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# Neural networks

## 1. Kazan National Research Technical University named after A.N. Tupolev German-Russian Institute of Advanced Technologies (GRIAT)

NEURAL NETWORKSby Dr. Igor Anikin

## 2. Table of contents

The basic concepts of neural networksSingle layer neural networks

Artificial neural networks.

The structure of an artificial neuron.

Activation functions.

Basic paradigms of neural networks.

Fundamentals of learning and training samples.

Using neural networks in practice

Rosenblatt's single layer perceptron.

Learning single layer neural networks.

Associative memory and its realization on single layer neural networks.

Using single layer neural networks for pattern recognition and time

series forecasing.

Multilayer perceptrons

The structure of multilayer perceptrons

Back propagation of error.

Using multilayer perceptrons for pattern recognition and time series

forecasing.

## 3.

Self-organizing mapsThe principle of unsupervised learning.

Kohonen self-organizing maps.

Learning Kohonen networks.

Practical using of Kohonen networks

Recurent neural networks

Neural networks with feedback.

Hopfield neural network.

Hamming neural network.

Training Hopfield and Hamming neural networks.

Practical using of Hopfield and Hamming neural networks.

Training and Testing

Training error and testing error.

## 4. References

1.2.

3.

4.

David Kriesel.

Neural

A

brief Introduction

networks

to

//

http://www.dkriesel.com/en/science/neural_net

works

Raul Rojas. Neural Networks. A Systematic

Introduction

//

http://www.inf.fu-berlin.de/inst/ag-ki/rojas_

home/documents/1996/NeuralNetworks/neuron.pdf

.

L.P.J. Veelenturf. Analysis and Application of

Artificial

Neural

Networks

//

http://www.ru.lv/~peter/zinatne/ebooks/Anal

ysis%20and%20Applications%20of%20Artificial

%20Neural%20Networks.pdf

Artificial Neural Networks – Methodological

Advances and Biomedical Applications //

## 5.

The basic concepts ofneural networks

## 6. Questions for motivation discussion

What tasks are machines good at doing thathumans are not?

What tasks are humans good at doing that

machines are not?

What tasks are both good at?

What does it mean to learn?

How is learning related to intelligence?

What does it mean to be intelligent?

Do you believe a machine will ever been

intelligent?

If a computer were intelligent, how would

you know?

## 7. Types of learning

Knowledge acquisition from expert.Knowledge acquisition from data:

Supervised learning – the system is supplied

with a set of training examples consisting of

inputs and corresponding outputs, and is

required to discover the relation or mapping

between them.

Unsupervised learning – the system is

supplied with a set of training examples

consisting only of inputs. It is required to

discover what appropriate outputs should be.

## 8. Artificial Neural Network

An extremely simplified model of thehuman’s brain

Transforms inputs into the best outputs

(some neural networks are the universal

function approximators).

## 9. Artificial Neural Networks

Development of Neural Networks date back to the early1940s.

It experienced an upsurge in popularity in the late 1980s

due to discovery of new techniques of NN training.

Some NNs are models of biological neural networks and

some are not, but historically, much of the inspiration for the

field of NNs came from the desire to produce artificial

systems capable of sophisticated, perhaps intelligent,

computations similar to those that the human brain

routinely performs, and thereby possibly to enhance our

understanding of the human brain.

Most NNs have some sort of training rule. In other words,

NNs learn from the examples (as children learn to recognize

dogs from examples of dogs) and exhibit some capability for

generalization beyond the training data.

## 10. ANN vs Computers

Computershave

programmed

to

be

explicitly

Analyze the problem to be solved.

Write the code in a programming language.

Neural networks learn from the examples

No requirement of an explicit description of the problem.

No need for a programmer.

The neural computer adapts itself during a training

period, based on examples of similar problems even

without a desired solution to each problem. After

sufficient training the neural computer is able to relate

the problem data to the solutions, inputs to outputs, and

it is then able to offer a viable solution to a brand new

problem.

## 11. ANN vs Computers

Digital ComputersDeductive Reasoning. We

apply known rules to input

data to produce output.

Computation

is

centralized, synchronous,

and serial.

Memory is literally stored,

and location addressable.

Not fault tolerant. One

transistor goes and it no

longer works.

Exact.

Static connectivity.

Applicable if well-defined

rules

accessible

with

precise input data.

Neural Networks

Inductive

Reasoning. We use

given input and output data

(training examples) to make a

reasoning.

