This work is supported by the AFOSR F49620-00-0163 and FA9550-06-1-0096, NSF DMR-99-76550 and DMR 03-25939 (MCC), DOE DEFG02-91ER45439 (MRL), and CSE fellowship, UIUC.

Genetic Algorithms and Genetic Programming for Multiscale Modeling: Applications in Materials

Science and Chemistry and Advances in Scalability

Kumara SastryIndustrial and Enterprise Systems Engineering

University of Illinois at Urbana-Champaignksastry@uiuc.edu

Outline

Background and motivation

Objective and proposed approach

Genetic algorithms (GAs) and Genetic programming (GP)

Applications:Multi-timescale modeling of alloy kineticsMultiscaling quantum-chemistry simulation

Advances in scalability:Population sizing for GPScalable GP designScalability limits of multiobjective GAs

Summary & Conclusions

Multiscaling is Ubiquitous in Science & Engineering

Many phenomena are inherently multiscalePhysics, chemistry, biology, materials science, biotech, etc.,

Need accurate and fast modeling methodsTo advance science and expedite synthesisAccommodate different system sizes and time scales

Time (picoseconds-minutes) & Space (Nanometers-Meters)

Existing modeling methodologies effective on single scaleElectronic structure calculations (Angstroms)Finite-element method (micro- to millimeters)

Bridging methods for effective multiscaling is non-trivialCurrent multiscale methods fall short

GAs and GP for Multiscale Materials Modeling

Need accurate and efficient multiscale methodsSparsely sample low-level modelsDevelop custom-made constitutive relationship

Use GAs & GP for bridging higher- and lower-level modelsRobust, competent, and efficientHandle multiple objectivesIndependent of representation

[Sastry et al (2004) IJMCE]

Genetic Programming

f = 0.5

f = 0.9f = 0.95

f = 0.1

f = 0.01

f = 0.11

Applications: Multiscale Modeling in Materials Science and Chemistry

Accuracy of modeling depends on accurate representation of potential energy surface (PES)

Both values and shape matter

Ab initio methods:Accurate, but slow (hours-days)Compute PES from scratch

Faster methods:Fast (seconds-minutes), accuracy depends on PES accuracy Need direct/indirect knowledge of PES

Known and unknown potential function/methodMultiscaling quantum chemistry simulations [Sastry et al (2006) GECCO; Sastry et al (2007) MMP]

Multi-timescaling alloy kinetics [Sastry et al (2006) Phys Rev B]

Molecular dynamics (MD): (~10–9 secs) many realistic processes are inaccessible.

Kinetic Monte Carlo (KMC): (~secs) need all diffusion barriers a priori. (God or compute)

Efficient Coupling of MD and KMCUse MD to get some diffusion barriers.Use KMC to span time.Use GP to regress all barriers from some barrier info.

Span 10–15 seconds to seconds (15 orders of magnitude)

Recap: Multi-timescale Modeling of Alloy Kinetics

Rea

l tim

e

Complexity

TableLookupKMC

On the fly KMC

SymbolicallyRegressed KMC

(sr-KMC)

ΔE calculated: ∼3% (256) configurationsLow-energy events: <0.1% prediction errorOverall events: <1% prediction error

Results: (001) Surface Vacancy-assisted Migration

Total 2nd n.n. Active configurations: 8192

Dramatic scaling over MD (109 at 300 K)

102 decrease in CPU time for calculating barriers

103-107 less CPU timethan on-they-fly KMC

chosen by the AIP editors as focused article of frontier research in Virtual Journal of NanoscaleScience & Technology, 12(9), 2005

Multiscaling Photochemical Reactions

Photochemistry is important for photosynthesis, vision, solar energy, pharmacology,…

Need methods with accurate description of excited states

Dynamics requires many electronic-energy evaluations

Ground stateHalorhodopsin Excited state

Silver “Humies” Medal in Human Competitive Results, Best paper award, Real-world applications track, GECCO-2006 (ACM SIGEVO conference)

Reaction Dynamics Over Multiple Timescales

Accurate but slow (hours-days)Can calculate excited states

Fast (secs.-mins.), accuracy depends on parameters.Calculate integrals from fit parameters.

Ab InitioQuantum Chemistry

Methods

SemiempiricalMethods

TuneSemiempiricalParameters

Accurate excited-state surfaces with semiempirical methodsPermits dynamics of larger systems: proteins, nanotubes, etc.,

Fitting/Tuning semiempirical potentials is non-trivial

Energy & shape of the PES matterEspecially around ground and excited states

Choose few ground- and excited-state configurationsFitness #1: Errors in energy

For each configuration, compute energy difference via ab initio and semiempiricalmethods

Fitness #2: Errors in energy gradientFor each configuration, compute energy gradient via ab initio and SE methods

Fitness: Errors in Energy and Energy Gradient

ab initio SE method

ab initio SE method

*O OO

Current Reparameterization Methods Fall Short; Need Multiobjective Optimization

Current Method: Staged single-objective optimizationFirst minimize error in energiesSubsequently minimize weighted error in energy and gradient

Multiple objectives and highly multimodalDon’t know the weights of different objectivesLocal search gets stuck in low-quality optima

Simultaneously obtain “Pareto-optimal” solutions.

