I/ iultiobjective Genetic Algorithms for
iviultiscaling Excited-State Direct
Dynamics in Photochemistry

Kumara Sastry‘,  ...
Multiscaling Photochemical Reaction Dynamics

ozo Multi-scale modeling is ubiquitous in science & engineering
0 Phenomena ...
Outline

~: o Introduction:  Background and Purpose

oz» Method for multiscaling reaction dynamics
+ Limitations of existi...
Photochemical Reaction

Vision
h’V 0<l(»-lls

  net‘. .. 0   
<: :~

_ _, / /  ,. §’
gdfl
Ground Estate Excited state  ‘ , ...
Accurate Simulation of Reaction Dynamics Isn't Easy

~: » Ab inizio quantum chemistry methods: 
+ Ab initio multiple spawn...
Why Does This Matter? 

-: ~ Multiscaling speeds all modeling of physical problems: 
0 Solids,  fluids,  thermodynamics,  ...
Methodology:  Limited Ab Initio and Expermiental
Results to Tune Semiempirical Parameters

-: ~ Perform ab initio computat...
Current Reparameterization Methods Fall Short

~: ~ Staged single objective optimization
o First minimize error in energie...
Fitness:  Errors in Energy and Energy Gradient

~: o Choose a few ground- and excited-state configurations

-: » Fitness #...
Chromosome:  Real-Valued Encoding of
Semiempirical Parameters

~: » Consists of 11 semiempirical parameters for carbon
A M...
Multiobjective GA:  NSGAII with Binary
Tournament,  SBX,  and Polynomial Mutation
~: ~ Non-dominated sorting GA-ll (NSGA-l...
‘o

0

00

O

00

Overview of NSGA-ll

Initialize Population
Evaluate fitness of individuals

Selection:  “Survival of the...
Q0

0

O
0.0

00

Non-dominated Sorting Procedure

Indentify the best non-dominated set

o A set of solutions that are not...
Crowding In NSGA-II for Niche Preservation

oz» Crowding (niche-preservation) in objective space

oz» Each solution is ass...
Elitist Replacement in NSGA-II

oz» Combine parent and offspring population

oz» Select better ranking individuals and use...
Test Molecules:  Ethylene and Benzene

oz» Tune semiempirical potentials for ethylene and benzene
oz» Fundamental building...
Population Size of 800 Yields Good Solutions

oz» 5 independent runs with n =  2000 for 200 generations
«to Best non-domin...
Population Size of 800 Yields Good Solutions

Population size,  n =  100

Population size,  n =  200

-3 —I
-‘ N) A
,  {"1...
Run Duration of 100 Generations ls Appropriate

oz» 10 independent runs with n =  800
9 Rapid improvement up to gen.  25

...
Ethylene:  GA Finds Physical and Accurate PES

1.8

0 - -~ - :  ' . 

e4 16 5 3 : '.§g2.ii.3:§: §:Lai oz» 226% lower error...
GA Optimized Semiempirical Potentials are Physical

oz» Dynamics agree with ab initio results

oz» Energetics on untested,...
GA Optimized Semiempirical Potentials are Physical

oz» Energy difference between planar and twisted geometery should
be g...
Error in energy gradient,  5 (eVIA°)

Benzene:  GA Finds Physical and Accurate PES

(.0
0|

0 o NSGA-II

or Tonroetaiizooo...
Summary of Key Results

oz» Yields multiple parameter sets that are up to 226% lower
energy error and 87°/ o lower gradien...
Conclusions

oz» Broadly applicable in chemistry and materials science

4. Analogous applicability when multiscaling pheno...
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Multiobjective Genetic Algorithms for Multiscaling Excited State Direct Dyanmics in Photochemistry

