Molecular dynamics with on-the-fly machine learning of quantum
mechanical forces
James Kermode
Warwick Centre for Predictive Modelling
School of Engineering
University of Warwick
www.warwick.ac.uk/wcpm
Warwick Centre for Predictive Modelling Seminar Series
5th March 2015
Warwick Centre for Predictive Modelling
www.warwick.ac.uk/wcpm
2
Multiscale Materials Modelling
Accuracy & transferability
Materials failure process: complex chemistry and large systems
3
Matching Atomistic and Continuum
Cracks in ideal brittle materials are atomically sharp
Divergence of stress field
near a crack tip
1
⇠p
r
Stillinger-Weber potential, Si(111)
20 A = 2 nm
In situ AFM image of atomically sharp crack
tip in silica (Image: C. Marlière)
Process zone: ~2 nm
G Singh, JR Kermode, A De Vita and RW Zimmerman, Int. J. Fract. (2014)
5
Multiscale Materials Modelling
Simultaneous coupling
QM/MM – buffered force mixing
• Retain QM
forces only in
(green) core
region.
• Throw away
‘bad’ QM forces
on buffer region
close to
artificial surface
Abrupt force mixing and extended buffer region solve electronic
termination problem, giving accurate QM forces.
N Bernstein et al. Rep. Prog. Phys. 72 026501 (2009)
8
QM/MM – buffered force mixing
Covalent Materials
Oxides
Metals
160
n.n. core
EAM average error
n.n.n. core
EAM average error
(b)
140
percentage error wrt PBC DFT
(a)
(c)
[112] direction
120
100
av. force:
1.6 eV/Å
av. force:
0.4 eV/Å
80
60
40
20
0
Silicon bulk and surface ~ 7 A
Silica (quartz, amorphous) ~ 10 A
2.5
5
width of the buffer (Angstrom)
7.5
Dislocation core in Ni ~ 5 A
Figure 3: Force error convergence with Figure
respect to
size in crystalline
bulk silicon (b) hydroxylated amorph
4: buffer
(a) thermalised
alpha-quartz,
and its 001 surface. Only CE used here.
N Bernstein
et al.
Rep.inProg.
Phys. 72
026501
(2009)Individual force errors for C
the same
slab
contact
with
water.
A Peguiron
et al., J.dots)
Chem.and
Phys,average
142 064116
(2015)
EE (yellow
force
error
9
for CE (red line) and
‘Learn on the Fly’ scheme
a
100 Å
211
01̄1
Molecular
Mechanics
(empirical,
fast)
1̄11
c
b
Force-based hybrid approach (cf. QM/MM)
✓Get QM forces right on a moving region,
giving accurate trajectories
✓Reproduces brittle response without any
empirical crack growth criteria
10 Å
10Add
Å chemical complexity near tip: wide range of
✓
chemomechanical problems in reach
d
Quantum Mechanics
(accurate, slow)
Code: http://www.libatoms.org
A. De Vita and R. Car, MRS Proc 491, 473 (1998)
G Csányi et al. Phys. Rev. Lett. 93 175503 (2004)
10
ctioa
his
of
red
me
onage
the
ng
all
ta
ack
no
on
onof
del
der
out
ave
on
01]
to
11)
his
hat
me
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his
3a.
the
the
mesed
ereA
eak
the
eron.
rgy
as
nded,
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ons
on
nd
ng
laway
ent
of
Recent Chemomechanical Applications
B2
A1
A2
B4
A3
A4
H induced ‘SmartCut’
B3
B1
Dynamical instabilities in Si
5Å
Crack-dislocation interactions
b
110
110
001
10 Å
JR Kermode et. al.
Nature 455 1224 (2008)
c
d
Three dimensional effects
G Moras, L Ciachhi, C Elsässer,
P Gumbsch and A De Vita
Phys. Rev. Lett. 105, 075502 (2010)
Crack-impurity scattering
C Gattinoni, JR Kermode and
A De Vita, In prep
Stress corrosion cracking
e
5 mm
Figure 3 | The (110)[1 10]
· crack system. a, Geometry of the crack tip
propagating straight at low speeds by sequential breaking of type A bonds
(blue). Red atoms are described using quantum mechanics, yellow atoms
using an interatomic potential. At each propagation step, the A bond is
weaker than the corresponding B bond because of a neighbouring undercoordinated atom (indicated by an arrow for the A1–B1 pair). b, At higher
speed an instability occurs. At this point any slight shear disturbance in the
stress field reverses the relative stability of A and B bonds, and the crack is
deflected onto a (111) plane. The energy release rate for this system is
G 5 6.7 J m22. In addition to the main tensile load, this includes a small
G 5 0.24 J m22 shear contribution to break the symmetry in the y direction.
JR Kermode,
A. Glazier,
Ga Kovel,
L Pastewka,
Both exposed
(111) crack surfaces
undergo
2 3 1 reconstruction.
