Constrained molecular dynamics as a search and optimization tool
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Constrained Molecular Dynamics as a Search and Optimization Tool. Riccardo Poli Department of Computer Science University of Essex Christopher R. Stephens Instituto de Ciencias Nucleares UNAM. Introduction. Search and optimization algorithms take inspiration from many areas of science:

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Constrained molecular dynamics as a search and optimization tool

Constrained Molecular Dynamics as a Search and Optimization Tool

Riccardo Poli

Department of Computer Science

University of Essex

Christopher R. Stephens

Instituto de Ciencias Nucleares

UNAM


Introduction

R. Poli - University of Essex

Introduction

  • Search and optimization algorithms take inspiration from many areas of science:

    • Evolutionary algorithms biological systems

    • Simulated annealing physics of cooling

    • Hopfield neural networks physics of spin glasses

    • Swarm algorithms social interactions


Lots of other things in nature know how to optimise

R. Poli - University of Essex

Lots of other things in nature know how to optimise!


Minimisation by marbles

R. Poli - University of Essex

Minimisation by Marbles


Minimisation by buckets of water

R. Poli - University of Essex

Minimisation by Buckets of Water


Minimisation by buckets of water1

R. Poli - University of Essex

Minimisation by Buckets of Water


Minimisation by buckets of water2

R. Poli - University of Essex

Minimisation by Buckets of Water


Minimisation by buckets of water3

R. Poli - University of Essex

Minimisation by Buckets of Water


Minimisation by waterfalls

R. Poli - University of Essex

Minimisation by Waterfalls


Minimisation by skiers

R. Poli - University of Essex

Minimisation by Skiers


Minimisation by molecules

R. Poli - University of Essex

Minimisation by Molecules


Constrained molecular dynamics

R. Poli - University of Essex

Constrained Molecular Dynamics

  • CMDis an optimisation algorithm inspired to multi-body physical interactions (molecular dynamics).

  • A population of particles are constrained to slide on the fitness landscape

  • The particles are under the effects of gravity, friction, centripetalacceleration, and couplingforces (springs).


Some math because it looks good

R. Poli - University of Essex

Some math (because it looks good  )

  • Kinetic energy of a particle


Some more math

R. Poli - University of Essex

Some more math

  • Equation of motion for a particle


Forces for courses no forces

R. Poli - University of Essex

Forces for Courses: No forces

  • If v=0 then CMD=kind of random search


Forces for courses no forces1

R. Poli - University of Essex

Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature

1/6


Forces for courses no forces2

R. Poli - University of Essex

Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature

2/6


Forces for courses no forces3

R. Poli - University of Essex

Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature

3/6


Forces for courses no forces4

R. Poli - University of Essex

Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature

4/6


Forces for courses no forces5

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Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature

5/6


Forces for courses no forces6

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Forces for Courses: No forces

  • If v0 then CMD=parallel search guided by curvature.

6/6


Forces for courses gravity

R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

1/5


Forces for courses gravity1

R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

2/5


Forces for courses gravity2

R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

3/5


Forces for courses gravity3

R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

4/5


Forces for courses gravity4

R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour.

5/5


Forces for courses gravity5

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

1/11


Forces for courses gravity6

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

2/11


Forces for courses gravity7

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

3/11


Forces for courses gravity8

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

4/11


Forces for courses gravity9

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

5/11


Forces for courses gravity10

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

6/11


Forces for courses gravity11

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

7/11


Forces for courses gravity12

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

8/11


Forces for courses gravity13

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

9/11


Forces for courses gravity14

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour

10/11


Forces for courses gravity15

R. Poli - University of Essex

Forces for Courses: Gravity

  • If E big  skier-type,local-optima-avoiding behaviour.

11/11


Forces for courses interactions

R. Poli - University of Essex

Forces for Courses: Interactions

  • Particle-particle interactions (springs)

    • Springs integrate information across the population of particles (a bit like crossover in a GA).

    • Without friction  oscillatory/exploratory search behaviour (similar to PSOs)

    • With friction  exploration focuses (like in a GA)


Forces for courses interactions1

R. Poli - University of Essex

Forces for Courses: Interactions

1/12


Forces for courses interactions2

R. Poli - University of Essex

Forces for Courses: Interactions

2/12


Forces for courses interactions3

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Forces for Courses: Interactions

3/12


Forces for courses interactions4

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Forces for Courses: Interactions

4/12


Forces for courses interactions5

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Forces for Courses: Interactions

5/12


Forces for courses interactions6

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Forces for Courses: Interactions

6/12


Forces for courses interactions7

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Forces for Courses: Interactions

7/12


Forces for courses interactions8

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Forces for Courses: Interactions

8/12


Forces for courses interactions9

R. Poli - University of Essex

Forces for Courses: Interactions

9/12


Forces for courses interactions10

R. Poli - University of Essex

Forces for Courses: Interactions

10/12


Forces for courses interactions11

R. Poli - University of Essex

Forces for Courses: Interactions

11/12


Forces for courses interactions12

R. Poli - University of Essex

Forces for Courses: Interactions.