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This paper studies the effectiveness of multiobjective genetic and evolutionary algorithms in multiscaling excited state direct dynamics in photochemistry via rapid reparameterization of semiempirical methods. Using a very limited set of ab initio and experimental data, semiempirical parameters are reoptimized to provide globally accurate potential energy surfaces, thereby eliminating the need for a full-fledged ab initio dynamics simulations, which is very expensive. Through reoptimization of the semiempirical methods, excited-state energetics are predicted accurately, while retaining accurate ground-state predictions. The results show that the multiobjective evolutionary algorithm consistently yields solutions that are significantly better—up to 230% lower in error in energy and 86.5% lower in error in energy-gradient—than those reported in literature. Multiple high-quality parameter sets are obtained that are verified with quantum dynamical calculations, which show near-ideal behavior on critical and untested excited state geometries. The results demonstrate that the reparameterization strategy via evolutionary algorithms is a promising way to extend direct dynamics simulations of photochemistry to multi-picosecond time scales.

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Multiobjective Genetic Algorithms for Multiscaling Excited State Direct Dyanmics in Photochemistry

  1. 1. I/ iultiobjective Genetic Algorithms for iviultiscaling Excited-State Direct Dynamics in Photochemistry Kumara Sastry‘, D. D. Johnsonz, A. L. Thompson‘, D. E. Goldberg‘, T. J. Martinez3, J. Leiding3, J. Owens? ’ ‘Illinois Genetic Algorithms Laboratory, Industrial and Enterprise Systems Engineering 2Materials Science and Engineering 3Chemistry and Beckman Institute University of Illinois at Urbana-Champaign . _ , _f__, _’v , out’. . _ MRSE Supported by AFOSR F49620—O3—I—0129, NSF/ DMR at MCC Dl'. ~IR»O3—76550 1”‘ v~ {LL? (3’.1
  2. 2. Multiscaling Photochemical Reaction Dynamics ozo Multi-scale modeling is ubiquitous in science & engineering 0 Phenomena of interest are usually multiscale o Powerful modeling methodologies on single scale ~: - Multiscaling photochemical reaction dynamics Ab Initio Tune . . . . S ' ' ' I Quantum Chemlstry : > Semiempmcal 1? emlempmca Methods methods Potentials Accurate but slow (hours-days) Fast (secs-mins). Accuracy depends on semiempirical potentials -: - Use multiobjective genetic algorithm for tuning semiempirical potentials for multiscaling reaction dynamics
  3. 3. Outline ~: o Introduction: Background and Purpose oz» Method for multiscaling reaction dynamics + Limitations of existing methods -: o Problem formulation «: ~ Overview of NSGA-ll oz» Results and Discussion ~: ~ Summary and Conclusions
  4. 4. Photochemical Reaction Vision h’V 0<l(»-lls net‘. .. 0 <: :~ _ _, / / ,. §’ gdfl Ground Estate Excited state ‘ , E . E _ H luau ”“£'_°‘~_ Photosynthesis Pollution Solar energy Basic Photosynthesis Eroctron mmolor xx; accenool >Fur. tbn _, .— lioncttovv center cantor chlorophyll, . . . . . . ~<-eyft ‘ . . V .7 « _‘ N; ‘ . . ~ "X _ L. ‘I07 vv 4 _I , .,u- ' I . w-. v l u> «_rI. }-1 ‘ "« '. I r. ‘ ~. came, ” 3,. .1»~ '; _ ml’. g '* W" . ' , _ _', _ - .3,‘ 1; II -I M‘ rvtv-L . ~ ll , u.. -. M : --'r '. '--‘fr’ ‘ w , . . . . ,I , : ' . , v .4.. _, Am a , pigment I molecules ) r 7 : “ r 0
