constrained molecular dynamics as a search and optimization tool

Constrained Molecular Dynamics as a Search and Optimization ToolRiccardo PoliDepartment of Computer ScienceUniversity of EssexChristopher R. StephensInstituto de Ciencias NuclearesUNAM

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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

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Lots of other things in nature know how to optimise!

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Minimisation by Marbles

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Minimisation by Buckets of Water

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Minimisation by Buckets of Water

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Minimisation by Buckets of Water

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Minimisation by Buckets of Water

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Minimisation by Waterfalls

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Minimisation by Skiers

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Minimisation by Molecules

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Constrained Molecular Dynamics CMD is 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, centripetal acceleration, and coupling forces (springs).

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Some math (because it looks good ) Kinetic energy of a particle

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Some more math Equation of motion for a particle

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Forces for Courses: No forces If v=0 then CMD=kind of random search

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature

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Forces for Courses: No forces If v0 then CMD=parallel search guided by

curvature.

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Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour

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Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour

2/5

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Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour

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Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour

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Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour.

5/5

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour

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Forces for Courses: Gravity If E big skier-type, local-optima-avoiding

behaviour.

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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)

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions

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Forces for Courses: Interactions.

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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

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CMD in PracticeWe calculate the force acting on each particle and

numerically integrate the motion equations for the system

repeatfor i =1 to Population Sizeai = Force ( x, v, fitness surface )

vi = vi + ∆ ai

xi = xi + ∆ vi

next iend repeat

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CMD is Similar to PSOs…Particle swarm optimisers are inspired to bird flocks foragingfor i =1 to Population Size (N)

for j = 1 to Dimension Size (d)aij = 1(pij – xij ) + 2(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

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…but… In CMD particles don’t fly, they slide No memory and no explicit

“intelligence” No random forces Simulation is physically realistic

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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.

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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

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Test problems De Jong’s F1

Unimodal Easy

De Jong’s F2 Unimodal Hard

Rastrigin’s Highly multimodal Very hard

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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

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Results: F2 Gravity is needed Springs are more

beneficial Friction less beneficial

(long narrow valley) Too few particles

convergence not guaranteed (oscillations)

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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

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VRML Demos

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Conic fitness function, one particle, gravity, no friction

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

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Quadratic fitness function, 5 particles, gravity, friction

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

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

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Multimodal fitness function, 5 particles, gravity, friction

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

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

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Rastrigin fitness function, 20 particles, gravity, friction

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

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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.