Computation

is

collective,

asynchronous, and parallel.

Memory

is

distributed,

internalized, short term and

content addressable.

Fault tolerant, redundancy, and

sharing of responsibilities.

Inexact.

Dynamic connectivity.

Applicable if rules are unknown

or complicated, or if data are

noisy or partial.

## 12. Biological neuron

## 13. Biological neuron

Many“neurons”

co-operate

perform the desired function

Basic elements:

Axon

Dendrite

Synapse

to

## 14. Artificial Neuron Structure

The output of a neuron is a function of theweighted sum of the inputs plus a bias

n

S x j w j b,

j 1

y f (S )

## 15. Common activation functions

## 16.

## 17. Examples of ANN topologies

Single layer ANNMultilayer ANN

ANN with one recurrent layer

## 18. Fundamentals of learning and training samples

The weights in a neural network are themost important factor in determining its

function.

A training set is a set of training patterns,

which we use to train our neural net.

Training is the act of presenting the

network with some sample data and

modifying

the

weights

to

better

approximate the desired function

## 19. Fundamentals of learning and training samples

There are two main types of trainingSupervised Training

Supplies the neural network with inputs and the

correct outputs (results).

We can estimate a error vector for certain input.

Response of the network to the inputs is measured.

The weights are modified to reduce the difference

between the actual and desired outputs

Unsupervised Training

The training set only consists of input patterns.

The neural network adjusts its own weights so that

similar inputs cause similar outputs. The network

identifies the patterns and differences in the inputs

without any external assistance

## 20. Fundamentals of learning and training samples

A training pattern is an input vector pwith the components x1, x2, . . . , xn

whose desired output is known.

By entering the training pattern into

the network we receive an output that

can be compared with the desired

output.

The set of training patterns is called P.

It contains a finite number of ordered

pairs (p, t) of training patterns with

corresponding desired output t.

## 21. Fundamentals of learning and training samples

Teaching input. Let j be an output neuron.The teaching input tj is the desired and

correct value j should output after the input of

a certain training pattern.

Analogously to the vector p the teaching

inputs t1, t2, . . . , tn of the neurons can also be

combined into a vector t. This vector always

refers to a specific training pattern p and

contained in the set P of the training patterns.

## 22. Fundamentals of learning and training samples

Error vector. For several output neuronsΩ1,Ω2, . . . ,ΩO the difference between

output vector and teaching input under a

training input p is referred to as error

vector.

t1 y1

E p ...

t y

O

O

## 23. Fundamentals of learning

Let P be the set of training patters. Inlearning procedure we realize finite

number of iterations or epochs.

Epoch – single presentation of the

entire data to the neural network.

Typically many epochs are required to

train the neural network

Iteration - the process of providing the

network with an single input and

updating the network's weights

## 24. General learning procedure

Let P be the set of n training patters pnFor i=1 to n

begin

1.

2.

We calculate NN output vector yi for the training

pattern pi.

We compare yi with desired output ti. Then we

calculate the error of output and make modification

of weights.

end

3.

If total error for the training set P more

than some threshold then go to the step 2

## 25. Using training samples

We have to divide the set of training samplesinto two subsets:

one training set really used to train;

one verification set to test our progress of learning.

The usual division relations are, 70% for

training data and 30% for verification data

(randomly chosen).

We can finish the training process when the

network provides the good results on the

training data as well as on the verification

data.

## 26. Learning curve

The learning curve indicates the progressof the error, which can be determined in

various ways. This curve can indicate

whether the network is progressing or not.

## 27. Error measurement

Let Ω be the output neuron and O be theset of output neurons.

The specific error Errp is based on a single

training sample.

The total error Err is based on all training samples.

Err

Errp

p P

## 28. When do we stop learning?

Generally, the training process isstopped when the user in front of the

learning computer "thinks" the error is

small enough.

## 29. Using neural networks in practice (discussion)

Classificationin marketing: consumer spending pattern classification

In defence: radar and sonar image classification

In medicine: ultrasound and electrocardiogram image

classification, EEGs, medical diagnosis

Recognition and identification

In general computing and telecommunications : speech, vision and

handwriting recognition

In finance: signature verification and bank note verification

Assessment

In engineering: product inspection monitoring and control

In defence: target tracking

In security: motion detection, surveillance image analysis and

fingerprint matching

Forecasting and prediction

In finance: foreign exchange rate and stock market forecasting

In agriculture: crop yield forecasting

In marketing: sales forecasting

In meteorology: weather prediction

## 30.

Single layer neural networks## 31. Single layer network with binary threshold activation function

ny j F S j F wij xi T j

i 1

w11

w21

W

...

w

n1

Matrix form

S W T X T

w12

w22

...

wn 2

... w1m

... w2 m

... ...

