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)