6
c–e, Photographs
fromDthree-point
bending
(reprinted
G Csányi,
Sherman
and experiments
A. De Vita,
In prepwith
permission): (110) fracture surface for low-speed cleavage (grey; c); fracture
surface of a specimen with the crack deflected from the (110) plane (white) to
the (111) plane (black) for intermediate-speed cleavage (d); (111) fracture
surface obtained for high-speed cleavage (e).
hers Limited. All rights reserved
Fig 2. Subcritical crack advance on Si(110), catalysed by oxygen (red atoms). Left: the
system before chemisorption of second O2, Right: crack advance after dissociative
adsorption.
JR Kermode, L Ben-Bashat, F Atrash, JJ Cilliers, D Sherman and A. De Vita., Nat. Commun. 4 2441 (2013)
Machine-­learning of QM forces
Zhenwei Li’s PhD project has advanced considerably during this reporting period. In this
part of the overall project, we are working to improve the informational efficiency of our
modelling approach, particularly on the idea of building a database of previous QM results
“on-­the-­fly”. By reusing (rather than recalculating) the information stored in the database, it
is in principle possible to dramatically reduce the computational cost of our simulations,
while still retaining our target high (QM) accuracy. With this goal in mind, Zhenwei has
developed an advanced interpolation scheme for QM forces based on Gaussian Process
regression, with an accuracy which systematically improves as more data is added to the
database of reference configurations. The scheme has been so far tested both on MD and,
more stringently, on the phonon band structure of silicon [3].
A Glazier, G Peralta, JR Kermode, A De Vita and D Sherman, Phys. Rev. Lett, 112 115501 (2014).
Interaction between cracks and point defects
Our project on modelling the interaction between crack propagating on the (111) cleavage
plane in silicon and isolated substitutional boron defects has been concluded, and the
resulting paper is currently under peer review [4].
Our QM simulations predict three regimes: at low speed we see deflection of cracks in
perfect silicon crystals;; at intermediate speeds, crack deflection occurs at defect sites, at a
rate proportional to the linear defect concentration. At higher speeds the crack-­defect
interaction is dynamically hindered, and perfectly smooth fracture surfaces are recovered.
These predictions are completely consistent with experimental results obtained by our
11
g
n
i
k
c
a
St
lt
u
fa
e
n
a
l
p
QM region 1:
Crack Tip
QM region 2:
Dislocation Core
12
shielding configuration, the red circle indicates the location of
the dislocation core, and the red line the stacking fault. The
colour code shows the yy component of the stress tensor, in
units of GPa. Red/yellow represent tension, and dark blue
compression.
Crack/dislocation – coarse-graining
y/Å
FIG. 5: Snapshot from a LOTF simulation of System A, with
the dislocation above the crack, at the end of a 75 ps simulation. Away from the dislocation core and its associated
stacking fault (shown with red line), the fracture propagates
relatively straight along the [001] direction, with a reduced
velocity see text, not sure about this due to the antishielding e↵ect of the dislocation. Atom colouring is as in
Fig. ??.
Stress / GPa
Dislocation core below crack
Stress / GPa
Dislocation core above crack
This work was funded by the Rio Tinto Centre for Advanced Mineral Recover at Imperial College, London and
FIG.
SnapshotCommission
from a LOTFADGLASS
simulation of
System
B, where
the 4:
European
FP7
project.
We
the
dislocation
core is Adrian
located Sutton
below the
this pure
would
like to thank
for crack.
useful For
discussions.
shielding configuration, the red circle indicates the location of
the dislocation core, and the red line the stacking fault. The
colour code shows the yy component of the stress tensor, in
units of GPa. Red/yellow represent tension, and dark blue
compression.
⇤
Electronic address: c.gattinoni@imperial.ac.uk
x/Å
x/Å
Quantifying loss of information during coarse-graining: uncertainty propagation from atomistic to continuum
C. Gattinoni, J.R. Kermode and A. De Vita, in prep.
13
Microstructural Uncertainty
• Most of our work so far has only considered
deterministic materials failure problems
• Stochastic microstructural effects are also
extremely important
• e.g. fracture in glass, a prototypical amorphous
material, where microstructural variation plays
key role in determining fracture response
Goal: average over
representative microstructural
configurations to produce
effective mean-field response.
Probabilistic ‘error bars’
essential.
MathSys MSc Project proposal with C. Ortner (Maths)
http://www2.warwick.ac.uk/fac/sci/mathsys/courses/msc/mscprojects/projects2015
14
Target Application Areas
Covalent Materials
Oxides
Metals
Background
Hydrogen embrittlement (HE) is the most devastating and unpredictable, yet least u
failure experienced by engineering components. Hydrogen can be introduced into a
sources: eg from decomposition of water during processing, moisture in the environ
reactions in a corrosive (eg marine) environment or electrocoating (such as automo
decomposition of oil or grease, or direct exposure to H in storage vessels. The pres
severe degradation in mechanical properties and consequently a loss in structural i
metals and alloys. In particular, high-strength ferritic steels used for pipelines, press
weight vehicles, gears and in hydrogen storage are prone to hydrogen cracking. It i
strength of the steel, the more prone it is to HE. Despite the considerable research
over the last 30 years the mechanisms responsible for the embrittling process are n
considerable disagreement in the scientific literature concerning the underlying proc
hydrogen embrittlement, even in simple, nominally pure, material systems.
