Supported by the NSF Division of Materials Research.ppt
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1、,Supported by the NSF Division of Materials Research,The Materials Computation Center Duane D. Johnson and Richard M. Martin (PIs) Funded by NSF DMR 03-25939,D.D. Johnson, D.E. Goldberg, and P. Bellon Students: Kumara Sastry (MSE/GE), Jia Ye (MSE) Departments of Materials Science and Engineering and
2、 General Engineering University of Illinois at Urbana-Champaign,Multiscale Modeling Methods for Materials Science and Quantum Chemistry,Multiscaling via Symbolic Regression,Genetic Programming: Machine-Learning Method for Multiscale Modeling,Ab Initio Accurate Semiempirical Quantum Chemistry Potenti
3、als via Multi-Objective GAs,First, what is a Genetic Programming (GP)? A Genetic Program is a genetic algorithm that evolves computer programs, requiring: Representation: programs represented by trees Internal nodes contain functionse.g., +, -, *, /, , log, exp, sin, AND, if-then-else, for Leaf node
4、s contain terminalse.g., Problem variables, constants, Random numbers Fitness function: Quality measure of the program Population: Candidate programs (set of individuals) Genetic operators: Selection: “Survival of the fittest”. Recombination: Combine parents to create offspring. Mutation: Small rand
5、om modification of offspring.,Goal: Evolve constitutive “law” between macroscopic variables from stress-strain data with multiple strain-rates for use in continuum finite-element modeling. Flow stress vs. temperature-compensated strain rate for AA7055 Aluminum Padilla, et al. (2004).GP fits both low
6、- and high strain-rate data well by introducing (effectively) a step-function between different strain-rate even though no knowledge of two sets of strain-rate data were indicated to GP. Automatically identified transition point via a complex relation, g, which models a step function between strain-
7、rates involved.GP identifies “law” with two competing mechanisms5-power law modeling known creep mechanism4-power law for as-yet-unknown creep mechanism.,1. Evolving Constitutive Relations,2. Multi-Timescale Kinetics Modeling Goal: To advance dynamics simulation to experimentally relevant time scale
8、s (seconds) by regressing the diffusion barriers on the PES as an in-line function.Molecular Dynamic (MD) or Kinetic Monte Carlo (KMC) based methods fall short 39 orders of magnitude in real time.Unless ALL the diffusion barriers are known in a look-up table.Table KMC has109 increase in “simulated t
9、ime” over MD at 300K.Our new “Symbolically-Regressed” KMC (sr-KMC) Use MD to get some barriers. Machine learn via GP all barriers as a regressed in-line function call, i.e. “table-look-up” KMC is replaced by function.,Summary Our results indicate that GP-based symbolic regression is an effective and
10、 promising tool for multiscaling. The flexibility of GP makes it readily amenable to hybridization with other multiscaling methods leading to enhanced scalability and applicability to more complex problems. Unlike traditional regression, GP adaptively evolves both the functional relation and regress
11、ion constants for transferring key information from finer to coarser scales, and is inherently parallel.,GP predicts all barriers with 0.1% error from explicit calculations of only 3% of the barriers. (Standard basis-set regressions fail.)GP symbolic-regression approach yields:102 decrease in CPU ti
12、me for barrier calculations.102 decrease in CPU over table-look-ups (in-line function call).104107 less CPU time per time-step vs. on-the-fly methods (note that each barrier calculation requires 10 s with empirical potential, 1800 s for tight-binding, and first-principles even more).(Future) Could c
13、ombine with pattern-recognition methods (e.g., T. Rahman et al.), or temperature-accelerated MD, to model more complex cooperative dynamics.(Current) Utilize the GP in-line table function obtain from tight-binding potential in a kinetic Monte Carlo simulation for this surface alloy vacancy-assisted
14、diffusion.,D.D. Johnson, T.J. Martinez, and D.E. Goldberg, Students: Kumara Sastry (MSE/GE) and Alexis L. Thompson (Chemistry) Departments of Materials Science and Engineering, Chemistry, and General Engineering University of Illinois at Urbana-Champaign,Goal: Functional augmentation and rapid multi
15、-objective reparameterization of semi-empirical methods to obtain reliable pathways for excited-state reaction chemistry.Ab Initio methods: accurate, but highly expensive.Semi-Empirical (SE) methods: approximate, but very inexpensive.Reparameterization based on few ab initio calculated data sets inv
16、olving excitations of a molecule, rather than low-energy (Born-Oppenheimer) states, e.g. use MNDO-PM3 Hamiltonian and find the MNDO parameters specific to particular molecular system.Involves optimization of multiple objectives, such as fitting simultaneously limited ab initio energy and energy-grad
17、ients of various chemical excited-states or conformations.(Future) Augmentation of functions may be needed.Propose: Multi-objective GAs for reparameterizationNon-dominate solutions represent physically allowed solutions, whereas dominant solutions can lead to unphysical solutions.Obtain set of Paret
18、o non-dominate solutions in parallel, not serially.Avoid potentially irrelevant pathways, arising from SE-forms, so as to reproduce more accurate reaction paths.(Future) Use Genetic Programming for functional augmentation, e.g., symbolic regression of core-core repulsions.Advantages of GA/GP Multi-O
19、bjective Optimizations, method is:robust, and yields good quality solutions quickly, reliably, and accurately, converges rapidly to Pareto-optimal ones, maintain diverse populations, suited to finding diverse solutions, niche-preserving methods may be employed, implicitly parallel search method, unl
20、ike applications of classic methods.,Kumara Sastry, D.D. Johnson, D.E. Goldberg, and P. Bellon, Int. J. of MultiScale Computational Engineering 2 (2), 239-256 (2004).,Analytic Estimate of Population Size vs. Empirical Results: Population size (no. of solutions kept to evolve) is a critical factor to
21、 ensure reliable solution. Shown is the probability that at least one copy of all raw subcomponents appear in population vs population size, n, for different tree sizes =2h, for the later diffusion example. Finding: population of 150-200 is enough.,Getting the Problems Measure of Fitness Problem-dep
22、endent choice: e.g., for diffusion, choose weighted (wi) least-squares fit of GP-derived vs. M calculated barriers, where wi = |EMD|1 as lower-energy barriers are more accessible than high-energy ones. Fit could to experimental data, too.,Getting the Problems Optimal Population Size,Application: Sur
23、face-vacancy-assisted diffusion in segregating CuxCo1-x.Using Molecular Dynamics based on density-functional, tight-binding, or empirical potentials, we calculate M (un)relaxed saddle-point energies E(xi) for atoms surrounding a vacancy with first and second neighbor environment denoted by 0 or 1 (f
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