Avoid potentially irrelevant and unphysical pathways.

Multiobjective optimization

Multiobjective Optimization

Unlike single-objective problems, multiobjective problems involve a set of optimal solutions often termed as Pareto-optimal solutions.

Notion of non-domination [Goldberg, 1989]

• A dominates C• A and B are non-dominant• B is more crowded than A

Solution A dominates C if:A is no worse than C in all objectivesA is better than C in at least one objective

Two goals:Converge onto the Pareto-optimal solutions (best non-dominated set )Maintain as diverse a distribution as possible

[Deb, 2002; Coello Coello et al, 2002]

Multi- vs. single-objective optimization

MOGA Finds High Quality SE Parameters

Not even one Pareto-optimal solution obtained with multiple weighted single-objective optimizations

MOGA results have significantly lower errors than current results.

Are all MOGA solutions good from chemists perspective?

MOGA vs. published results

Each point is a set of 11 parameters

MOGA Population Quantifies Parameter Stability

GA population contains useful data that can be minedE.g., 80,000 candidate solutions sampled in each GA run.

Example: stability/sensitivity of parameter setsRMS deviation of the error in energy and energy gradients of solutions within 1% of the Pareto-optimal solutions

Stable parameter set has low RMS deviations.

How to set the threshold?Perturbation of 1% around PM3 set reveals

RMS deviation in error in energy: 0.99RMS deviation in error in energy gradient: 0.023

On-Line Parameter Stability Analysis in MOGA

Analysis of quality of solutions around Pareto-optimal set yields a good measure of the SE parameter stability.

Online analysis is efficient and reliable

MOGA Results Yield Accurate Engergies

MOGA-optimized parameters yield accurate energies for untested, critical configurations.

Parameter sets with lower error in energy are preferable

Pyramidalized

E

D2d twisted

+ E(Pyr-CI S1)∗ E(D2d S1) - E(D2d S0)x E(D2d S1) - E(Pyr-CI S1)

Check potential energy surface with dynamics

Population transfer determined using 50 initial conditions for each parameter setParameter sets with lower error in energy gradient values have lifetimes close to ab initio value

[Quenneville, Ben-Nun & Martinez, 2001]

ab initio value: 180±50 fs

Semiempirical Parameter Interaction Identification

Multiple high-quality parameter sets

Symbolic regression of semiempirical parameters via GP

Interpretable optimized semiempiricial methods

Advances in Scalability: GP Population Sizing, Competent GP, and Scalability of MOGAs

Premium on competent and efficient GAs and GPNeed to understand their scalabilityGA and GP designs that solve hard problems quickly, reliably, and accurately

Population Sizing in GP [Sastry, et al (2003) GPTP; Sastry, O’Reilly, & Goldberg (2004) GPTP]

Building-block supply and decision-making grounds

Competent GP design [Sastry & Goldberg (2003). GPTP]

Automatically identify and exchange key building blocks

Scalability of competent MOGAs [Sastry, Pelikan, & Goldberg (2005) CEC; Pelikan, Sastry & Goldberg (2006) SOPM]

Reliably maintain diverse Pareto-optimal solutions

Recap: How do we size populations?

Tradeoff: Expense = # runs x popsize x # generations

Valuable tool for GP-theorists and practitionersInsight on GP mechanismsPractical guide to set population sizeCompetency: If it’s not a problem with pop size, it must be something elseScalability: How does population size scale with problem size?Efficiency: How to choose the tradeoff?

Limited attention paid in GP theoryCorrect population sizing is critical to GP success

[Sastry, et al (2003) GPTP; Sastry, O’Reilly, & Goldberg (2004) GPTP]

Population Sizing for Building Block Supply in GP

At least one copy of every competing schema in initial population:

Bigger the primitive set, greater the population size

Bigger the trees, smaller the population size

Expression also matters

[Holland, 1975; Goldberg, 1989; Reeves, 1993; Goldberg, Sastry, & Latoza, 2001; Sastry et al, 2004]

Tree size

# competing schemas

Error tolerance

Alphabet cardinalityExpression probability

GP Population Sizing: Decision-Making Model

Make correct decision between competing BBsStatistical in natureFitness is measure of the entire chromosome [Goldberg (1989); Goldberg & Rudnick (1992); Goldberg, Deb & Clark (1992);Harik, Cantú-Paz, Goldberg, & Miller (1997)]