  5. 5. Accurate Simulation of Reaction Dynamics Isn't Easy ~: » Ab inizio quantum chemistry methods: + Ab initio multiple spawning methods [Ben-Nun & Martinez, 2002] + Solve nuclear and electronic Schrodinger’s equations. ~ Accurate, but prohibitively expensive (hours-days) «: o Semiempirical methods: + Solve Schrodinger’s equations with expensive parts replaced with parameters. « Fast (secs-mins), accuracy depends on semiempirical potentials + Tuning semiempirical potentials is non-trivial + Energy & shape of energy landscape matter + Two objectives at the bare minimum :0: Minimizing errors in energy and energy gradient
  6. 6. Why Does This Matter? -: ~ Multiscaling speeds all modeling of physical problems: 0 Solids, fluids, thermodynamics, kinetics, etc. , o Example: GP used for multi-timescaling Cu-Co alloy kinetics [Sastry, et a] (2006), Physical Review B] oz» Here we use MOGA to enable fast and accurate modeling o Retain ab initio accuracy, but exponentially faster -: - Enabling technology: Science and Synthesis 9 Fast, accurate models permit larger quantity of scientific studies 9 Fast, accurate models permit synthesis via repeated analysis -: o This study potentially enables: o Biophysical basis of vision o Biophysical basis of photosynthesis o Protein folding and drug design o Rapid design of functional materials (zeolites, LCDs, etc. ,)
  7. 7. Methodology: Limited Ab Initio and Expermiental Results to Tune Semiempirical Parameters -: ~ Perform ab initio computations for a fewconfigurations o Both excited- and ground-state configurations o Augmented with experimental measurements -: - Standard parameter sets don’t yield accurate potential energy surfaces (PESS) o Example: AM1, PM3, MNDO, CNDO, INDO, etc. o Accurately describe ground-state properties o Yields wrong description of excited-state dynamics -: ~ Parameter sets need to be reoptimized 9 Maintain accurate description of ground-state properties o Yield globally accurate PES and hence physical dynamics
  8. 8. Current Reparameterization Methods Fall Short ~: ~ Staged single objective optimization o First minimize error in energies 0 Subsequently minimize weighted error in energy and gradient -: - Reparameterization involves multiple objectives 9 Don't know the weights of different objectives ~: - Reparameterization is highly multimodal o Local search gets stuck in low—quaIity optima -: - Current methods still fall short o Often doesn’t yield globally accurate PES o Yield uninterpretable semiempirical potentials o Semiempirical potentials are not transferable 4- Use parameters optimized for simple molecules in complex environments without complete reoptimization.
  9. 9. Fitness: Errors in Energy and Energy Gradient ~: o Choose a few ground- and excited-state configurations -: » Fitness #1: Error in energy and geometry o For each configuration, compute energy and geometry an Via ab initio and semiempirical methods . ,,, ,, f1 = Z | AEa, -_ — AE5e| +Differences in geometry 1 -: - Fitness #2: Error in energy gradient o For each configuration compute energy gradient 2: Via ab initio and semiempirical methods nu f2 = Z lVEaz'. — VEsel 1
  10. 10. Chromosome: Real-Valued Encoding of Semiempirical Parameters ~: » Consists of 11 semiempirical parameters for carbon A Most important parameters affecting excited-state PES »: Semiempirical parameters for hydrogen is not reoptimized + Set to their PM3 values »: ~ Core-core repulsion parameters are not optimized + Set to their PM3 values oz» Real-valued encoding of chromosomes -: - Variable ranges: 20-50% around PM3 values Retain reasonable representation of ground state PES
  11. 11. Multiobjective GA: NSGAII with Binary Tournament, SBX, and Polynomial Mutation ~: ~ Non-dominated sorting GA-ll (NSGA-ll) [Deb et (11, 2000] o: Binary tournament selection (s = 2) [Goldberg, Deb, 8. Korb, 1989] ~: « Simulated binary crossover (SBX) (r; ,.=5, pf = 0.9) [Deb 8. Agarwal, 1995; Deb 8. Kumar, 1995] oz» Polynomial Mutation (/7,, ,=1O, pm = 0.1) [Deb etal, 2000] «: ~ Results reported are best over 5, 10, and 30 NSGA-ll runs
