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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 EssexIntroduction
  • 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
constrained molecular dynamics
R. Poli - University of EssexConstrained 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 more math
R. Poli - University of EssexSome more math
  • Equation of motion for a particle
forces for courses no forces1
R. Poli - University of EssexForces 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 EssexForces 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 EssexForces 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 EssexForces for Courses: No forces
  • If v0 then CMD=parallel search guided by curvature

4/6

forces for courses no forces5
R. Poli - University of EssexForces for Courses: No forces
  • If v0 then CMD=parallel search guided by curvature

5/6

forces for courses no forces6
R. Poli - University of EssexForces for Courses: No forces
  • If v0 then CMD=parallel search guided by curvature.

6/6

forces for courses gravity
R. Poli - University of EssexForces for Courses: Gravity
  • Minimum seeking behaviour
  • If E small + friction  hillclimbing behaviour

1/5

forces for courses gravity1
R. Poli - University of EssexForces for Courses: Gravity
  • Minimum seeking behaviour
  • If E small + friction  hillclimbing behaviour

2/5

forces for courses gravity2
R. Poli - University of EssexForces for Courses: Gravity
  • Minimum seeking behaviour
  • If E small + friction  hillclimbing behaviour

3/5

forces for courses gravity3
R. Poli - University of EssexForces for Courses: Gravity
  • Minimum seeking behaviour
  • If E small + friction  hillclimbing behaviour

4/5

forces for courses gravity4
R. Poli - University of EssexForces for Courses: Gravity
  • Minimum seeking behaviour
  • If E small + friction  hillclimbing behaviour.

5/5

forces for courses gravity5
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

1/11

forces for courses gravity6
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

2/11

forces for courses gravity7
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

3/11

forces for courses gravity8
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

4/11

forces for courses gravity9
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

5/11

forces for courses gravity10
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

6/11

forces for courses gravity11
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

7/11

forces for courses gravity12
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

8/11

forces for courses gravity13
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

9/11

forces for courses gravity14
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour

10/11

forces for courses gravity15
R. Poli - University of EssexForces for Courses: Gravity
  • If E big  skier-type,local-optima-avoiding behaviour.

11/11

forces for courses interactions
R. Poli - University of EssexForces 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 friction
R. Poli - University of EssexForces 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 EssexCMD 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 EssexCMD 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

slide54
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 EssexCMD 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 EssexExperiments
  • 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 EssexTest problems
  • De Jong’s F1
    • Unimodal
    • Easy
  • De Jong’s F2
    • Unimodal
    • Hard
  • Rastrigin’s
    • Highlymultimodal
    • Very hard
results f1
R. Poli - University of EssexResults: 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 EssexResults: 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 EssexResults: 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
conclusions
R. Poli - University of EssexConclusions
  • 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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