... wnm

## 32. Single layer network with binary threshold activation function

1, S 0y

0, S 0

S w11 x1 w21 x2 T1

w11 x1 w21 x2 T1 0

x2

T1 w11

x1

w21 w21

## 33. Practice with single layer neural network

1.2.

Performing a calculations in single layer

neural networks with using direct and matrix

form. Using various activation functions.

Using single layer neural networks with

binary threshold activation function as linear

classifier. Adjusting the linear classifier

based on training samples.

## 34. Hebbian learning rule

-Introduced by Donald Hebb in his 1949 book “The Organizationof Behavior”.

-Describes a basic mechanism for synaptic plasticity

wij t 0 0, i, j

wij t 1 wij t xi y j , where t time

S w11 x1 w21 x2 T

w11 t 1 w11 t x1 y1

w21 t 1 w21 t x2 y1

T t 1 T t y1

## 35. Hebbian learning rule (matrix form)

x112

x1

X

...

L

x1

x12

x22

...

x2L

... x1n

... xn2

, where X i x1i ,..., xni input pattern

... ...

L

... xn

S XW

Y F (S )

W X T Y hebbian learning rule

## 36. Practice with hebbian learning rule

Construction the neural network basedon hebbian learning rule for modeling

OR logical operator

## 37. Delta rule (Widrow-Hoff rule)

1. The delta rule is a gradient descent learning rule forupdating the weights of the inputs to artificial

neurons in single-layer neural network

2. The goal is to minimize the error between the actual

outputs and the target outputs in the training data

3. For each (input/output) training pair, the delta rule

determines the direction you need to adjust wij to

reduce the error for that training pair.

4. Derivatives are used for teaching

## 38. Delta rule (Widrow-Hoff rule)

ADALINE (ADAptive LINear Element) networkn

y1 w j1 x j T

j 1

L

1 L k k 2

E E (k ) ( y1 t )

2 k 1

k 1

1

E (k ) ( y1k t k ) 2

2

## 39. Delta rule (Widrow-Hoff rule)

Gradient descent method: findthe steepest

way

down

the

slope from where you are, and

take a step in that direction

E (k )

w j1 (t 1) w j1 (t )

w j1 (t )

E

E y1k

k

( y1k t k ) x kj

w j1 (t ) y1 w j1

E (k )

T (t 1) T (t )

T (t )

E

E y1k

k

( y1k t k )

T (t ) y1 T

## 40. Delta rule algorithm

1.2.

3.

4.

5.

Define 0<a<1 and Emin

Initialize the weights with some

small random value

Take input pattern and calculate

output vector.

Modify weights and bias according

delta rule.

Do steps 3-4 until E<Emin

## 41. Linear classifiers

## 42. Practice with delta rule

Construction the ADALINE neuralnetwork

(linear

classifier

with

minimum error value) based on given

training patterns.

## 43. Rosenblatt's single layer perceptron

The perceptron is an algorithm forsupervised classification of an input

into one of several possible nonbinary outputs.

It is a type of linear classifier.

Was invented in 1957 by Frank

Rosenblatt as a machine for image

recognition.

## 44. Rosenblatt's single layer perceptron

1, s 0f (s)

1, s 0

Learning rule

## 45. Rosenblatt's learning algorithm

1.2.

3.

4.

5.

Initialise the weights and the threshold.

Weights may be initialised to 0 or to a

small random value.

Take input pattern x from X and

calculate output vector y from Y.

If yi=tj then wij will not change.

If yi≠tj then wij(t+1) = wij (t) + xi tj

Do steps 2-4 until yi=tj for whole

training set

## 46. Rosenblatt's single layer perceptron

Itwas

quickly

proved

that

perceptrons could not be trained to

recognize many classes of patterns.

It is linear classifier. For example, it

is impossible for these classes of

network to learn an XOR function.

## 47. Practice with Rosenblatt's perceptron

Construction the linear classifier (Rosenblatt’s neuralnetwork perceptron) based on given training patterns.

