Constrained molecular dynamics as a search and optimization tool
Sponsored Links
This presentation is the property of its rightful owner.
1 / 72

Constrained Molecular Dynamics as a Search and Optimization Tool PowerPoint PPT Presentation


  • 59 Views
  • Uploaded on
  • Presentation posted in: General

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:

Download Presentation

Constrained Molecular Dynamics as a Search and Optimization Tool

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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 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


R. Poli - University of Essex

Lots of other things in nature know how to optimise!


R. Poli - University of Essex

Minimisation by Marbles


R. Poli - University of Essex

Minimisation by Buckets of Water


R. Poli - University of Essex

Minimisation by Buckets of Water


R. Poli - University of Essex

Minimisation by Buckets of Water


R. Poli - University of Essex

Minimisation by Buckets of Water


R. Poli - University of Essex

Minimisation by Waterfalls


R. Poli - University of Essex

Minimisation by Skiers


R. Poli - University of Essex

Minimisation by Molecules


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).


R. Poli - University of Essex

Some math (because it looks good  )

  • Kinetic energy of a particle


R. Poli - University of Essex

Some more math

  • Equation of motion for a particle


R. Poli - University of Essex

Forces for Courses: No forces

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


R. Poli - University of Essex

Forces for Courses: No forces

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

1/6


R. Poli - University of Essex

Forces for Courses: No forces

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

2/6


R. Poli - University of Essex

Forces for Courses: No forces

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

3/6


R. Poli - University of Essex

Forces for Courses: No forces

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

4/6


R. Poli - University of Essex

Forces for Courses: No forces

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

5/6


R. Poli - University of Essex

Forces for Courses: No forces

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

6/6


R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

1/5


R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

2/5


R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

3/5


R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour

4/5


R. Poli - University of Essex

Forces for Courses: Gravity

  • Minimum seeking behaviour

  • If E small + friction  hillclimbing behaviour.

5/5


R. Poli - University of Essex

Forces for Courses: Gravity

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

1/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

2/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

3/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

4/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

5/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

6/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

7/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

8/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

9/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

10/11


R. Poli - University of Essex

Forces for Courses: Gravity

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

11/11


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)


R. Poli - University of Essex

Forces for Courses: Interactions

1/12


R. Poli - University of Essex

Forces for Courses: Interactions

2/12


R. Poli - University of Essex

Forces for Courses: Interactions

3/12


R. Poli - University of Essex

Forces for Courses: Interactions

4/12


R. Poli - University of Essex

Forces for Courses: Interactions

5/12


R. Poli - University of Essex

Forces for Courses: Interactions

6/12


R. Poli - University of Essex

Forces for Courses: Interactions

7/12


R. Poli - University of Essex

Forces for Courses: Interactions

8/12


R. Poli - University of Essex

Forces for Courses: Interactions

9/12


R. Poli - University of Essex

Forces for Courses: Interactions

10/12


R. Poli - University of Essex

Forces for Courses: Interactions

11/12


R. Poli - University of Essex

Forces for Courses: Interactions.

12/12


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


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


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


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 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.


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


R. Poli - University of Essex

Test problems

  • De Jong’s F1

    • Unimodal

    • Easy

  • De Jong’s F2

    • Unimodal

    • Hard

  • Rastrigin’s

    • Highlymultimodal

    • Very hard


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


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)


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


R. Poli - University of Essex

VRML Demos


R. Poli - University of Essex

Conic fitness function, one particle, gravity, no friction


R. Poli - University of Essex

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


R. Poli - University of Essex

Quadratic fitness function, 5 particles, gravity, friction


R. Poli - University of Essex

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


R. Poli - University of Essex

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


R. Poli - University of Essex

Multimodal fitness function, 5 particles, gravity, friction


R. Poli - University of Essex

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


R. Poli - University of Essex

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


R. Poli - University of Essex

Rastrigin fitness function, 20 particles, gravity, friction


R. Poli - University of Essex

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


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.


  • Login