Diamond
160
140
percentage error wrt PBC DFT
Rocks
120
n.n. core
EAM average error
n.n.n. core
EAM average error
[112] direction
100
80
60
With increasing demands for advanced and ultra high strength steels, the steel indu
senstitivity to hydrogen embrittlement (HE). The concerns over HE are preventing t
using these materials more extensively, which is contributing to their failure to meet
emissions by up to 35 g/km. TWIP, TRIP and complex phase steels having UTS of
designed but cannot be put into service because of the risks of HE during manufac
mild humid conditions. It is therefore of significant technological importance to deve
understanding of hydrogen embrittlement and to use this understanding to underpin
generation of ultra high strength steels, that are resistant to hydrogen embrittlemen
40
The focus of this research programme will be on Hydrogen Embrittlement in Steels
The presence of hydrogen in interstitial sites results in dilation of the material. A co
wants to diffuse to regions of high tensile hydrostatic stress in a material, such as to
0
2.5
5
7.5
in Fig 1. The local high hydrogen concentration in the vicinity of the crack tip has a
width of the buffer (Angstrom)
stru
(a)
(b)
abo
this
Figure 1: (a) Convergence of QM forces near the core of a dislocation in the phase at room temperature
the
(solid lines, for nearest neighbours, blue n.n., and next nearest neighbours, red n.n.n. of the dislocation
muc
core). The dashed lines indicate the percentage error that the EAM potential makes with respect to DFT.
fact
Quantum calculations are only strictly needed for the nearest neighbours of a dislocation core. (b) AtomAn
istic model of a quadrupole of screw dislocations in a Ni-based superalloy. Inset: dissociation of one of
emb
the screw dislocations into Shockley partials, which can be tracked by two separate mobile QM regions
in th
(red circles).
Hyd
(HE
loca
Hyd
tran
Key Scientific Goals. The key targets of this project are to study the glide of screw dislocations
enh
in the phase (initially bulk, then closer to the / 0 interface), to evaluate the relevant diffusion
vac
0
the
mechanisms and barriers, and to study dislocation climb
at the / showing
interface,
including
the role
Fig 1 Schematic,
the diffusion
of hydrogen
towards a crack tip.
mec
Understanding
the
effect
of
hydrogen
on
crack
propagation
played by vacancies in this process. These overall targets can be decomposed into four work requires
20
Wednesday, 9 October 13
Glass
Silicon Photovoltaics
Superalloys
2
Project Structure and Resource Management
•
previous generation Intrepid BG/P machine).
QUIP ensemble )parallelism
performance. Ensemble parallelism
b( )a( over independent QM clusters
leads to weak scaling in the time-to-solution for computing all the QM forces (Fig. 9b), which is
the rate-determining
step of
forQM
our QM/MM dynamical simulations. The benchmark time shown
Force-mixing
allows work
QUIP
includes alltothe
currently required for communication between QUIP and VASP, although
calculations
beI/O
distributed
we plan to reduce this overhead using an improved interfacing strategy in future. Note that while
Automatically split large QM
in principle perfect weak-scaling should be obtained, our pilot implementation of theQMensemble
Box 1
regions
into
smaller
convex
parallelism is based on a proxy process running on the FEN as schematically illustrated in Fig
clusters
graphleads
partitioning
QUIP
8.proxy,bywhich
to a slight drop off in scaling at the largest partition
size of 131,072 cores
proxy
Box 2
Individual
QM calculations
(8192 nodes).
This is duerun
to in
the inability of a single FEN to keep up with the I/OQMworkload
parallel
neededonto~1024
servicecores
this each
many simultaneous calculations (one execution thread is required to
handle upeach
sub-block
Scalable
to ~100,000
corepartition). Efficiency improvements which replace file-based
communication
with a new solution using sockets are under active development, and
we are
Petascale
machines
QM Box N
confident this will recover the close-to-ideal weak scaling observed up to 65,536 cores.
HPC Scalability
BG/Q Execute Nodes
Front End Node
N clusters
N forces
•
Cluster 1
Force 1
•
1024 cores
QM calculation on Cluster 1
Cluster 2
1024 cores
QM calculation on Cluster 2
...