## 48. Associative memory

Associative memory (computer science) - adata-storage device in which a location is

identified by its informational content rather

than by names, addresses, or relative

positions, and from which the data may be

retrieved. This memory enable one to retrieve

a piece of data from only a tiny sample of itself.

Associative memory (psychology) - recalling

a previously experienced item by thinking of

something that is linked with it, thus invoking

the association

## 49. Associative memory

Autoassociativememories

are

capable of retrieving a piece of data

upon presentation of only partial

information from that piece of data

Heteroassociative

memories

can

recall an associated piece of datum

from one category upon presentation

of data from another category.

## 50. Autoassociative memory based on sign activation function

Neuralstructure:

Activation function

network

1, s 0

f (s)

1, s 0

Number of neurons

in the input layer =

Number of neurons

in the output layer

Learning rule

(adopted hebbian rule)

W XT X

Example:

## 51. Practice with autoassociative memory

Realization of the associative memorybased on sign activation function.

Working with multiple patterns.

Recognition of the original and noisy

patterns.

Investigation of the properties and

constraints of the associative memory

based on sign activation function.

## 52. Using single layer neural networks for time series forecasting

A time series points, measuredin time spaced

intervals

sequence of data

typically at points

at uniform time

## 53. Using single layer neural networks for time series forecasting

Training samplesx ( 2)

x(1)

x(3)

x ( 2)

X

...

...

x(m p ) x(m p 1)

x( p )

... x( p 1)

...

...

... x(m 1)

...

x( p 1)

x( p 2)

Y

...

x ( m)

## 54. Practice with time series forecasting

Using ADALINE neural networks forcurrency forecasting:

Creation the training set from the raw

data (www.val.ru).

Learning the ADALINE.

Training ADALINE network with using

delta rule and estimation the error.

## 55. Multilayer perceptron

## 56. Multilayer perceptron

A multilayerperceptron (MLP)

is

a feed

forward artificial neural network model that maps

sets of input data onto a set of appropriate

outputs.

Consists of multiple layers (input, output, one or

several hidden layers) of nodes in a directed

graph, with each layer fully connected to the next

one.

Neurons with a nonlinear activation function.

Utilizes

a supervised

learning technique

called backpropagation of error.

Typical structure

## 57. Multilayer perceptron

Structure (2 hidden layers)Calculation the output Y for input vector X

## 58. Multilayer perceptron

Activation function is not a thresholdFunction approximator

Usually a sigmoid function

Not limited to linear problems

Information flows in one direction

The outputs of one layer act as inputs to

the next layer

## 59. Classification ability

A single layer network can only finda linear discriminant function.

It can divide the input space by

means of hyperplane (straight lines

in two-dimensional space)

## 60. Classification ability

Universal Function Approximation TheoremMLP with one hidden layer can approximate

arbitrarily closely every continuous function that

maps intervals of real numbers to some output

interval of real numbers

f:[0,1]n->[0,1]

2n+1 neurons in hidden layer.

Can form single convex

decision regions

One hidden layer is sufficient

for the large majority of problems

## 61. Classification ability

Any function can be approximated to arbitraryaccuracy by a network with two hidden layers

MLP with two hidden layers can classify sets of

any form. It can form arbitrary disjoint decision

regions

## 62. Backpropagation algorithm

D. Rumelhart, G. Hinton, R. Williams (1986)Most common method of obtaining the

weights in the multilayer perceptron

A form of supervised training

The basic backpropagation algorithm is

based on minimizing the error of the

network using the derivatives of the error

function

Backpropagation of error generalizes the

delta rule

## 63. Basic steps

Forward propagation of a trainingpattern's input through the neural

network in order to generate the

propagation's output activations.

Backward propagation of the

output’s error through the neural

network using the training pattern

target in order to generate the deltas

of all output and hidden neurons.

## 64. Backpropagation

## 65. Backpropagation

We use gradient descent method forminimizing the error

## 66. Backpropagation

Theorem. For any hidden layer i of the neuralnetwork, error of the neuron i calculates by

recursive way through the errors of neurons of

the next layer j.

m

i j F ( S j ) wij

j 1

where m – number of neurons in the next layer j

wij – weights between neuron i and neurons in the

next layer j

Sj – weighted sum for the neuron j in next layer.

Proof

## 67. Backpropagation

Theorem. We can calculate derivatives of errorE through the weights w and bias T by

following way.

