Artificial Neural Networks:

An Alternative Approach toRisk – Based Design

By George Mermiris

Introduction

Inspiration from the study of the human brain and physical neuronsResponse speed for physical neurons is 10-3 s compared to electrical circuits with 10-9 sMassive parallel structure: 1011 neurons with 104 connections per neuronThe efficiency of the brain is directly dependent on the accumulated experience new connections are established which determine our capabilities

The Biological Model

Cell Body

Dendrites

Synapses

Axon

Artificial Neural Networks (ANN):Basic Forms, Feed-Forward Networks

a = f(n) = f(wp + b)

1,1 1 1,2 2 1,R Rn w p w p ... w p b

a = f(n) = f(wp + b)

General pattern:

• p: input vector• w: weight matrix• b: bias vector• n: net output of the neuron• : activation function• a: output vector of the network

Multi-Neuron, Single-Layer ANN

a = f(n) = f(Wp + b)

Multi-Layer, Multi-Neuron Network

Abbreviated Form of a Network

Activation Functions

xe11)x(f

xx

xx

eeee)x(f

f (x) x

Linear Function

Log – Sigmoid Function

Hyperbolic Tangent Sigmoid Function

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 concept1. Performance Index: Approximate Square

Error: F(x) = (t - a)T(t – a) = eTeThe Steepest Descent Algorithm for function F and modifications:

k 1 kFw ww

k 1 kFb bb

k 1 k k k x x g g: gradient

The Backpropagation Algorithm2. Chain Rule of

Calculus:

3. Calculation of the first derivatives of the performance index starting from the last layer and backpropagating to the first (!)

F F nw n w

F F nb n b

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

pW

(105)

51

b

(101)

+

logs

ig

W (121

0)b

(121)

logs

ig

+

W (211

0)b

(211)

pure

lin

+

a (101)

a (121)

a (211)

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”

“Testing the network”

“Network output”

Example 2: Section Areas Curve (Set1)

Example 2: Section Areas Curve (Set2)

Example 2: Section Areas Curve (Set3)

1. Readily applicable to any stage of the design process, especially at the preliminary design where rough approximations are necessary

2. Potential to include different design parameters in the training set and avoid iterations

3. Results are obtained very fast with high accuracy4. No highly sophisticated mathematical technique

is involved, only basic concepts of Linear Algebra and Calculus

5. Very short computer times in common PC’s

Strong points of ANN

1. Basic requirement is the existence of historical data for the creation of training set

2. Not readily applicable to novel ship types3. 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

4. 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

Weak points of ANN

Other networks and training algorithms: recurrent ANNFuture Work

Suitable database for creating the training set for different applicationsApplication to the Global Ship Design including Risk Data and Human Reliability Data

Thank You!