1. 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

2. R. Poli - University of Essex
February 19
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

3. Lots of other things in nature know how to optimise!
February 19
R. Poli - University of Essex
Lots of other things in nature know how to optimise!

4. Minimisation by Marbles
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R. Poli - University of Essex
Minimisation by Marbles

5. Minimisation by Buckets of Water
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R. Poli - University of Essex
Minimisation by Buckets of Water

6. Minimisation by Buckets of Water
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R. Poli - University of Essex
Minimisation by Buckets of Water

7. Minimisation by Buckets of Water
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R. Poli - University of Essex
Minimisation by Buckets of Water

8. Minimisation by Buckets of Water
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R. Poli - University of Essex
Minimisation by Buckets of Water

9. Minimisation by Waterfalls
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R. Poli - University of Essex
Minimisation by Waterfalls

10. Minimisation by Skiers
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R. Poli - University of Essex
Minimisation by Skiers

11. Minimisation by Molecules
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R. Poli - University of Essex
Minimisation by Molecules

12. Constrained Molecular Dynamics
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R. Poli - University of Essex
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).

13. Some math (because it looks good  )
February 19
R. Poli - University of Essex
Some math (because it looks good  )
Kinetic energy of a particle

14. R. Poli - University of Essex
February 19
R. Poli - University of Essex
Some more math
Equation of motion for a particle

15. Forces for Courses: No forces
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R. Poli - University of Essex
Forces for Courses: No forces
If v=0 then CMD=kind of random search

16. Forces for Courses: No forces
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R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature
1/6

17. Forces for Courses: No forces
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R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature
2/6

18. Forces for Courses: No forces
February 19
R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature
3/6

19. Forces for Courses: No forces
February 19
R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature
4/6

20. Forces for Courses: No forces
February 19
R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature
5/6

21. Forces for Courses: No forces
February 19
R. Poli - University of Essex
Forces for Courses: No forces
If v0 then CMD=parallel search guided by curvature.
6/6

22. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
Minimum seeking behaviour
If E small + friction  hillclimbing behaviour
1/5

23. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
Minimum seeking behaviour
If E small + friction  hillclimbing behaviour
2/5

24. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
Minimum seeking behaviour
If E small + friction  hillclimbing behaviour
3/5

25. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
Minimum seeking behaviour
If E small + friction  hillclimbing behaviour
4/5

26. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
Minimum seeking behaviour
If E small + friction  hillclimbing behaviour.
5/5

27. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
1/11

28. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
2/11

29. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
3/11

30. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
4/11

31. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
5/11

32. Forces for Courses: Gravity
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R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
6/11

33. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
7/11

34. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
8/11

35. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
9/11

36. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour
10/11

37. Forces for Courses: Gravity
February 19
R. Poli - University of Essex
Forces for Courses: Gravity
If E big  skier-type, local-optima-avoiding behaviour.
11/11

38. Forces for Courses: Interactions
February 19
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)

39. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
1/12

40. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
2/12

41. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
3/12

42. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
4/12

43. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
5/12

44. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
6/12

45. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
7/12

46. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
8/12

47. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
9/12

48. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
10/12

49. Forces for Courses: Interactions
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R. Poli - University of Essex
Forces for Courses: Interactions
11/12

50. Forces for Courses: Interactions.
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R. Poli - University of Essex
Forces for Courses: Interactions.
12/12

51. Forces for Courses: Friction
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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

52. R. Poli - University of Essex
February 19
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

53. R. Poli - University of Essex
February 19
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

54. R. Poli - University of Essex
February 19
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

55. CMD is Similar to Gradient Descent
February 19
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.

56. R. Poli - University of Essex
February 19
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

57. R. Poli - University of Essex
February 19
R. Poli - University of Essex
Test problems
De Jong’s F1
Unimodal
Easy
De Jong’s F2
Hard
Rastrigin’s
Highly multimodal
Very hard

58. R. Poli - University of Essex
February 19
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

59. R. Poli - University of Essex
February 19
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)

60. R. Poli - University of Essex
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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

61. R. Poli - University of Essex
February 19
R. Poli - University of Essex
VRML Demos

62. Conic fitness function, one particle, gravity, no friction
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R. Poli - University of Essex
Conic fitness function, one particle, gravity, no friction

63. Conic fitness function, 5 particles, gravity, springs (ring), friction
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R. Poli - University of Essex
Conic fitness function, 5 particles, gravity, springs (ring), friction

64. Quadratic fitness function, 5 particles, gravity, friction
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R. Poli - University of Essex
Quadratic fitness function, 5 particles, gravity, friction

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

66. R. Poli - University of Essex
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R. Poli - University of Essex
Quadratic fitness function, 5 particles, gravity, friction, springs (ring)

67. Multimodal fitness function, 5 particles, gravity, friction
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R. Poli - University of Essex
Multimodal fitness function, 5 particles, gravity, friction

68. Multimodal function, 5 particles, gravity, friction, springs (ring)
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R. Poli - University of Essex
Multimodal function, 5 particles, gravity, friction, springs (ring)

69. R. Poli - University of Essex
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R. Poli - University of Essex
Multimodal function, 5 particles, gravity, friction, springs (all connected)

70. Rastrigin fitness function, 20 particles, gravity, friction
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R. Poli - University of Essex
Rastrigin fitness function, 20 particles, gravity, friction

71. R. Poli - University of Essex
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R. Poli - University of Essex
Rastrigin function, 20 particles, gravity, friction, springs (all connected)

72. R. Poli - University of Essex
February 19
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.