How neural networks can help in kinetic Monte Carlo simulations in - PowerPoint PPT Presentation

International Workshop on “Beyond Molecular Dynamics: Long - time atomic scale simulations” Max Planck Institute for the Physics of Complex Systems - Dresden, Germany, 26-29 March, 2012 How neural networks can help in kinetic Monte Carlo simulations in alloys N. Castin, G. Bonny, D. Terentyev, L. Malerba Structural Materials Modelling and Microstructure Nuclear Materials Science Institute SCK  CEN, Mol – Belgium lmalerba@sckcen.be

Objective of this presentation  Show how the use of neural networks can be helpful to develop kMC models that are truly extensions of MD models  Emphasise that neural networks are a powerful numerical tool which does not, however, replace (or worse, obliterate) physics 2 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

Which processes are we interested in? Point-defect diffusion-driven processes occurring under irradiation <011> <110> Class “1.5” processes : We do not know where the system will go … … but we hope it will end up sufficiently close to the experiment we want to simulate And we know the system will get there via elementary processes for each of which it is generally possible to know initial and final state Main problem : in alloys, the actual rates of these elementary processes depend on the combinatorially large number of possible local atomic configurations 3 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

Atomistic KMC simulations to extend MD Point-defect diffusion-driven processes take too long for MD  Atomistic KMC simulations are widespread techniques to go beyond MD Example: BCC iron at 600 K <011> <110> Vacancy migration: 1 jump / 4 ns SIA migration: 1 jump / 4 ps Fundamental equations:    Diffusion jumps are thermally i E       m exp   event k activated processes / frequencies  i 0   k T B are used as probabilities Monte Carlo algorithm  i  k  N  1  2   i 0 1 Random number extraction, R n  [0,1] å G D µ Residence time algorithm t 1 i i Most physics is contained in the migration energies, E m 4 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

Migration energy calculation takes time i-j E SP def E j     i j i j E E E i-j E m m SP i E i saddle-point (SP) Point-defect migration energies can be calculated very accurately: DFT, interatomic potentials , … - Drag method, nudged elastic-band method , … - Problem : kMC simulations require migration energy in chemically changing enviroments to be calculated for each possible point-defect jump, at each timestep, to choose the event  The more accurate the calculation, the less effective the timescale extension as compared to MD  Approximations are used to speed up on-the-fly calculation 5 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

i are functions of the local environment E m i-j E SP def      i j i j E E E f ( c , c ) m SP i i SP  0 E j i-j E m c i = atomic configuration (positions & chemical nature) in state i  E=E j -E i  E/2 E i Frequent simplifying assumptions:  e . g . E  0 = constant, determined by      i j 1 ) E f ( c , c ) m i j 0 chemical nature of jumping atom 2 E is typically calculated - As summation of pair energy constants extended to closest neighbours (1 st & 2 nd ) or as cluster expansion – parameters nowadays fitted to DFT - From interatomic potentials used on rigid lattice (relaxation is also possible, but computationally costly) Problem : irrespective of how accurately  E is computed, correlation between thermodynamics (initial & final states) and kinetics is implicitly assumed 6 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

i are functions of the local environment E m i-j E SP def      i j i j E E E f ( c , c ) m SP i i SP E j i-j E m c i = atomic configuration (positions & chemical nature) in state i E i Frequent simplifying assumptions: nn = config. limited to 1 st -2 nd neighbour shell c X   i j nn 2 ) E f ( c ) SP SP  Typically f & g are sums of pair energy nn E g ( c ) i i constants (nowadays fitted to DFT) Problems : - Medium-to-long range chemical interactions and strain field effects are disregarded (especially serious for E SP ) - How reliable can a summation of close neighbour pair energy constants be to reproduce complex many-body interactions from DFT (think of concentrated alloys …)? 7 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

i are functions of the local environment E m i-j E SP def      i j i j  E E E f ( c , c ) m SP i i SP  E j i-j E m c i = atomic configuration (positions & chemical nature) in state i  E=E j -E i  E/2 E i Less frequent assumption: fitted coeff.  = any cluster cluster ex pansion       3 ) f ( c ) K K defined in lattice   SP 0 with atom at SP    i = on-site variable that takes a different value     i depending on the type of atom at site i of cluster    i cluster ex pansion   E           E f ( c ) E E i j E  i i 0 m 2  fitted coeff. Problem : Choice of clusters, where to truncate expansion, …: - convergence with cluster expansion is not easy matter! 8 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

i are functions of the local environment E m i-j E SP def      i j i j E E E f ( c , c ) m SP i i SP E j i-j E m c i = atomic configuration (positions & chemical nature) in state i E i Our assumption: A = sufficiently large cluster including all         i j 4 ) E f ; i A atoms that define the local environment m i around the migrating defect       ,   A Problem : - We do not know how this multivariable function looks like Solution : - Resort to powerful universal regression tool to build an approximation to this function: artificial neural network 9 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

How does it work in practice Local atomic environment as vector of on-site variables,  i A r 1 1 2 1 2 1 1 2 1 1 1 2 2 1 1 1 2 1 2 1 A r Artificial Neural Network Accurate NEB calculation with interatomic potential in separate box allowing full relaxation AKMC simulation box - rigid lattice Training ± 100 s ± 100 m s E SP i-j E j i-j E m Migration energy Database E i (10 4 points) (if error committed by ANN is estimated to be large …) Migration energy 10 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba

How an ANN looks like Artificial neural networks are considered as universal approximation tools : in theory they can reproduce any mapping between input and output variables { x i , i =1, …, N } Input layer Network of simple processing units arranged in layers (signal propagated Hidden layer(s) from top to bottom) Output layer æ ö æ ö H I ç ÷ o = tanh w o 0 + å w oj × tanh å x i × w ji + w j 0 { o j , j =1, …, n } ç ÷ ç ÷ è ø è ø 1 in our case j = 1 i = 1 Network “ synapses ” Input signals (on-site variables) to be found by training. 11 BEMOD12 – MPI-PKS, Dresden, 26-29 March, 20102 – L. Malerba