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Constructing interatomic potentials from first principles using machine learning: the example of tungsten
Gábor Csányi Engineering Laboratory
Quantum mechanics is many-body
-1
0
1
2
3
4
5
6
7
12 14 16 18 20 22
Ener
gy [e
V / a
tom
]
Volume [A3 / atom]
Tungsten, Finnis-Sinclairbccfccsc
bccfccsc
DFT
Tungsten: DFT, Embedded Atom ModelCarbon: tight binding
Quantum mechanics has some locality
force F
r
neighbourhood
far field
Force errors around Si self interstitial
Force errors around O in water with QM/MM
r
L̂i = qi + pi ·⇥i + . . .Vel
(R1
, R2
, . . .) ⇡atomsX
i
"(R1
�Ri, R2
�Ri, . . .) +1
2
X
ij
L̂iL̂j1
Rij+
�ij
|Rij |6
Finite range atomic energy function
Traditional ideas for functional forms
• Pair potentials: Lennard-Jones, RDF-derived, etc.
• Three-body terms: Stillinger-Weber, MEAM, etc.
• Embedded Atom (no angular dependence)
• Bond Order Potential (BOP) Tight-binding-derived attractive term with pair-potential repulsion
• ReaxFF: kitchen-sink + hundreds of parameters
These are NOT THE CORRECT functions. Limited accuracy, not systematic
"i =1
2
X
j
V2(|rij |)
"i = ��P
j ⇢(|rij |)�
given byGOAL: potentials based on quantum mechanics
Wishlist
• Representation of atomic neighbourhood
• Interpolation of functions
• Databaseof configurations
smoothness, faithfulness, continuity
flexible but smooth functional form, few sensible parameters
predictive power non-domain specific
Basic function fitting with basis functions
Fit a function based on observations at y ⌘ {yi} {xi}f(x)
yj =NX
i=1
↵ik(xi, xj)
f(x) =NX
i=1
↵ik(xi, x)
regularised fit:
e.g. k(x, x0) = �
2w
e
�|x�x
0|2/2�2
k ⌘ k(xi, x)↵ = C�1y
[K]ij ⌘ k(xi, xj)
f(x) = kTC�1y
C ⌘ K+ �2⌫I
yj =NX
i=1
↵i
�k(xi, xj) + �
2⌫�ij
�
y = (K+ �2⌫I)↵
arbitrary σ,σw,σν
f(x) = kTC�1y
Representing the atomic neighbourhood in strongly bound materials
• What are the arguments of the atomic energy ε ? Need a representation, i.e. a coordinate transformation
- Exact symmetries: • Global Translation
• Global Rotation
• Reflection
• Permutation of atoms
- Faithful: different configurations correspond to different representations
- Continuous, differentiable, and smooth (i.e. slowly changing with atomic position) (“Lipschitz diffeomorphic”)
• Rotational invariance by itself is easy: q ≡ Rij = ri ⋅ rj (Weyl)
- Complete, but not invariant permutationally
- Not continuous with changing number of neighbours
r1
r2
r3
r4
Machine learning framework: a variety of kernels
Gaussian Process Regression:
• Linear regression:
• Neural networks
• Gaussian kernel
"(q(i)) =NX
k
↵kK�q(i),q(k)
�
"(q(i)) =X
j
q(i)j
NX
k
↵kq(k)j = q(i) · �KDP(q
(i),q(k)) = q(i) · q(k)
KSE
�q(i),q(k)
�= exp
⇣�X
j
(q(i) � q(k))2
2�2j
⌘
KNN
�q(i),q(k)
�= �|q(i) � q(k)|2 + const.
Construct smooth similarity kernel directly
• Overlap integral
k(⇢i, ⇢i0) =X
n,n0,l
p(i)nn0lp(i0)nn0l
• After LOTS of algebra: SOAP kernel pnl = c†nlcnl
pnn0l = c†nlcn0l
S(⇢i, ⇢i0) =
Z⇢i(r)⇢i0(r)dr,
k(⇢i, ⇢i0) =
Z ���S(⇢i, R̂⇢i0)���2dR̂ =
ZdR̂
����Z
⇢i(r)⇢i0(R̂r)dr
����2
cutoff: compact support• Integrate over all 3D rotations:
Gaussian Approximation Potential: Remarks• Cannot observe atomic energy ε in Quantum Mechanics
- Total energies, forces, stresses: sums of ε and ∂ε at different locations
• Many data point locations are very similar - Automatic sparsification of data, remove similar configurations
• Probabilistic model: noise control
• Computational cost: ~ 0.01 sec/atom/cpu core
• Essentially no “truly free” parameters, but:
• What physics do we get for what we put in the database?
Fitting to derivatives:
Fitting to sums:
"
⇤(x) =X
n
↵n@
@x2K(x, xn)
"
⇤(x) =X
n
↵n[K(x, x0n) +K(x, x00
n)]
Building up databases for tungsten (W)
Existing potentials for tungsten (W)
DFT reference
Error
Peierls barrier for screw dislocation glide
Vacancy-dislocation binding energy
(~100,000 atoms in 3D simulation box)
Outstanding problems
• Accuracy on database accuracy in properties?
• Database contents region of validity ?
• Systematic treatment of long range effects
- polarisable multipole electrostatics
- many-body dispersion
• Electronic temperature
- potential explicitly dependant on “local” kinetic temperature ?
The grand plan
Fast, accurate potentials
conformation exploration and
analysis (“sampling”)
High-throughput materials discovery
Databases
Efficient Markov chain mixing
Machine learning
Quantum mechanics
Bottom-up prediction of
materials properties
Numerical analysis
Link-up to higher length scale models
The team & friends
Albert P. Bartók (Cambridge)
Wojciech Szlachta (Cambridge)
Risi Kondor (Chicago)
Mike Payne (Cambridge)
Livia B. Pártay (Cambridge)
Christoph Ortner (Warwick)
James Kermode (KCL)
Alessandro De Vita (King’s College London)
Peter Gumbsch (IWM)
Noam Bernstein (NRL)
Robert Baldock (Cambridge)
Letif Mones (Cambridge)
Jeff Hammond (Argonne)
Thomas Stecher (Cambridge)
Peter Pinski (Cambridge)
Sebastian John (Cambridge)
Dov Sherman (Technion)
Alan Nichol (Cambridge)
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