Abstract:
Effective and efficient multiscale modeling is essential to advance both the science and synthesis in a wide array of fields such as physics, chemistry, materials science, biology, biotechnology and pharmacology. This study investigates the efficacy and potential of using genetic algorithms for multiscale materials modeling and addresses some of the challenges involved in designing competent algorithms that solve hard problems quickly, reliably and accurately. In particular, this thesis demonstrates the use of genetic algorithms (GAs) and genetic programming (GP) in multiscale modeling with the help of two non-trivial case studies in materials science and chemistry.

The first case study explores the utility of genetic programming (GP) in multi-timescaling alloy kinetics simulations. In essence, GP is used to bridge molecular dynamics and kinetic Monte Carlo methods to span orders-of-magnitude in simulation time. Specifically, GP is used to regress symbolically an inline barrier function from a limited set of molecular dynamics simulations to enable kinetic Monte Carlo that simulate seconds of real time. Results on a non-trivial example of vacancy-assisted migration on a surface of a face-centered cubic (fcc) Copper-Cobalt (CuxCo1-x) alloy show that GP predicts all barriers with 0.1% error from calculations for less than 3% of active configurations, independent of type of potentials used to obtain the learning set of barriers via molecular dynamics. The resulting method enables 2–9 orders-of-magnitude increase in real-time dynamics simulations taking 4–7 orders-of-magnitude less CPU time.

The second case study presents the application of multiobjective genetic algorithms (MOGAs) in multiscaling quantum chemistry
simulations. Specifically, MOGAs are used to bridge high-level quantum chemistry and semiempirical methods to provide accurate representation of complex molecular excited-state and ground-state behavior. Results on ethylene and benzene—two common building-blocks in organic chemistry—indicate that MOGAs produce high-quality semiempirical methods that (1) are stable to small perturbations, (2) yield accurate configuration energies on untested and critical excited states, and
(3) yield ab initio quality excited-state dynamics. The proposed method enables simulations of more complex systems to realistic multi-picosecond timescales, well beyond previous attempts or expectation of human experts, and 2–3 orders-of-magnitude reduction in computational cost.

While the two applications use simple evolutionary operators, in order to tackle more complex systems, their scalability and limitations have to be investigated. The second part of the thesis addresses some of the challenges involved with a successful design of genetic algorithms and genetic programming for multiscale modeling. The first issue addressed is the scalability of genetic programming, where facetwise models are built to assess the population size required by GP to ensure adequate supply of raw building blocks and also to ensure accurate decision-making between competing building blocks.

This study also presents a design of competent genetic programming, where traditional fixed recombination operators are replaced by building and sampling probabilistic models of promising candidate programs. The proposed scalable GP, called extended compact GP (eCGP), combines the ideas from extended compact genetic algorithm (eCGA) and probabilistic incremental program evolution (PIPE) and adaptively identifies, propagates and exchanges important subsolutions of a search problem. Results show that eCGP scales cubically with problem size on both GP-easy and GP-hard problems.

Finally, facetwise models are developed to explore limitations of scalability of MOGAs, where the scalability of multiobjective algorithms in reliably maintaining Pareto-optimal solutions is addressed. The results show that even when the building blocks are accurately identified, massive multimodality of the search problems can easily overwhelm the nicher (diversity preserving operator) and lead to exponential scale-up. Facetwise models are developed, which incorporate the combined effects of model accuracy, decision making, and sub-structure supply, as well as the effect of niching on the population sizing, to predict a limit on the growth rate of a maximum number of sub-structures that can compete in the two objectives to circumvent the failure of the niching method. The results show that if the number of competing building blocks between multiple objectives is less than the proposed limit, multiobjective GAs scale-up polynomially with the problem size on boundedly-difficult problems.

