitude
ANN Architecture:
1-4-3-1
Activation Function:
Log – Sigmoid for
hidden layers and
Linear for output
layer
Example 1: Resistance Experiment
Example 1: Resistance Experiment
Case 2: 2 cm wave
amplitude
ANN Architecture:
1-3-2-1
Activation Function:
Log – Sigmoid for
hidden layers and
Linear for output
layer
Example 1: Resistance Experiment
Example 2: Section Areas Curve
Input: L, Amax, , LCB, Cp
ANN Architecture: 5-10-12-21
Activation Function: Log – Sigmoid for hidden layers and
Linear for output layer
b
(101)
51
+
W
(1210)
b
(121)
+
a (211)
W
(2110)
b
(211)
+
purelin
W
(105)
a (121)
logsig
p
logsig
a (101)
Example 2: Section Areas Curve
-
Training Set
L=[153 156 159 … 180], in m
Amax=[335 345 355 … 425], in m2
=[36000 37000 38000 … 45000] , in m3
LCB=[-2.4 –2.5 –2.6 … -3.3], in m
Cp=[0.702 0.688 0.660 …0.588]
Ordinates of SA curves for each combination
Generalisation Sets [L Amax  LCB Cp]
- Set1=[160 360 38500 –2.65 0.6664]
- Set2=[178.5 420 44500 –3.25 0.594]
- Set3=[150 325 35000 –2.3 0.718]
“Network input”
“Network output”
“Testing the network”
Example 2: Section Areas Curve
(Set1)
Example 2: Section Areas Curve
(Set2)
Example 2: Section Areas Curve
(Set3)
Strong points of ANN
1. Readily applicable to any stage of the design
2.
3.
4.
5.
process, especially at the preliminary design where
rough approximations are necessary
Potential to include different design parameters in
the training set and avoid iterations
Results are obtained very fast with high accuracy
No highly sophisticated mathematical technique is
involved, only basic concepts of Linear Algebra and
Calculus
Very short computer times in common PC’s
Weak points of ANN
1. Basic requirement is the existence of historical data
2.
3.
4.
for the creation of training set
Not readily applicable to novel ship types
The results are very sensitive to the network’s
architecture and the training method selected each
time, although these two parameters are very
easily adjusted
There is no specific network architecture for a
specific calculation: different architectures can
provide the same results. The general rule is to use
the simplest possible network
Future Work
Other networks and training algorithms: recurrent ANN
Suitable database for creating the training set for
different applications
Application to the Global Ship Design including Risk
Data and Human Reliability Data
Thank You!