Interatomic potentials with error bars. Gábor Csányi Engineering Laboratory
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1 Interatomic potentials with error bars Gábor Csányi Engineering Laboratory
2 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
3 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
4 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
5 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
6 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
7 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)
8 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)
9 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) f(x) =k T C 1 y
10 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) ) j
11 1D example 1 f(x) GP splines f(x) r Gaussians basis functions are wide! f(x) =k T C 1 y
12 Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...)
13 Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...)
14 Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...)
15 Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...)
16 Samples from Gaussian posterior P (y N+1 y) / Normal(k T C 1 y,...)
17 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 etc Basis functions adapt to input data, e.g. for derivative x 0 )/@x Can deal with gradient data, sums, etc.
18 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 )
19 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 ) Representation is implicit
20 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 ) Representation is implicit These are NOT THE CORRECT functions. Limited accuracy, not systematic
21 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 ) Representation is implicit These are NOT THE CORRECT functions. Limited accuracy, not systematic given by GOAL: potentials based on quantum mechanics
22 Fundamental limitation of analytic functional form Energy [ev / atom] Tungsten, Embedded Finnis-Sinclair Atom Model bcc fcc sc DFT bcc fcc sc Carbon (diamond) Brenner DFT Volume [A 3 / atom] Elastic constants: Brenner
23 Construct smooth similarity kernel directly cutoff: compact support
24 Construct smooth similarity kernel directly Overlap integral Z S( i, i 0)= i (r) i 0(r)dr, cutoff: compact support
25 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
26 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
27 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
28 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 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?
29 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
30 Building up databases for tungsten (W)
31 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
32 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
33 Peierls barrier for screw dislocation glide
34 Vacancy-dislocation binding energy (~100,000 atoms in 3D simulation box)
35 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
36 He-Vacancy interaction in tungsten 2-body SOAP-GAP Needs more He data, other defects Do the same with with Takuji Oda
37 Self-aware potentials: predicting the error Bulk silicon: Diamond and Beta-tin phases Predicted error correctly signals where GAP is unreliable Energy Error
38 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
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