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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

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

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).

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature

1/6

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature

2/6

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature

3/6

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature

4/6

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature

5/6

R. Poli - University of EssexForces for Courses: No forces

- If v0 then CMD=parallel search guided by curvature.

6/6

R. Poli - University of EssexForces for Courses: Gravity

- Minimum seeking behaviour
- If E small + friction hillclimbing behaviour

1/5

R. Poli - University of EssexForces for Courses: Gravity

- Minimum seeking behaviour
- If E small + friction hillclimbing behaviour

2/5

R. Poli - University of EssexForces for Courses: Gravity

- Minimum seeking behaviour
- If E small + friction hillclimbing behaviour

3/5

R. Poli - University of EssexForces for Courses: Gravity

- Minimum seeking behaviour
- If E small + friction hillclimbing behaviour

4/5

R. Poli - University of EssexForces for Courses: Gravity

- Minimum seeking behaviour
- If E small + friction hillclimbing behaviour.

5/5

R. Poli - University of EssexForces for Courses: Gravity

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

1/11

R. Poli - University of EssexForces for Courses: Gravity

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

2/11

R. Poli - University of EssexForces for Courses: Gravity

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

3/11

R. Poli - University of EssexForces for Courses: Gravity

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

4/11

R. Poli - University of EssexForces for Courses: Gravity

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

5/11

R. Poli - University of EssexForces for Courses: Gravity

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

6/11

R. Poli - University of EssexForces for Courses: Gravity

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

7/11

R. Poli - University of EssexForces for Courses: Gravity

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

8/11

R. Poli - University of EssexForces for Courses: Gravity

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

9/11

R. Poli - University of EssexForces for Courses: Gravity

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

10/11

R. Poli - University of EssexForces for Courses: Gravity

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

11/11

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)

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

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

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

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

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.

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

R. Poli - University of EssexTest problems

- De Jong’s F1
- Unimodal
- Easy
- De Jong’s F2
- Unimodal
- Hard
- Rastrigin’s
- Highlymultimodal
- Very hard

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

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)

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

R. Poli - University of EssexQuadratic fitness function, 5 particles, gravity, no friction, springs (ring)

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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