constrained molecular dynamics as a search and optimization tool
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
Constrained Molecular Dynamics as a Search and Optimization ToolRiccardo PoliDepartment of Computer ScienceUniversity of EssexChristopher R. StephensInstituto de Ciencias NuclearesUNAM
Apr 22, 2023 R. Poli - University of Essex 2
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
Apr 22, 2023 R. Poli - University of Essex 3
Lots of other things in nature know how to optimise!
Apr 22, 2023 R. Poli - University of Essex 4
Minimisation by Marbles
Apr 22, 2023 R. Poli - University of Essex 5
Minimisation by Buckets of Water
Apr 22, 2023 R. Poli - University of Essex 6
Minimisation by Buckets of Water
Apr 22, 2023 R. Poli - University of Essex 7
Minimisation by Buckets of Water
Apr 22, 2023 R. Poli - University of Essex 8
Minimisation by Buckets of Water
Apr 22, 2023 R. Poli - University of Essex 9
Minimisation by Waterfalls
Apr 22, 2023 R. Poli - University of Essex 10
Minimisation by Skiers
Apr 22, 2023 R. Poli - University of Essex 11
Minimisation by Molecules
Apr 22, 2023 R. Poli - University of Essex 12
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).
Apr 22, 2023 R. Poli - University of Essex 13
Some math (because it looks good ) Kinetic energy of a particle
Apr 22, 2023 R. Poli - University of Essex 14
Some more math Equation of motion for a particle
Apr 22, 2023 R. Poli - University of Essex 15
Forces for Courses: No forces If v=0 then CMD=kind of random search
Apr 22, 2023 R. Poli - University of Essex 16
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature
1/6
Apr 22, 2023 R. Poli - University of Essex 17
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature
2/6
Apr 22, 2023 R. Poli - University of Essex 18
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature
3/6
Apr 22, 2023 R. Poli - University of Essex 19
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature
4/6
Apr 22, 2023 R. Poli - University of Essex 20
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature
5/6
Apr 22, 2023 R. Poli - University of Essex 21
Forces for Courses: No forces If v0 then CMD=parallel search guided by
curvature.
6/6
Apr 22, 2023 R. Poli - University of Essex 22
Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour
1/5
Apr 22, 2023 R. Poli - University of Essex 23
Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour
2/5
Apr 22, 2023 R. Poli - University of Essex 24
Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour
3/5
Apr 22, 2023 R. Poli - University of Essex 25
Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour
4/5
Apr 22, 2023 R. Poli - University of Essex 26
Forces for Courses: Gravity Minimum seeking behaviour If E small + friction hillclimbing behaviour.
5/5
Apr 22, 2023 R. Poli - University of Essex 27
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
1/11
Apr 22, 2023 R. Poli - University of Essex 28
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
2/11
Apr 22, 2023 R. Poli - University of Essex 29
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
3/11
Apr 22, 2023 R. Poli - University of Essex 30
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
4/11
Apr 22, 2023 R. Poli - University of Essex 31
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
5/11
Apr 22, 2023 R. Poli - University of Essex 32
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
6/11
Apr 22, 2023 R. Poli - University of Essex 33
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
7/11
Apr 22, 2023 R. Poli - University of Essex 34
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
8/11
Apr 22, 2023 R. Poli - University of Essex 35
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
9/11
Apr 22, 2023 R. Poli - University of Essex 36
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour
10/11
Apr 22, 2023 R. Poli - University of Essex 37
Forces for Courses: Gravity If E big skier-type, local-optima-avoiding
behaviour.
11/11
Apr 22, 2023 R. Poli - University of Essex 38
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)
Apr 22, 2023 R. Poli - University of Essex 39
Forces for Courses: Interactions
1/12
Apr 22, 2023 R. Poli - University of Essex 40
Forces for Courses: Interactions
2/12
Apr 22, 2023 R. Poli - University of Essex 41
Forces for Courses: Interactions
3/12
Apr 22, 2023 R. Poli - University of Essex 42
Forces for Courses: Interactions
4/12
Apr 22, 2023 R. Poli - University of Essex 43
Forces for Courses: Interactions
5/12
Apr 22, 2023 R. Poli - University of Essex 44
Forces for Courses: Interactions
6/12
Apr 22, 2023 R. Poli - University of Essex 45
Forces for Courses: Interactions
7/12
Apr 22, 2023 R. Poli - University of Essex 46
Forces for Courses: Interactions
8/12
Apr 22, 2023 R. Poli - University of Essex 47
Forces for Courses: Interactions
9/12
Apr 22, 2023 R. Poli - University of Essex 48
Forces for Courses: Interactions
10/12
Apr 22, 2023 R. Poli - University of Essex 49
Forces for Courses: Interactions
11/12
Apr 22, 2023 R. Poli - University of Essex 50
Forces for Courses: Interactions.
12/12
Apr 22, 2023 R. Poli - University of Essex 51
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
Apr 22, 2023 R. Poli - University of Essex 52
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
Apr 22, 2023 R. Poli - University of Essex 53
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
Apr 22, 2023 R. Poli - University of Essex 54
…but… In CMD particles don’t fly, they slide No memory and no explicit
“intelligence” No random forces Simulation is physically realistic
Apr 22, 2023 R. Poli - University of Essex 55
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.
Apr 22, 2023 R. Poli - University of Essex 56
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
Apr 22, 2023 R. Poli - University of Essex 57
Test problems De Jong’s F1
Unimodal Easy
De Jong’s F2 Unimodal Hard
Rastrigin’s Highly multimodal Very hard
Apr 22, 2023 R. Poli - University of Essex 58
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
Apr 22, 2023 R. Poli - University of Essex 59
Results: F2 Gravity is needed Springs are more
beneficial Friction less beneficial
(long narrow valley) Too few particles
convergence not guaranteed (oscillations)
Apr 22, 2023 R. Poli - University of Essex 60
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
Apr 22, 2023 R. Poli - University of Essex 61
VRML Demos
Apr 22, 2023 R. Poli - University of Essex 62
Conic fitness function, one particle, gravity, no friction
Apr 22, 2023 R. Poli - University of Essex 63
Conic fitness function, 5 particles, gravity, springs (ring), friction
Apr 22, 2023 R. Poli - University of Essex 64
Quadratic fitness function, 5 particles, gravity, friction
Apr 22, 2023 R. Poli - University of Essex 65
Quadratic fitness function, 5 particles, gravity, no friction, springs (ring)
Apr 22, 2023 R. Poli - University of Essex 66
Quadratic fitness function, 5 particles, gravity, friction, springs (ring)
Apr 22, 2023 R. Poli - University of Essex 67
Multimodal fitness function, 5 particles, gravity, friction
Apr 22, 2023 R. Poli - University of Essex 68
Multimodal function, 5 particles, gravity, friction, springs (ring)
Apr 22, 2023 R. Poli - University of Essex 69
Multimodal function, 5 particles, gravity, friction, springs (all connected)
Apr 22, 2023 R. Poli - University of Essex 70
Rastrigin fitness function, 20 particles, gravity, friction
Apr 22, 2023 R. Poli - University of Essex 71
Rastrigin function, 20 particles, gravity, friction, springs (all connected)
Apr 22, 2023 R. Poli - University of Essex 72
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.