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Hopfield Neural Networks for Optimization
虞台文 大同大學資工所 智慧型多媒體研究室
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Content Introduction A Simple Example Race Traffic Problem
Example A/D Converter Example Traveling Salesperson Problem
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Hopfield Neural Networks for Optimization
Introduction 大同大學資工所 智慧型多媒體研究室
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Energy Function of a Hopfield NN
Interaction btw neurons Interaction to the external constant Running a Hopfield NN asynchronously, its energy is monotonically non-increasing.
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Solving Optimization Problems Using Hopfield NNs
Reformulating the cost of a problem in the form of energy function of a Hopfield NN. Build a Hopfield NN based on such an energy function. Running the NN asynchronously until the NN settles down. Read the answer reported by the NN.
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Hopfield Neural Networks for Optimization
A Simple Example Race Traffic Problem 大同大學資工所 智慧型多媒體研究室
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A Simple Hopfield NN 1 2 I1 I2
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The Race Traffic Problem
+1 1 v1 v2
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The Race Traffic Problem
1 2 1
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The Race Traffic Problem
1 2 1 1 1 1 Stable State
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The Race Traffic Problem
1 2 1 1 1 1 Stable State
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The Race Traffic Problem
1 2 1 1 1 How about if to run synchronously?
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Hopfield Neural Networks for Optimization
Example A/D Converter 大同大學資工所 智慧型多媒體研究室
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Reference Tank, D.W., and Hopfield, J.J., “Simple "neural" optimization networks: An A/D converter, signal decision circuit and a linear programming circuit,” IEEE Transactions on Circuits and Systems, Vol. CAS-33 (1986)
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I A/D Converter Using Unipolar Neurons v0 A/D v1 Analog v2 v3 20 21 22
23 I Analog Using Unipolar Neurons
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A/D Converter Using Unipolar Neurons
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A/D Converter 1 2 3 v0 v1 v2 v3 I0 I1 I2 I3
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Hopfield Neural Networks for Optimization
Example Traveling Salesperson Problem 大同大學資工所 智慧型多媒體研究室
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Reference J. J. Hopfield and D. W. Tank, “Neural” computation of decisions in optimization problems, ” Biological Cybernetics, Vol. 52, pp , 1985.
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Traveling Salesperson Problem
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Traveling Salesperson Problem
Given n cities with distances dij, what is the shortest tour?
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Traveling Salesperson Problem
2 3 4 1 5 9 6 11 8 10 7
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Traveling Salesperson Problem
Distance Matrix Find a minimum cost Hamiltonian Cycle.
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Search Space Find a minimum cost Hamiltonian Cycle.
Assume we are given a fully connection graph with n vertices and symmetric costs (dij=dji). The size of search space is Find a minimum cost Hamiltonian Cycle.
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Problem Representation Using NNs
Time 1 2 3 4 5 2 1 1 2 3 4 5 4 City 3 5
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Problem Representation Using NNs
The salesperson reaches city 5 at time 3. Time 1 2 3 4 5 2 1 1 2 3 4 5 4 City 3 5
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Problem Representation Using NNs
Goal: Find a minimum cost Hamiltonian Cycle. Problem Representation Using NNs Time 1 2 3 4 5 2 1 1 2 3 4 5 4 City 3 5
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The Hamiltonian Constraint
Goal: Find a minimum cost Hamiltonian Cycle. The Hamiltonian Constraint Time 1 2 3 4 5 Each row and column can have only one neuron “on”. For a n-city problem, n neurons will be on. 2 1 1 2 3 4 5 4 City 3 5
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Cost Minimization Goal: Find a minimum cost Hamiltonian Cycle. Time 1
2 3 4 5 The total distance of the valid tour have to be very low. 2 1 1 2 3 4 5 d35 d54 d42 d25 d51 4 City The summation of these dij’s is very low. 3 5
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Indices of Neurons i Time vxi 1 2 3 4 5 1 2 3 4 5 City x
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Hamiltonian-Cycle Satisfaction
Energy Function Hamiltonian-Cycle Satisfaction Cost Minimization
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Energy Function n neurons ‘on’ Each row one or zero neuron ‘on’
Each column one or zero neuron ‘on’
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Total distance of the tour
Energy Function Total distance of the tour
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Energy Function
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Build NN for TSP Mapping Energy function of a 2-D neural network
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Analog Hopfield NN for 10-City TSP
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Analog Hopfield NN for 10-City TSP
The shortest path
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Analog Hopfield NN for 10-City TSP
The shortest path
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Analog Hopfield NN for 30-City TSP
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Analog Hopfield NN for 30-City TSP
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