[Goldberg, Deb, and Clark (1992)]

Population Sizing in GP for Good Decision Making

Population size increasesSignal-noise-ratio decreasesError tolerance decreasesTree size decreasesAlphabet size increasesBuilding block size increasesProblem size increasesBloat increases

Probabilistic safety factorSignal-to-noise ratio

Sub-componentcomplexity

Number ofsub-components

[Sastry, O’Reilly, & Goldberg (2004)]

Scalable GP Design

Design a competent GPSolve hard problems quickly, reliably, and accuratelyIdentify and exchange substructures effectivelyPolynomial scale-up on boundedly difficult problem

Combine ideas from GAs & GPExtended compact GA (eCGA) [Harik, 1999]

Probabilistic incremental program evolution (PIPE)[Salustowicz & Schmidhuber, 1997]

Study scale-up of competent GPGP-easy and GP-hard problems

[Sastry & Goldberg (2003). GPTP]

Extended Compact Genetic Algorithm (eCGA)

A Probabilistic model building GA [Harik, 1999]

Builds models of good solutions as linkage groups

Key idea:Good probability distribution ≡ Linkage learning

Key components:Representation: Marginal product model (MPM)

Marginal distribution of a gene partition

Quality: Minimum description length (MDL)Occam’s razor principleAll things being equal, simpler models are better

Search Method: Greedy heuristic search

PIPE: Model Representation & Sampling

Univariate model: Independent nodes [Salustowicz & Schmidhuber, 1997]

Start with root node, depth first, left-to-right traversalChoose either a function or terminal based on model

Extended Compact Genetic Programming: eCGP

Complete n-ary tree (similar to PIPE)Marginal product model (similar to eCGA)

Partition tree-nodes into clustersMarginal probability distribution of each cluster

Scalability of eCGP: GP-easy and GP-hard Problems

GP-Hard problem

eCGP scales as O(λk3)

Advantageous when linkage-learning is critical

Do such problems exist in GP domain?Broader issue of problem difficulty in GPOptimization vs. System identification

GP-Easy problem

[Sastry, & Goldberg, 2003]

Scalability of Multiobjective GAs

Multiobjective estimation of distribution algorithms (EDAs) [Bosman & Thierens, 2002, Khan et al, 2002, Ocenasek, 2002, Ahn, 2005]

Model-building and model-sampling of EDAsSelection and replacement of multiobjective GAs

Outperform MOGAs on boundedly-difficult additively and hierarchically decomposable problems

Limited scalability analysisDemonstrated scalability is a strong point of EDAs

How do MOEDAs scale with problem size on additively separable boundedly-difficult problems?

[Sastry, Pelikan, & Goldberg (2005) CEC; Pelikan, Sastry & Goldberg (2006) SOPM]

Usual Scalability Game

Boundedly-difficult problem where linkage learning is criticalScalability as a function of # building blocks, mUse similar approach for MOEDAs

Overwhelming The Nicher is Easy!

Building blocks are accurately identified and sampled

Test problem has exponential # Pareto-optimal solutions Optimal solutions composed by 0000 and 1111 blocks2m solutions in the Pareto-optimal setm+1 distinct points in the Pareto-optimal set

At least one copy of all 2m solutions

O(2m)

OneMax-ZeroMax

MOEDAs Outperform Simple MOGAs

[Pelikan, Sastry, Goldberg (2005)]

Practical population sizes can’t yield all optimal solutionsCapture representative solutions on the Pareto-optimal frontEDAs outperform simple MOEAs [Pelikan, Sastry, Goldberg (2005)]

Controlled Growth of Competing Building Blocks

Understand the fundamental limit of growth in # Pareto-optimal solutions that can yield polynomial scale-up

Controlledgrowth of competing

building blocks

O(m2)

[Sastry, Pelikan, & Goldberg, 2005]

TrapTrapTrap Trap Trap

Inv TrapTrapTrap Trap Inv Trap

Obj #1

Obj #2

# competing BBs = 2

eCGA pop. size:

Niching pop. size:

Ensuring polynomialscalability:

Summary and Conclusions

Broadly applicable in chemistry and materials scienceAnalogous applicability in modeling multiscaling phenomena.

Facilitates fast and accurate materials modelingAlloys: Kinetics simulations with ab initio accuracy.

102-1015 increase in simulation time.104-107 times faster than current methods

Chemistry: Reaction-dynamics with ab initio accuracy.102-105 increase in simulation time10-103 times faster than current methods.

Lead potentially to new drugs, new materials, fundamental understanding of complex chemical phenomena

Science: Biophysical basis of vision, and photosynthesisSynthesis: Pharmaceuticals, functional materials