Abstract:

A bottleneck for multi-timescale thermally-activated dynamics is the computation of the potential energy surface (PES). We explore the use of genetic programming (GP) to symbolically regress a mapping of the saddle-point barriers from only a few calculated points via molecular dynamics, thereby avoiding explicit calculation of all barriers. The GP-regressed barrier function enables use of kinetic Monte Carlo (KMC) to simulate real-time kinetics (seconds to hours) based upon realistic atomic interactions. To illustrate the concept, we apply a GP regression to vacancy-assisted migration on a surface of a concentrated binary alloy (from both quantum and empirical potentials) and predict the diffusion barriers within ~0.1% error from 3% (or less) of the barriers. We discuss the significant reduction in CPU time (4 to 7 orders of magnitude), the efficacy of GP over standard regression, e.g., polynomial, and the independence of the method on the type of potential.

This paper discusses scalability of standard genetic programming (GP) and the probabilistic incremental program evolution (PIPE). To investigate the need for both effective mixing and linkage learning, two test problems are considered: ORDER problem, which is rather easy for any recombination-based GP, and TRAP or the deceptive trap problem, which requires the algorithm to learn interactions among subsets of terminals. The scalability results show that both GP and PIPE scale up polynomially with problem size on the simple ORDER problem, but they both scale up exponentially on the deceptive problem. This indicates that while standard recombination is sufficient when no interactions need to be considered, for some problems linkage learning is necessary. These results are in agreement with the lessons learned in the domain of binary-string genetic algorithms (GAs). Furthermore, the paper investigates the effects of introducing unnecessary and irrelevant primitives on the performance of GP and PIPE.

We propose the use of genetic programming (GP)—a genetic algorithm that evolves computer programs—for bridging simulation methods across multiple scales of time and/or length. The effectiveness of genetic programming in multiscale simulation is demonstrated using two illustrative, non-trivial case studies in science and engineering. The first case is multi-timescale materials kinetics modeling, where genetic programming is used to symbolically regress a mapping of all diffusion barriers from only a few calculated ones, thereby avoiding explicit calculation of all the barriers. The GP-regressed barrier function enables use of kinetic Monte Carlo for realistic potentials and simulation of realistic experimental times (seconds). Specifically, a GP regression is applied to vacancy-assisted migration on a surface of a binary alloy and predict the diffusion barriers within 0.1–1\% error using 3\% (or less) of the barriers. The second case is the development of constitutive relation between macroscopic variables using measured data, where GP is used to evolve both the function form of the constitutive equation as well as the coefficient values. Specifically, GP regression is used for developing a constitutive relation between flow stress and temperature-compensated strain rate based on microstructural characterization for an aluminum alloy AA7055. We not only reproduce a constitutive relation proposed in literature, but also develop a new constitutive equation that fits both low-strain-rate and high-strain-rate data. We hope these disparate example applications exemplify the power of GP for multiscaling at the price, of course, of not knowing physical details at the intermediate scales.

Abstract:
This paper derives a population sizing relationship for genetic programming (GP). Following the population-sizing derivation for genetic algorithms in Goldberg, Deb, and Clark (1992), it considers building block decision making as a key facet. The analysis yields a GP-unique relationship because it has to account for bloat and for the fact that GP solutions often use subsolutions multiple times. The population-sizing relationship depends upon tree size, solution complexity, problem difficulty and building block expression probability. The relationship is used to analyze and empirically investigate population sizing for three model GP problems named ORDER, ON-OFF and LOUD. These problems exhibit bloat to differing extents and differ in whether their solutions require the use of a building block multiple times.

Abstract:
This paper describes a probabilistic model building genetic programming (PMBGP) developed based on the extended compact genetic algorithm (eCGA). Unlike traditional genetic programming, which use fixed recombination operators, the proposed PMBGA adapts linkages. The proposed algorithms, called the extended compact genetic programming (eCGP) adaptively identifies and exchanges non-overlapping building blocks by constructing and sampling probabilistic models of promising solutions. The results show that eCGP scales-up polynomially with the problem size (the number of functionals and terminals) on both GP-easy problem and boundedly difficult GP-hard problem.

Abstract:
This paper analyzes building block supply in the initial population for genetic programming. Facetwise models for the supply of a single schema as well as for the supply of all schemas in a partition are developed. An estimate for the population size, given the size (or size distribution) of trees, that ensures the presence of all raw building blocks with a given error is derived using these facetwise models. The facetwise models and the population sizing estimate are verified with empirical results.