  12. 12. ‘o 0 00 O 00 Overview of NSGA-ll Initialize Population Evaluate fitness of individuals Selection: “Survival of the non-dominated” o Non-dominated sorting 9 Individual comparison Recombination: Combine traits of parents Mutation: Random walk around an individual Evaluate offspring solutions Replacement: Best among parents and offspring
  13. 13. Q0 0 O 0.0 00 Non-dominated Sorting Procedure Indentify the best non-dominated set o A set of solutions that are not dominated by any individual in the population Discard them from the population temporarily Identify the next best non-dominated set Continue till all solutions are classified Fl: Rank :1 Fl: Rank = l F3: Rank = 2
  14. 14. Crowding In NSGA-II for Niche Preservation oz» Crowding (niche-preservation) in objective space oz» Each solution is assigned a crowding distance + Crowding distance = front density in the neighborhood 4. Distance of each solution from its nearest neighbors _ O .5.» 2 G . . 9: I '. >< (7 Solution B is more crowded than A 29 I 2 33lo- °é’ II : X 5 '. ,io >( L: I OQ: Error in energy
  15. 15. Elitist Replacement in NSGA-II oz» Combine parent and offspring population oz» Select better ranking individuals and use crowding distance to break the tie Non-dominated sorting F, ---------------------------- -- Population Parent F2 . ... ... ... ... ... ... ... ... ... .. in next . population F I generation 3 ---------- -- Offspring population Crowding distance sorting
  16. 16. Test Molecules: Ethylene and Benzene oz» Tune semiempirical potentials for ethylene and benzene oz» Fundamental building blocks of organic molecules oz» Play important role in photochemistry of aromatic systems oz» Extensively studied both theoretically and experimentally oz» Simple and thus amenable to rapid analysis + Verification using ab initio results + Exhaustive dynamics simulations - Transferability to more complex molecules 4. Expect less complex results thus easy interpretability oz» Complex enough and thus can test for validity of semiempirical methods on untested, yet critical configurations
  17. 17. Population Size of 800 Yields Good Solutions oz» 5 independent runs with n = 2000 for 200 generations «to Best non-dominated set assumed to be true Pareto front oz» Mahfoud’s population-sizing model suggests n = 750 + Maintain at least one copy of each optimum with 98% probability oz» Empirical results agree with the model prediction + 10 independent runs with n = 50, 100, 200,400, and 800 + Fixed total number of evaluations at 80,000 + With n = 800, NSGA-II finds almost all Pareto-optimal solutions oz» Suggests operators are appropriate for the search problem
  18. 18. Population Size of 800 Yields Good Solutions Population size, n = 100 Population size, n = 200 -3 —I -‘ N) A , {"15 > 9.0.00 i)Aa>a: 30:12 « 1'): 0:0: 1 2 3 Error in energy. eV Population size, n = 400 0 Error in energy. eV Population size, n = 800 . ". X 1.4 1.4? 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 20 1 2 3 0'20 1 2 3 Error in energy gradient, eV/ /9 Error in energy gradient, eVl/9 Error in energy gradient, eV/ A’ Error in energy gradient, eVlA’ Error in energy, eV Error in energy, eV
  19. 19. Run Duration of 100 Generations ls Appropriate oz» 10 independent runs with n = 800 9 Rapid improvement up to gen. 25 o Gradual improvement up to gen. 100 1.8 -6-Gen. 0 -13- Gen. 10 — -+>-Gen. 25 0 Gen. 50 -0FGen. 100 _| O) l —l is _ N) 0 on Error in energy gradient, (eVIA°) .0 .0 A 9’) -3 0 N J. is é 10 Error in energy, f1, (eV) 0 M