Proof

## 68. Backpropagation

Backpropagation rule## 69. Backpropagation algorithm

the training speed (0< <1) anddesired minimal error Em

2.Initialize the weights and biases by random

way.

3.Take consequently all input patterns x from X.

1.Define

y j F ( vector

wij yi T j ) y by following way

Calculate output

i

Realize backpropogation shceme by following

way

ij

j

i

Modify ijweights

and j biases

by following way

w (t 1) w (t ) F ( S ) y

T j (t 1) T j (t ) j F ( S j )

## 70. Backpropagation algorithm

4. Calculate overall error for all patterns1 L

E ( y kj t kj ) 2

2 k 1 j

5. If E>Em then go to the step 3.

## 71. Practice. Calculation delta-rule expressions for various activation functions

## 72. Some problems

The learning rate is importantToo small

Convergence extremely slow

Too large

May not converge

The result may

converge to

a local minimum.

Possible decision:

Using adaptive

learning rate

## 73. Some problems

OverfittingThe number of hidden neurons is very important, it defines the

complexity of the decision boundary:

Too few

Underfit the data – it does not have enough free

parameters to fit the training data well.

Too many

Overfit the data – NN learns the insignificant details

Try different number and use validation set to choose the

best one.

Start small and increase the number until satisfactory results

are obtained.

## 74.

What constitutes a “good” trainingset?

Samples must represent the general

population

Samples must contain members of each

class

Samples in each class must contain a

wide range of variations or noise effect

## 75. Practice with multilayer perceptron

1.2.

Using MLP for noisy digits

recognition &

Using MLP for time series

forecasting.

- Training set preparation.

- MLP learning in Deductor software.

- Estimation the error.

## 76. Recurrent neural networks

Capable to influence to themselvesby means of recurrences, e.g. by

including the network output in the

following computation steps.

Hopfield neural network

Hamming neural network

## 77. Hopfield network

1. Invented by John Hopfield in 1982.2. Content-addressable memory with binary threshold nodes (-1,1 or 0,1)

3. wij=wji, wii=0

yi t 1 F w ji y j t Ti

j 1

j i

Y t 1 F S t ; S t W T Y t T

S S1 ,..., S n Y y1 ,..., yn

T

T T1 ,..., Tn

w11

w

W 21

...

wn1

T

T

w12

w22

...

wn 2

... w1n

... w2 n

wii 0

... ...

... wnn

## 78. Hopfield network

## 79. Hopfield network as associative memory

## 80. Using hopfield network as associative memory

y1 y112 2

y

y

Y 1

... ...

L

y L y1

y12

y22

...

y2L

y1n

2

... yn

... ...

L

... yn

...

y i {0;1}

Hebbian rule

W (2Y I )T (2Y I ) I

1 1 1

L 0 0

where I 1 1 1 I 0 L 0

1 1 1

0 0 L

## 81. Hopfield network as associative memory

1.2.

Take noisy pattern y

Realize iterations

yi (t 1) F w ji y j (t )

j

1, S 0

sign( S )

0, S 0

3.

Until we will not reach stable state

(attractor)

## 82. Example

## 83. Practice with Hopfield network

Realization of the associative memorybased on Hopfield Neural Network

Working with multiple patterns.

Recognition of the original and noisy

patterns.

Investigation of the properties and

constraints of the associative memory

based on Hopfield network.

## 84. Hamming network

R. Lippman (1987)Hamming network is two-network bipolar classifier. The first

layer is single-layer perceptron. It calculates hamming distance

between the vectors. The second network is Hopfield network.

## 85. Hamming network

X 1 x11 ,..., x1n X 2 x12 ,..., xn2 X m x1m ,..., xnmwij xij / 2 , T j n / 2

yj dj

d j Hamming distance between input pattern and j stored pattern

1, if k j

vkj

, e const ,0 e 1 / m

e, if k j

S j , S j 0

z j F S j

0, S j 0

## 86. Hamming network working algorithm

Define weights wij, TjGet input pattern and initialize

Hopfield weights

Make iterations in Hopfield network

until we get stable output.

Take output neuron with 1 value.

## 87. Self-organizing maps

## 88. Self-organizing maps

Unsupervised TrainingThe training set only consists of input

patterns.

The neural network adjusts its own weights

so that similar inputs cause similar outputs.

The network identifies the patterns and

differences in the inputs without any

external assistance

## 89. Self-organizing maps (SOM)

A self-organizing map (SOM) is a type ofartificial neural network that is trained using

unsupervised learning to

produce

a

lowdimensional

(typically

two-dimensional),

discretized representation of the input space of

the training samples, called a map.

