Intelligent Numerical Computation1 Graph bisection  Problem statement  Mathematical framework  Methods  Numerical results

Intelligent Numerical Computation2 T = zeros(N,N); for i = 1:N for j = 1:N if rand(1,1) > 0.5 & j~=i T(i,j) = 1; T(j,i)=1; end Graph generation

Intelligent Numerical Computation3 Cost 5 Cost 6 Graph Bipartition

Intelligent Numerical Computation4 Graph bisection  Operation: partition all nodes to two sets  Objective Minimization of cut size Cut size means the number of connections between two sets  Constraints Two sets have equal size

Intelligent Numerical Computation5 Representation of Edges and Memberships

Intelligent Numerical Computation6 MINIMIZE SUBJECT TO Mathematical framework

Intelligent Numerical Computation7 Energy function

Intelligent Numerical Computation8 Spin model  Graph bisection

Intelligent Numerical Computation9 A physical-like random system  Boltzmann assumption S is regarded as a random vector  Free energy

Intelligent Numerical Computation10 Entropy  Entropy of the whole system  Sum of individual entropies

Intelligent Numerical Computation11 Mean field approximation  Individual entropy

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Intelligent Numerical Computation14 Mean field approximation

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Intelligent Numerical Computation16 Mean field equation

Intelligent Numerical Computation17 Mean field equations

Intelligent Numerical Computation18 1. Set near zero randomly 2. Set beta to a sufficiently small value 3. Employ mean field equations to determine a fixed point 4. If the halting condition holds, halt 5. Increase beta by an annealing schedule 6. Go to step 3 Procedure of mean field annealing

Intelligent Numerical Computation19 Flow chart Set v(i) near zero for all i beta sufficiently small and alpha near one mean(v.^2)<0.98 exit Update all vi asynchronously v(i)=tanh(beta*sum(w(i,:)*v)) beta=beta/alpha

Intelligent Numerical Computation20 Halting condition  Mean of squares of all exceeds a predetermined threshold value

Intelligent Numerical Computation21 Annealing schedule  Set beta to a sufficiently small value  Increase beta carefully

Intelligent Numerical Computation22 Hopfield neural networks for TSP  Problem statement  Mathematical framework  Mean field annealing  Numerical results

Intelligent Numerical Computation23 KL(Kullback-Leberler) Divergence  Boltzmann distribution  Normalization

Intelligent Numerical Computation24 Intractable Partition function

Intelligent Numerical Computation25 Factorial distribution  An approximation

Intelligent Numerical Computation26 KL divergence  Semi-distance between two pdfs = {x}

Intelligent Numerical Computation27 Step 1

Intelligent Numerical Computation28 Step 2

Intelligent Numerical Computation29 Step 3

Intelligent Numerical Computation30 Step 4