Force 2
•
QM/MM MD
with N QM atoms
Cluster N
Force N
) d(
1024 cores
QM calculation on Cluster N
)c(
Monday, 7 October 13
lc )011(iS eht no erutcarf fo noitalumis peed reyal 01 a morf tohspans )a( :krow gniogno fo sthgilhgiH .2 .giF
size and #
fo nrettap eht etoN .level TFD eht ta delledom tnorf kcarc eht gnola smota 005~ htiw ,1.1MWeak
ot tnascaling:
veler ,enproblem
alp
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f eht ggrow
nola together
sdnob
u esolC )c( .1.1M ot tnaveler ,)eulb ni dethgilhgih( smota MQ 002~ htiw ,ztrauq ni erutcarf elttirb fo noitalumis
erroc smota deruoloC .2.1M ot tnaveler ,)ssalg( acilis suohproma ni erutcarf fo noitalumis a ni noiger pit kcarc
bus edon 821 a no gninnur noitaluclac TFD tnereffid a ot gnidnopserroc ruoloc hcae htiw ,snoiger MQ 51 eht ot
ehcs gninoititrap fo sliated erom rof 2.3.1 noitceS ees( ereh sedon 8402 fo ezis noititrap latot a rof ,noititrap
.)txet ees( gninraeL enihcaM ecrof MQ rof gnisu selucelo4,096
m fo tecores
s elpmaxe
65,536 cores
4 QM clusters 64 QM clusters
M rof snoitaluclac noitcudorP .stceffe lanoisnemid-­eerht .1 -­ krow gniogno fo sthgilhgiH
d eerht fo snoitalumis desab-­TFD wen edulcni dna ,etelpmoc yltsom era )erutcarf enillatsyrc
uohtiw elbisaaef neeb evah ton dluow taht noitagaporp kcarc detavitca byllamreht fo msinahcem
elcun eht yb sdeeps wol yrev ta etagaporp nac skcarc taht dnif eW .secruoser ssalc pihsredaeL
17
s tsrif eht era Fig.
eseht9. ,eScaling
gdelwoplots
nk rufor:
o oaT VASP
.sgojcalculations
noitacolsidfor
ota ssingle
uogola200
na ,atom
sknikSiO2
fo nQM
oitarcluster.
gim b Ensemble parallelism
performance. For each data point, QUIP runs on one 1024 core sub-block partition, and each SiO2 QM cluster
‘Learn on the Fly’ predictor/corrector
‘Learn on the Fly’ scheme adds ‘springs’ tuned to reproduce QM forces
k1
k2
k3
k4
U (R) = Uclassical (R) +
n rn
springs n
LOTF vs. buffered force mixing:
✓ Both give correct
trajectories
✓ LOTF allows speed up with
Fit points
predictor/corrector
dynamics
Cheap classical forces + adj. springs
Expensive QM potential
G Csányi et al. Phys. Rev. Lett. 93 175503 (2004)
N Bernstein et al. Rep. Prog. Phys. 72 026501 (2009)
18
‘Learn on the Fly’ predictor/corrector
Extrap
Interp
Cheap classical forces + adj. springs
Fit points
Typically ΔT ~ 10 Δt
Expensive QM potential
G Csányi et al. Phys. Rev. Lett. 93 175503 (2004)
N Bernstein et al. Rep. Prog. Phys. 72 026501 (2009)
19
Towards Optimal Information Efficiency
• Recent trend of automatic, non-parametric force field construction using Machine Learning
techniques (e.g. NNs [1], GAPs [2]), extending force-matching approach
• Our Idea: rather than learning whole PES once-and-for-all, use same technique, Gaussian
Process regression, to predict QM forces as a function of local atomic environment
• Keeping “in touch” with QM reduces risk of extrapolating outside of fit domain
Past N
Accelerate QM/MM with MLbased scheme where forces on
atoms are:
• Either predicted using
Past 2
Past 1
QM calculation
Interpolation point
Present
Future
Time
Gaussian Process
regression
• Or computed on-the-fly
and added to growing ML
database
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
21
n function f (x), here any force component associated with the atomistic configuratio
aussian covariance function C(x, x0 , cov ) is a priori assumed to express the correlatio
f (x) and f (x0 ) when the distance between x and x0 is comparable with cov . Given
Gaussian
Process(y(GP)=regression:
use
Bayesian
inference
to
compute
most likely
functionto
values
et
{y}
of
values
f
(x
),
k
=
1,
N
)
which
the
force
component
is
known
take, withi
olecular
Dynamics
k
kwith On-The-Fly Machine Learning
given data and a prior distribution for model parameters
certainty err ,Mechanical
over a teaching Forces:
set {x} of Supplementary
atomic-configuration Material
data points (xk , k = 1, N )
Quantum
1D Example:
dicted
“posterior”
distribution
for the value
yN +1 at a new
Zhenwei
Li, James
R. Kermode
and Alessandro
Depoint
VitaxN +1 has mean an
Posterior
Gaussian Process Regression
ussian Process
∝
T
y
=
k
C
N +1
Regression
1
y
p(parameters|data)
L(data|parameters) p(parameters)
(S1
Likelihood
Prior
=  kT C 1 k
(S2
Process (GP) provides a nonparametric way to carry out Bayesian inference of an
f(x) vector’
ction
any used
forcetocomponent
associated
withBlack:
the function
atomistic
configuration
(x, x0 ,f (x),
has been
generate (i)
the N -dimensional
‘feature
k describin
cov ) here
an
covariance
function
C(x, x0configurations
, cov ) is a priori
to
express
the
correlation
riance
between
the database
{x} assumed
and the
new
‘test’
atomic
configuratio
Blue: observations 0
0
i)and
thefcovariance
between
the between
test configuration
itself and
(iii)
the cov
N⇥
covarianc
(x ) when the
distance
x and x and
is comparable
with
. N
Given
a
yi = f(xi)
Cofdescribing
the data
in {x}[1]: is known to take, within
values (ykthe
= foverlap
(xk ), k between
= 1, N ) which
thepoints
force component
Red:
GP mean,
std. dev.
