Diffusion benchmark by Peterson

In Petersons original publication on machine learning accelerated nudged elastic band the method is first tested on a simple benchmark of a gold atom moving on a Al-Pt slab.

The initial guess for the trajectory visualized above can be generated using the following snippet:

from ase import io
from ase.build import fcc100, add_adsorbate
from ase.constraints import FixAtoms
from ase.calculators.emt import EMT
from ase.optimize import QuasiNewton
from ase.neb import NEB

# 2x2-Al(001) surface with 3 layers and an
# Au atom adsorbed in a hollow site:
initial = fcc100('Al', size=(3, 2, 3))
# Replace two Al atoms by Pt:
initial[12].symbol = 'Pt'
initial[15].symbol = 'Pt'
add_adsorbate(initial, 'Au', 1.7, 'hollow')
initial.center(axis=2, vacuum=4.0)

# Fix all substrate atoms.
mask = [atom.tag > 0 for atom in initial]
constraint = FixAtoms(mask=[atom.tag > 0 for atom in initial])
initial.set_constraint(constraint)

# Use EMT potential:
initial.set_calculator(EMT())

# Initial state:
qn = QuasiNewton(initial)
qn.run(fmax=0.05)

# Final state:
final = initial.copy()
final.set_calculator(EMT())
final[-1].x += final.get_cell()[0, 0] / 3
qn = QuasiNewton(final)
qn.run(fmax=0.05)

images = [initial]
for i in range(5):
    image = initial.copy()
    image.set_calculator(EMT())
    image.set_constraint(constraint)
    images.append(image)
images.append(final)

neb = NEB(images)
neb.interpolate()

io.write('diffusion.traj', images)
io.write('diffusion.gif', images, rotation='-80x,-30y,-5z')

Standard nudged elastic band

A calculation with the ASE standard NEB method can be run in a few lines:

from ase.io import read
from ase.calculators.emt import EMT
from ase.neb import NEB
from ase.optimize import FIRE

images = read('diffusion.traj', index=':')

for image in images:
    image.set_calculator(EMT())

neb = NEB(images)

opt = FIRE(neb)
opt.run()

and converges after 24 iterations:

      Step     Time          Energy         fmax
*Force-consistent energies used in optimization.
FIRE:    0 17:30:33        6.043962*       3.7816
FIRE:    1 17:30:33        5.912006*       3.2047
FIRE:    2 17:30:33        5.722624*       2.2477
...
FIRE:   22 17:30:37        5.490102*       0.0570
FIRE:   23 17:30:37        5.490091*       0.0514
FIRE:   24 17:30:37        5.490079*       0.0446

Machine learning accelerated nudged elastic band

Using Gaussian process regression with a squared exponential kernel

from ase.io import read
from ase.calculators.emt import EMT
from ase.neb import NEB
from mlpot.mlneb import run_mla_neb
from mlpot.calculators.gprcalculator import GPRCalculator
from mlpot.kernels import RBFKernel

images = read('diffusion.traj', index=':')

for image in images:
    image.set_calculator(EMT())

neb = NEB(images)

kernel = RBFKernel(constant=100.0, length_scale=1.0)
ml_calc = GPRCalculator(kernel=kernel, C1=1e8, C2=1e8, opt_restarts=1)

run_mla_neb(neb, ml_calc)

convergence is reached after just 3 evaluations of the band:

r_max = 1.49
Maximum force per image after 1 evaluations of the band:
[0.29568641 2.40409944 3.78158447 2.86624199 0.90700111]
Fit called with 7 geometries.
Starting hyperparameter optimization 1/1 with parameters:  [1.]
Finished hyperparameter optimization after 6 iterations  with value:  312.7611772253583  and parameters: [0.67832091]
Fit finished. Final RMSE energy = 0.000132, RMSE force = 0.000014.
Starting machine learning neb run.
      Step     Time          Energy         fmax
*Force-consistent energies used in optimization.
FIRE:    0 17:32:14        6.043930*       3.7816
...
FIRE:    7 17:32:14        5.109532*       0.4979
Switching to climbing image NEB
Starting machine learning neb run.
      Step     Time          Energy         fmax
*Force-consistent energies used in optimization.
FIRE:    0 17:32:14        5.109532*       0.4979
...
FIRE:   31 17:32:16        5.096236*       0.0233
Maximum force per image after 2 evaluations of the band:
[0.64545486 1.40822419 1.53210775 1.30412201 0.69528136]
Fit called with 12 geometries.
Starting hyperparameter optimization 1/1 with parameters:  [0.67832091]
Finished hyperparameter optimization after 9 iterations  with value:  368.2374911559936  and parameters: [0.77545366]
Fit finished. Final RMSE energy = 0.000409, RMSE force = 0.000040.
Starting machine learning neb run.
      Step     Time          Energy         fmax
*Force-consistent energies used in optimization.
FIRE:    0 17:32:23        6.043569*       3.7816
...
FIRE:    4 17:32:23        5.441238*       0.4421
Switching to climbing image NEB
Starting machine learning neb run.
      Step     Time          Energy         fmax
*Force-consistent energies used in optimization.
FIRE:    0 17:32:23        5.441238*       0.4421
...
FIRE:   11 17:32:24        5.439436*       0.0241
Maximum force per image after 3 evaluations of the band:
[0.01156111 0.03813662 0.01763575 0.00411534 0.01259674]
Converged after 3 evaluations of the band.