Interatomic potentials with error bars. Gábor Csányi Engineering Laboratory

Interatomic potentials with error bars Gábor Csányi Engineering Laboratory

What makes a potential Ingredients Desirable properties Representation of atomic neighbourhood smoothness, faithfulness, continuity Interpolation of functions flexible but smooth functional form, few sensible parameters Database of configurations predictive power non-domain specific

What makes a potential Ingredients Desirable properties {Functional form Representation of atomic neighbourhood Interpolation of functions smoothness, faithfulness, continuity flexible but smooth functional form, few sensible parameters Database of configurations predictive power non-domain specific

Function fitting with basis functions Fit a function f(x) based on observations y {y i } at {x i } f(x) = NX i k(x i,x) e.g. k(x, x 0 )= we 2 x x0 2 /2 2 i=1

Function fitting with basis functions Fit a function f(x) based on observations y {y i } at {x i } f(x) = y j = NX i k(x i,x) e.g. k(x, x 0 )= we 2 x x0 2 /2 2 NX i k(x i,x j ) i=1 i=1

Function fitting with basis functions Fit a function f(x) based on observations y {y i } at {x i } f(x) = y j = NX i k(x i,x) e.g. k(x, x 0 )= we 2 x x0 2 /2 2 i=1 NX regularised fit: i k(x i,x j )+ 2 ij arbitrary σ,σw,σν i=1

Function fitting with basis functions Fit a function f(x) based on observations y {y i } at {x i } f(x) = y j = NX i=1 NX i=1 y =(K + = C 1 y i k(x i,x) i k(x i,x j )+ 2 ij 2 I) e.g. k(x, x 0 )= 2 we x x0 2 /2 2 regularised fit: arbitrary σ,σw,σν [K] ij k(x i,x j ) C K + 2 I k k(x i,x)

Function fitting with basis functions Fit a function f(x) based on observations y {y i } at {x i } f(x) = y j = NX i=1 NX i=1 y =(K + = C 1 y i k(x i,x) f(x) =k T C 1 y i k(x i,x j )+ 2 ij 2 I) e.g. k(x, x 0 )= 2 we x x0 2 /2 2 regularised fit: arbitrary σ,σw,σν [K] ij k(x i,x j ) C K + 2 I k k(x i,x)

Machine learning framework: Kernel regression "(q (i) )= NX k k K q (i), q (k) Linear regression: K DP (q (i), q (k) )=q (i) q (k) Neural networks "(q (i) )= X j K NN q (i), q (k) = q (i) q (k) 2 + const. q (i) j NX k k q (k) j = q (i) Gaussian kernel K SE q (i), q (k) =exp X j (q (i) q (k) ) 2 2 2 j

1D example 1 f(x) GP splines 0.5 0 f(x) -0.5-1 1.5 2 2.5 3 3.5 4 r Gaussians basis functions are wide! f(x) =k T C 1 y

Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...) 2 0-2 -4-2 0 2 4

Generalised input data Often direct measurements of yi not available Consider any linear operator Compute P (y N+1 L(y) L(y) L can be built using Σ, @/@x etc Basis functions adapt to input data, e.g. for derivative data @k(x, x 0 )/@x Can deal with gradient data, sums, etc.

Traditional ideas for functional forms Pair potentials: Lennard-Jones, RDF-derived, etc. Three-body terms: Stillinger-Weber, MEAM, etc. " i = 1 2 X j V 2 ( r ij ) Embedded Atom (no angular dependence) Bond Order Potential (BOP) Tight-binding-derived attractive term with pair-potential repulsion ReaxFF: kitchen-sink + hundreds of parameters " i = P j ( r ij )

Fundamental limitation of analytic functional form Energy [ev / atom] 7 6 5 4 3 Tungsten, Embedded Finnis-Sinclair Atom Model bcc fcc sc DFT bcc fcc sc Carbon (diamond) Brenner DFT 2 1 0-1 12 14 16 18 20 22 Volume [A 3 / atom] Elastic constants: Brenner 1061 133 736 717

Construct smooth similarity kernel directly cutoff: compact support

Construct smooth similarity kernel directly Overlap integral Z S( i, i 0)= i (r) i 0(r)dr, cutoff: compact support

Construct smooth similarity kernel directly Overlap integral Z S( i, i 0)= i (r) i 0(r)dr, Integrate over all 3D rotations: Z k( i, i 0)= S( i, ˆR i 0) 2 d ˆR = Z d ˆR cutoff: compact support Z i (r) i 0( ˆRr)dr 2 After LOTS of algebra: X SOAP kernel k( i, i 0)= n,n 0,l p (i) nn 0 l p(i0 ) nn 0 l nl p nn 0 l = c nl c n 0 l

How to generate databases? Target applications: large systems Capability of full quantum mechanics (QM): small systems QM MD on representative small systems: sheared primitive cell elasticity large unit cell phonons surface unit cells surface energy gamma surfaces screw dislocation vacancy in small cell vacancy vacancy @ gamma surface vacancy near dislocation Iterative refinement 1. QM MD Initial database 2. Model MD 3. QM Revised database What is the acceptable validation protocol? How far can the domain of validity be extended?

Existing potentials for tungsten Error 15% 50% BOP MEAM FS DFT reference 0 C11 C12 C44 Elastic const. Vacancy energy (100) (110) (111) (112) Surface energy

Building up databases for tungsten (W)

Existing potentials for tungsten Error 15% 50% BOP MEAM FS DFT reference 0 C11 C12 C44 Elastic const. Vacancy energy (100) (110) (111) (112) Surface energy

Existing potentials for tungsten Error DFT reference 15% 0 50% GAP BOP MEAM FS C11 C12 C44 Elastic const. Vacancy energy (100) (110) (111) (112) Surface energy

Peierls barrier for screw dislocation glide

Vacancy-dislocation binding energy (~100,000 atoms in 3D simulation box)

He-Vacancy interaction in tungsten - preliminary Collaboration with Duc Nguyen-Manh (CCFE) Add He-W interaction on top of W potential 200 training configurations of He@V, 44 test energy error force error

He-Vacancy interaction in tungsten 2-body SOAP-GAP Needs more He data, other defects Do the same with H@W, with Takuji Oda

Self-aware potentials: predicting the error Bulk silicon: Diamond and Beta-tin phases Predicted error correctly signals where GAP is unreliable Energy Error

Outstanding problems Accuracy on database accuracy in properties? Database contents region of validity? Alloys - permutational complexity? Chemical variability? Systematic treatment of long range effects Electronic temperature