nty err , over a teaching set {x} of✓atomic-configuration
data
points
(xk , k = 1, N ),
◆
2
d
2 point xN +1 has mean and
“posterior”
for=the
a new
Covariance distribution
N +1 at
+
(S3
Cmn
expvalue ymn
mn err .
2
(smoothness of prior)
2 cov
2
y,N +1
T
1
Covariance
matrix
C
note
that, as
an alternative
the
rotationally
invariant
internal
vector
(IV)
representatio
yNto
=
k
C
y
(S1)
Posterior
mean
+1
Feature vector
k
d in
the
main
text,
more
general
representations
could
be
chosen
requiring
the determi
2
T
1
Posterior variance
=  k C k
(S2)
Self-covariance κ
y,N +1
f the optimal rotational alignment of pairs of atomic configurations. Some work in thi
n, revealed
that
this
could
be
achieved
by
a
search
in
the
3D
rotations
space
using
a
stan
)
has
been
used
to
generate
(i)
the
N
-dimensional
‘feature
vector’
k
describing
cov
22
aternion
exploit the better
topology
of the
SUatomic
(2) manifold
through it
e betweenrepresentation
the database to
configurations
{x} and
the new
‘test’
configuration
QM
benchmark
(Fig.
Force
Gaussian Processes
for Fracture
at the Atomic
range reflecting the
decay rate/interaction
range configurations
of
forces
“internal”
representation for atomic
and
isSimulations
the value
usedScale
in1),
thishere
work.ob
Functional
(DFTB)
force vectors
[24]. Afterthe
carrying
out MLaccuin this reprecally Tight
impose Binding
in the system. Crucially,
to improve
prediction
err = 0.05 eV/Å,
the following
equation:
we transform
the predicted
force
back
into spectrum
the
regularising
thefrom
linearthe
algebra
[2
obtained
Stilli
racy, this set cansentation,
be expanded
to include
any additional
Cartesian representation, so that they have the correct
components
of the
k-dimensiona
sical
potential
[29]
is
also
shown f
vector presumed orientation
to carry for
useful
information
on
target
′
′
′
MD trajectory integration.
𝐓 𝐫 = r1 𝐔1 𝐓 𝐫(Supplementary
+ r2 𝐓𝐔2 𝐓 𝐫 + rEq.
𝐓 𝐫 its Car
3 𝐓𝐔3S1),
Application
to
force
learning:
compute
most
probable
force
given
database
of
atomic
environments
QM forces. TheseFor
areeach
typically
vectors
obtained
atom, k force
independent
internal
vectors (IVs)
can be reconstructed as F = A(1+
and well-established
associated QM
forces
(plus aforce
prior
assumption
that
forces
atoms
Comparing
equationvary
10.3
10.4, approach
we obtainas
the result:
from
classical
fields
orrelative
from
QM
Vi can
be uniquely
defined
by the
positions
rq with
ofsmoothly
where move).
the absolute or
its neighbours, expensive
which makesthan
them the
invariant
under transconfiguration is provided by the
models
lessa measure
computationally
current
Requires
of “similarity
for force learning”
of
two
atomic configs,
1 3)
T
𝐔the
𝐫 = 𝐓𝐔𝑖 (A
𝐫)+ =(𝑖(A
=T1,A)
2, and
(1
lations
and
any
permutation
of
neighbours
of
A
.
𝑖 𝐓 same
reference
(e.g., an empirical
tight
binding forces are vectors!
invariant Hamiltonian
to translation,
permutation
and
rotation.
Complication:
chemical species. A possible choice is:
As a first, stringent test of th
model if the main QM model is DFT-based, see inset
racy of our matrix representatio
" ✓
#
◆
Nneighb
p(i)
of Fig. 2). The directions V̂i X
= Vi /||Vi || form
performance in handling highly s
rq an in(1)
Internal
vectors
systemVwhich
i =
urations involving very small or n
ternal
coordinate
forr̂qk exp
> 3 gives
an
overr
(i)
cut
q=1
(i =1… k)
nal vectors and target forces, by
determined description of vector quantities. A represenwhereconfiguration
each of these vectors
is defined
by di↵erent
tation for the atomic
x is given
by the
k ⇥k values dispersion of crystalline Si using
The predicted phonon spectra c
of
the
parameters
r
and
p,
chosen
within
a
suitable
T
cut
Feature
matrix
matrix
X with
elements Xij = Vi · V̂j = (V A )ij where
QM benchmark (Fig. 1), here o
range reflecting the decay rate/interaction range of forces
we have defined the
vector
and
direction
matrices
Functional Tight Binding (DFTB
in
the
system.