Self-organizing maps are different from other

artificial neural networks in the sense that they

use a neighborhood function to preserve the

topological properties of the input space.

The model was first described as an artificial

neural network by the Finnish professor Teuvo

Kohonen.

## 90. Self-organizing maps

We only ask which neuron is active at themoment.

We are not interested in the exact output of the

neuron but in knowing which neuron provides

output.

These networks widely used for clustering

SOMs (like our brain) decide the task of

mapping

a

high-dimensional

input

(N

dimensions) onto areas in a low-dimensional

grid of cells (G dimensions).

## 91.

## 92. Scheme of training of self-organizing map

## 93. Competitive learning

Competitive learning is a form of unsupervisedlearning in artificial neural networks, in which nodes

compete for the right to respond to a subset of the

input data

S j wij xi W j X T

i

where X x1 ,..., xn input pattern

W j w1 j ,..., wnj

winner take all rule

if S k max S j then

j

1, if j k

y j F S j

0, if j k

## 94. Competitive learning

Dj X Wjx1 w1 j 2 ... xn wnj 2

Neuron winner Dk min D j

j

Wk (t 1) Wk (t ) X (t ) Wk (t )

## 95. Vector quantization

It works by dividing a large set ofpoints (vectors) into groups having

approximately the same number of

points closest to them. Each group is

represented by its centroid point, as

in k-means and

some

other clustering algorithms.

## 96. Vector quantization

Choose random weights from[0;1].

t=1

Take all input patterns Xl,l=1,L

D lj X l W j

pattern

recognition

x1 w1 j 2 ... xn wnj 2

Neuron winner Dkl min D lj

j

Applications:

data compression

Video codecs

wij (t 1) wij (t ) (t ) xi wij (t ) , j k

QuickTime

wij (t 1) wij (t ), j k

(t ) 1 / t

Cinepak

Indeo etc.

t=t+1

Audio codecs

Ogg Vorbis

TwinVQ

DTS etc.

## 97. Kohonen Maps

## 98. Kohonen maps

Neightborh ood function for neuron winnerh p,k,t e

u k u p

2

2 2 ( t )

W p (t 1) W p (t ) t h p, k , t X (t ) W p (t ) for winner neuron

## 99. Kohonen maps learning procedure

1.Choose random weights from [0;1].

2.

t=1

3.

Take input pattern Xl and calculate Dij=(Xl-Wij),where i,j=1,m

4.

Detect winner neuron D(k1,k2)=min(Dij)

5.

Calculate for every output neuron

uk u p

h p,k,t e

6.

2

2 2 ( t )

Modify weights by following way

W p (t 1) W p (t ) t h p, k , t X (t ) W p (t ) for winner neuron

Repeat steps 3-6 for all input patterns

## 100. Training and Testing

## 101. Training

The goal is to achieve a balancebetween correct responses for the

training patterns and correct

responses for new patterns.

## 102. Training and Verification

The set of all known samples isbroken into two independent sets

Training set

A group of samples used to train the neural

network

Testing set

A group of samples used to test the

performance of the neural network

Used to estimate the error rate

## 103. Verification

Provides an unbiased test of the qualityof the network

Common error is to “test” the neural

network using the same samples that

were used to train the neural network.

The network was optimized on these

samples, and will obviously perform well on

them

Doesn’t give any indication as to how well

the network will be able to classify inputs

that weren’t in the training set

## 104. Summary (Discussion)

Artificial neural networks are inspired by the learningprocesses that take place in biological systems.

Artificial neurons and neural networks try to imitate

the working mechanisms of their biological

counterparts.

Learning can be perceived as an optimisation

process.

Biological neural learning happens by the

modification of the synaptic strength. Artificial neural

networks learn in the same way.

The synapse strength modification rules for artificial

neural networks can be derived by applying

mathematical optimisation methods.

## 105. Summary

Learning tasks of artificial neural networks canbe reformulated as function approximation

tasks.

Neural networks can be considered as nonlinear

function approximating tools (i.e., linear

combinations of nonlinear basis functions),

where the parameters of the networks should be

found by applying optimisation methods.

The optimisation is done with respect to the

approximation error measure.

In general it is enough to have a single hidden

layer neural network (MLP or other) to learn the

approximation of a nonlinear function.