  20. 20. Ethylene: GA Finds Physical and Accurate PES 1.8 0 - -~ - : ' . e4 16 5 3 : '.§g2.ii.3:§: §:Lai oz» 226% lower error in energy 3 Owens (2004): Physical 3. ‘ms " °"°”“"’°°“’i”""“"'°" ‘ oz» 32.5% lower error in energy .75 12: 5 gradient 8 g 1' oz» All solutions below 1.2eV § °'8 ‘? . x,, .. error in energy yield 5 06 "° "” ‘ I b II t PES §- 3 goayaccurae in 04 i "‘°'"°° 11+ +1- 4+ + 0 OK) 1 2 3 4 5 6 7 Error in energy, f1, ev oz» Significant reduction in errors oz» Globally accurate potential energy surfaces or Resulting in physical reaction dynamics oz» Evidence of transferability: “Holy Grail" in molecular dynamics
  21. 21. GA Optimized Semiempirical Potentials are Physical oz» Dynamics agree with ab initio results oz» Energetics on untested, yet critical configurations + cis-trans isomerization in ethylene + AM1, PM3, and other parameter sets yield wrong energetics ~ GA yields results consistent with ab illitin results E e’ _/ , Planar / . , . 1 . . ~: _/ Pyramidalized —‘ — — _ - ——-' ‘—— S _i / - I _ 7 -i Twisted » * . T Q _ (_('*. .,: >‘____ 61 1-) l . ——J _ - 1 - S _+ _.8 eV ‘ J/ , I _. _ _ _ _ 3 .1 _. 1 0 _l
  22. 22. GA Optimized Semiempirical Potentials are Physical oz» Energy difference between planar and twisted geometery should be greater than zero (ideally "2.8 eV) oz» Energy difference between pyramidalized and twisted geometry should be greater than zero (ideally "0.88 eV) 0 O NSGA-II Phys-cal ,5 + NSGA-llzunphyslcal E 0 Owensf2004)Pt1ysl: a| X Owensi2O04)Unonys1cal‘ Error in energy gradient f2. eVIA" O U. _. 125 2 Error in energy, 11, eV 0' M .5 (A3 30 3° A E(min so) « _ EIDM sl) (eV] I 5 g .1 E032‘ :1) - . Etpyi s1) (eV] 3 + *3 Q3 1 oz I I i O I 0 03 1 15 2 2T5 as 1 175 2 25 Error in energy ev Error in energy, ev
  23. 23. Error in energy gradient, 5 (eVIA°) Benzene: GA Finds Physical and Accurate PES (.0 0| 0 o NSGA-II or Tonroetaiizoooi oz» 46% lower error in energy 30 O 25% ° z» 86.5% lower error in energy °o gradient 20 . E * oz» All solutions below 8eV 15» 13% error in energy yield globally accurate PES _. O o 090 Ce» ll 6 8 10 Error in energy, f1 (eV) oz» Significant reduction in errors 01 ,0 12 70 oz» Globally accurate potential energy surfaces or. Resulting in physical reaction dynamics oz» Evidence of transferability: “Holy Grail" in molecular dynamics
  24. 24. Summary of Key Results oz» Yields multiple parameter sets that are up to 226% lower energy error and 87°/ o lower gradient error oz» Enables 102-105 increase in simulation time even for simple molecules oz» 10-103 times faster than the current methodology for tuning semiempirical potentials oz» Observed transferability is a very important to chemists + Enables accurate simulations without complete reoptimization 4. "Holy Grail" for two decades in chemistry & materials science. oz» Pareto analysis using rBOA and symbolic regression via GP 4» lnterpretable semiempirical potentials + New insight into multiplicity of models and why they exist.
  25. 25. Conclusions oz» Broadly applicable in chemistry and materials science 4. Analogous applicability when multiscaling phenomena is involved: Solids, fluids, thermodynamics, kinetics, etc. o: o Facilitates fast and accurate materials modeling . :» Alloys: Kinetics simulations with ab inirio accuracy. 104-107 times faster than current methods. . :. Chemistry: Reaction-dynamics simulations with ab initio accuracy.102-105 times faster than current methods. oz» Lead potentially to new drugs, new materials, fundamental understanding of complex chemical phenomena --. Science: Biophysical basis of vision, and photosynthesis o: o Synthesis: Pharmaceuticals, functional materials

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