Crucially,
to
improve
the
prediction
accu0
1
0
1
spectrum obtained from the Stil
0
1
0
1
|
| any additional
racy,| this| set can
| be expanded |to include
|. . . V̂ A on target sical potential [29] is also shown
|T = @
|vector
| presumed
T | @ |
A
A
=
to
carry
useful
information
V̂
V
.
.
.
V
V
1
k
1
T
Tk
@
A
@
A
. V
V = V1 . .QM
.| (2)obtained
k | ,A
These
typically
force
vectors
1 . . .| V̂
k|
| forces.
| = areV̂
Figure
10.1: QM
Schematic 2D projection of an atomic cluster within
from
|
|from |well-established classical
|
| force
| fields or
spherical radius r . Both external coordinate {x, y} and internal
Describing Atomic Environments
𝑐𝑢𝑡
models less computationally expensive thancoordinate
the current
{U(r01, m1), U(r02, m2)} are shown. Force F on the target
reference
(e.g., an under
empirical
tight
central binding
atom FIG.
(green ball)
indicated by the thickofblack
The Si
1. is Comparison
thearrow.
bulk
The feature matrix
X = Hamiltonian
V AT is invariant
transinternal
vectors
are functions of the positional vectors of the neighboring
model if the main QM model is DFT-based,
see
inset
lated
with DFTB (blue), and SW (
lations, permutations
and
rotations
of
the
corresponding
atoms (red balls) while the parameters of r0 and m are both tunable to
of Fig. 2). The directions V̂i = Vi /||Vi || form an inon-the-fly
approach (red), computed
generate different
internal vectors.
atomic configuration.
similarity
ternal The
coordinate
systembetween
which fortwo
k >atomic
3 gives an overment Parlinksi-Li-Kawazoe method
environments xmdetermined
and xn isdescription
expressedof by
thequantities.
distance
vector
A
represenIn order to establish a 3-D internal coordinate for an atomic cluster, at least three inte
eV/Å (dotted lines) and
err =0.05
(IVs)
are k
not⇥k
co-planar
have to be invented. Figure 10.1 demonstrates the IV
tation for the atomic configuration x isvectors
given
bythatthe
4
5
⇥
10
eV/
Å U(r
value
(solid
fo

a
two-dimensional
case,
where,
both
U(r
, m1) and
functionslines)
of the positi
T
01
02, m2) are
k
2
m
n
matrix
X
with
elements
X
=
V
·
V̂
=
(V
A
)
where
ij
i
j
ij
Li, JR Kermodevectors
and AofDe
Vita, Phys.
Rev. Lett, to
Inthe
Press
(2015)
23 fur
1 X Xij ZX
neighbouringrameter.
atoms with respect
central
one. However, was
several
ij
2 we have
The
ML
database
co
defined the vector and direction
matrices
(3)
dmn =
constraints have to be applied before the IVs are considered to be suitable for use in
atomic configuration. The similarity between
two
atomic
Force
Gaussian
Processes for Fracture Simulations at the Atomic Scale
ment Parlinksi-Li-Kawazoe method
environments xm and xn is expressed by the distance
=0.05database:
eV/Å (dotted lines) and
Scale factors
errover
the following equation:
4
5
⇥
10
eV/

k
N
k
m
n 2
m Å value
n 2(solid lines) f
X
X
X
Xij
Xij
Xijdatabase
)
(XThe
1
ij ML
2
rameter.
was c
2
′
′
′
(3)
dmn =
=
.
Distance metric
𝐓 i𝐫 = r1 𝐔1MD
𝐓 𝐫 trajectory.
+ r2 𝐓𝐔2 𝐓 𝐫 2+ r3 𝐓𝐔3 𝐓 𝐫
k i,j=1
N
i
i
n,m=1
Covariance
Cov(m, n) = exp
✓
d2mn
2
2 cov
“Curse of dimensionality” – QM is long range
• Forces depend mostly on atoms within
cutoff ~10 A from central atom
• Project target forces into internal
coordinate system (dim. k ~ 10-20)
• Perform GP regression there
• Overdetermined system - transform back
to find ‘best’ absolute Cartesian reference
frame in least squares sense
Z Li, JR
◆
j=1
(1
Comparing equation 10.3 with 10.4, we obtain the result:
+
2
mn err
𝐔𝑖 𝐓 𝐫 = 𝐓𝐔𝑖 (𝐫)
(𝑖 = 1, 2, and 3)
(1
F = AF
=
F
=
A
1
T Schematic 2DT projection of an atomic cluster within
Figure 10.1:
spherical radius r𝑐𝑢𝑡 . Both external coordinate {x, y} and internal
+ {U(r01, m1), U(r02, m2)} are shown. Force F on the target
coordinate
central atom (green ball) is indicated by the thick black arrow. The
internal vectors are functions of the positional vectors of the neighboring
atoms (red balls) while the parameters of r0 and m are both tunable to
generate different internal vectors.