12/12


Forces for courses friction

R. Poli - University of Essex

Forces for Courses: Friction

  • Friction “relaxes” a particle into a good position once an interesting region has been found in the landscape.

    • More friction less exploration

    • Less friction  more exploration

    • Similar to temperature in simulated annealing.

    • Similar to selection pressure in a GA


Cmd in practice

R. Poli - University of Essex

CMD in Practice

We calculate the force acting on each particle and numerically integrate the motion equations for the system

repeat

for i =1 to Population Size

ai = Force (x, v, fitness surface )

vi = vi + ∆ ai

xi = xi + ∆ vi

next i

end repeat


Cmd is similar to psos

R. Poli - University of Essex

CMD is Similar to PSOs…

Particle swarm optimisers are inspired to bird flocks foraging

for i =1 to Population Size (N)

for j = 1 to Dimension Size (d)

aij = f1(pij – xij ) + f2(pgj – xij )

vij = vij + aij

xij = xij + vij

next j

{if f(xi) < f(pi) then pi = xi // Intelligence

if f(pi) < f(pg) then g = i}

next i


Constrained molecular dynamics as a search and optimization tool

R. Poli - University of Essex

…but…

  • In CMD particles don’t fly, they slide

  • No memory and no explicit “intelligence”

  • No random forces

  • Simulation is physically realistic


Cmd is similar to gradient descent

R. Poli - University of Essex

CMD is Similar to Gradient Descent

But…

  • CMD uses multiple interacting particles which can pull each other out of bad areas

  • Particles have velocity and mass which help escape local optima

  • Particles “feel” the local shape of the fitness surface (centripetal acceleration) in addition to the slope.


Experiments

R. Poli - University of Essex

Experiments

  • Setups

    • n = 1, n = 2 and n = 10 particles

    • N = 1, N = 2 and N = 3 dimensions

    • Springs: no, ring, full

    • Gravity: no, yes

    • Friction: no, yes

  • 30 independent runs per setup

  • 5000 integration steps per run


Test problems

R. Poli - University of Essex

Test problems

  • De Jong’s F1

    • Unimodal

    • Easy

  • De Jong’s F2

    • Unimodal

    • Hard

  • Rastrigin’s

    • Highlymultimodal

    • Very hard


Results f1

R. Poli - University of Essex

Results: F1

  • More particles  better performance

  • Gravity is sufficient to guarantee near perfect results

  • Springs are not too beneficial

  • Friction helps settle in the global optimum


Results f2

R. Poli - University of Essex

Results: F2

  • Gravity is needed

  • Springs are more beneficial

  • Friction less beneficial (long narrow valley)

  • Too few particles convergence not guaranteed (oscillations)


Results rastrigin

R. Poli - University of Essex

Results: Rastrigin

  • Gravity is needed

  • Springs help, especially when fully connected

  • Friction helps settle in the global optimum

  • More particles are necessary to solve problem reliably


Vrml demos

R. Poli - University of Essex

VRML Demos


Conic fitness function one particle gravity no friction

R. Poli - University of Essex

Conic fitness function, one particle, gravity, no friction


Conic fitness function 5 particles gravity springs ring friction

R. Poli - University of Essex

Conic fitness function, 5 particles, gravity, springs (ring), friction


Quadratic fitness function 5 particles gravity friction

R. Poli - University of Essex

Quadratic fitness function, 5 particles, gravity, friction


Quadratic fitness function 5 particles gravity no friction springs ring

R. Poli - University of Essex

Quadratic fitness function, 5 particles, gravity, no friction, springs (ring)


Quadratic fitness function 5 particles gravity friction springs ring

R. Poli - University of Essex

Quadratic fitness function, 5 particles, gravity, friction, springs (ring)


Multimodal fitness function 5 particles gravity friction

R. Poli - University of Essex

Multimodal fitness function, 5 particles, gravity, friction


Multimodal function 5 particles gravity friction springs ring

R. Poli - University of Essex

Multimodal function, 5 particles, gravity, friction, springs (ring)


Multimodal function 5 particles gravity friction springs all connected

R. Poli - University of Essex

Multimodal function, 5 particles, gravity, friction, springs (all connected)


Rastrigin fitness function 20 particles gravity friction

R. Poli - University of Essex

Rastrigin fitness function, 20 particles, gravity, friction


Rastrigin function 20 particles gravity friction springs all connected

R. Poli - University of Essex

Rastrigin function, 20 particles, gravity, friction, springs (all connected)


Conclusions

R. Poli - University of Essex

Conclusions

  • CMD uses the physics of masses and forces to guide the exploration of fitness landscapes.

  • For now we have explored three forces:

    • Gravity provides the ability to seek minima.

    • Interaction via springs provides exploration.

    • Friction slows down and focuses the search.

  • The results are encouraging and we hope much more can come from CMD.


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