+
A A
A
A F
In order to establish a 3-D internal coordinate for an atomic cluster, at least three inte
vectors (IVs) that are not co-planar have to be invented. Figure 10.1 demonstrates the IV
a two-dimensional case, where, both U(r01, m1) and U(r02, m2) are functions of the positi
Kermodevectors
and AofDe
Vita, Phys. Rev. Lett, In Press (2015)
neighbouring atoms with respect to the central one. However, 24
several fur
constraints have to be applied before the IVs are considered to be suitable for use in
ML for forces – interpolation accuracy
Teaching database built from
~2000 configs at 20 fs intervals
along a reference DFT MD
trajectory in Si at 1000 K
Teaching points
Test points
Time
Measure interpolation ability: test
configs midway between
teaching points
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
25
ML for forces – interpolation accuracy
Teaching database built from
~2000 configs at 20 fs intervals
along a reference DFT MD
trajectory in Si at 1000 K
Teaching points
Time
Test points
Measure interpolation ability: test
configs midway between
teaching points
Force error / eV/A
Sorting by distance and selecting
only ~500 most relevant
environments sufficient to
predict accurate forces at test
points
Augmenting descriptor IVs with
classical potential force vectors
significantly improves accuracy
~6% force error wrt DFT
Database size
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
26
nal
get
ed
QM
ent
ng
set
ineren⇥k
ere
spectrum obtained from the Stillinger-Weber (SW) clasHigh
Accuracy
Spectrum
sical potential
[29] isForces:
also shownPhonon
for comparison.
In parComputed using forces learnt from 300 K MD trajectory (DFTB)
(2)
ns-
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
FIG. 1. Comparison of the bulk Si phonon spectrum calcu-
27
On-the-fly Machine Learning of QM Forces
a
b•
•
•
•
•
64 atom bulk Si test system; QM forces evaluated every step for testing
Configs added to database when average force error > 0.1 eV/A
Learning rate provides measure of complexity for MD force learning - ‘chemical novelty’
T = 200 K – long periods (> 1 ps) where no QM is necessary
T = 1000 K – QM calculation rate of 1/30 sufficient after initial phase
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
28
Quantifying Chemical Novelty
b
c
Total # QM calcs per cycle
Individual QM calculation
Z Li, JR Kermode and A De Vita, Phys. Rev. Lett, In Press (2015)
29
(s)
sq
(S4)
= in multi-component
r̂sq exp
for predicting QMV
forces
systems. For the
of this tests, the
descriptor
Zs ,Zpurpose
q
i
r
(i)
q=1
is of the same form as described
in the main cut
paper, while the definition of the IVs (Eq. 1) was
extended so that individual vectors are constructed only between atoms with the same species:
where s is the atom of interest, q is one of its neighbours and Zs and Zq are the species of atoms
" ✓ feature
#
◆p(i)
s and q. The IVs are next combined toNX
form an extended
matrix
Xij similar to the
neighb
r
sq
(s)
one described in the main paper V
(Eq.
2
and
following
text),
with
an additional
block structure
(S4)
=
r̂
exp
sq
Zs ,Zq
i
r
(i)
cut
reflecting
the presence of di↵erent atomic species:
Internal
vectors
q=1
✓
◆
T
T
where s is the atom of interest, q V
is Zone
of itsVneighbours
and Zs and (a)
Zq SiO
are2 the species of atoms
Z1 A Z 2
1 A Z1
X=
(S5)
Tto form an
T extended feature matrix X
s
and
q.
The
IVs
are
next
combined
similar
to the
V
A
V
A
ij
Z2 Z1
Z2 Z 2
Block feature matrix
one described in the main paper (Eq. 2 and following text), with an additional block structure
T
where the
matricesthe
Vpresence
VZ2 and
ATZ2 formed
reflecting
of di↵erent
atomic
species:from the IVs and normalized directions
Z1 , AZ1 and
for species Z1 and Z2 , respectively.
✓
◆
T
T
VZ1 Alevel
V
Z1 A
We selected crystalline -SiC describedX
at=the DFTB
and
amorphous
SiO2 described at the
Z1
Z2
(S5)
T
T
V
A
V
A
Z
Z
DFT (LDA) level as test systems, to provide a range2of Zbonding
2 and
Z2 long-range order properties.
1
In both cases, we find that the forces from reference 300 K QM molecular dynamics trajectories can
T
T
where
the
matrices
V
,
A
and
V
and
A
the IVs and normalized
directions
Z
Z
1
2
Z1using a database containing
Z2 formed from
be reproduced within a 10% accuracy
2000 configurations
(Figs. S1(a)
for species
Z2 , respectively.
1 and
and S1(b)).
For theZ300
K amorphous
SiO2 trajectory, we further investigated if and to what extent
We
crystalline
at the
level and
SiO2the
described
the accuracy
ofselected
the predicted
forces-SiC
can described
be improved
by DFTB
incorporating
inamorphous
the descriptor
force at the
DFT (LDA)
as test systems,
to provide
a range
of bonding andinteratomic
long-rangepotential,
order properties.
vector predicted
by a level
short-ranged
form of the
polarisable
Tangney-Scandolo
In give
bothacases,
we findaccurate
that the description
forces from reference
300 As
K QM
trajectories
can
known to
reasonably
of silica [5].
canmolecular
be seen indynamics
Fig. S2(a),
this
bethe
reproduced
withinerror
a 10%
accuracy with
using aa modest
database⇠1000
containing
2000 configurations
(Figs. S1(a)
decreases
relative force
achievable
configuration
database from
and(corresponding
S1(b)). For the to
300a K
amorphous
SiO2compared
trajectory,towe
further
investigated
and to what extent
10% to 6%
0.06
eV/Å error,
the
average
DFT forceif magnitude
theÅ).
accuracy
the close
predicted
forces
can beerror
improved
by incorporating
in the
descriptor
of 1.0 eV/
This isofvery
to the
standard
assumed
in the Gaussian
Process
for thethe force
predicted
form of the polarisable Tangney-Scandolo interatomic potential,
teachingvector
data by
setting by
thea short-ranged
err hyperparameter to 0.05 eVÅ.
known
give a we
reasonably
accuratethe
description
of silica
As can
be seen to
in 1000
Fig. S2(a),
this
For the
SiCtosystem,
next increased
temperature
of the[5].target
trajectory
K,
decreases
theconfigurations
relative force are
error
a modest
configuration
and found
that 2000
stillachievable
sufficient with
to stay
within a⇠1000
10% force
error, oncedatabase
more from
(a) SiO2
(b) SiCDFT force magnitude
6%sets
(corresponding
to a 0.06 eV/
error,
compared
to the
provided10%
thattothe
of2,IVs
are DFT
augmented
by Å
forces
from
an appropriate
classical
interatomic
Amorphous
SiO
target
SiC,average
target
DFTB
1.0 this
eV/Å).
is very close topotential
the standard
error assumed
in
Gaussian
Process
the com
Supplementary
Figure
S2: Relative
error
of ML-predicted
force for
magnitudes
potentialof (in
caseThis
a Stillinger-Weber
[6] re-parametrised
to the
match
the bulk
and
the average QM force
magnitude
MD trajectories of: (a) a 72-atom amorphous SiO
teaching data
by setting
the
to 0.05
eVÅ.for
err hyperparameter
elastic properties
predicted
by the
reference
DFTB Hamiltonian
(Fig.
S2(b)).
at 300 K, computed
with DFT (LDA);
a 64-atom
-SiC system
at 1000K,
K, compu
For the SiC system, we next increased
the temperature
of the(b)target
trajectory
to 1000
DFTB. The black curves show results from descriptors including only IVs from Eq. S4.
Z Li, JR
and A De
Vita, within
Phys. Rev.
Lett,force
In Press (2015)
and found that 2000 configurations
areKermode
still
to descriptors
stay
10%
once more
30 p
curves
are sufficient
obtained from
also aincluding
forceserror,
from classical
interatomic
provided that the sets of IVs are augmented
by forces
appropriate
classical
interatomic for SiC
Tangney-Scandolo
(TS) forfrom
SiO2 an
[5] and
Stillinger-Weber
(SW) reparameterised
Extension to Multi-species Systems
Summary
• Hybrid QM/MM approaches allow ‘chemomechanical’ materials failure
problems to be modelled with QM precision
• Addresses lengthscale limitations of QM approaches e.g. DFT
• Practical: e.g., experimentally verifiable predictions of fracture phenomena
• Timescale limitations – prohibitive to model many important processes
• On-the-fly Machine Learning of QM forces improves timescales accessible
• Target optimal information efficiency – only do QM when necessary
• Need representation of atomic environments suitable for force learning
• Progressively fewer QM calculations are required when same chemical process is
encountered again
• Chemical ‘novelty’ and uncertainty can be quantified
• Next Steps
•
•
•
•
Improved atomic environment descriptors, e.g. vector covariance kernel
Use GP variance predictions to quantify uncertainties
Integrate on-the-fly ML and hybrid QM/MM approaches
Propagation of uncertainties through coarse-graining
31
Acknowledgments
Federico Bianchini, Marco Caccin, Zhenwei Li and Alessandro De Vita
Albert Bartok-Partay, Gábor Csányi and Mike Payne
Anke Peguiron, Gianpietro Moras, Lars Pastewka and Peter Gumbsch
Gaurav Singh and Robert Zimmerman
Chiara Gattinoni
Noam Bernstein
Financial support and supercomputing resources:
32
32