Machine Learning for the Materials Scientist

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1 Machine Learning for the Materials Scientist Chris Fischer*, Kevin Tibbetts, Gerbrand Ceder Massachusetts Institute of Technology, Cambridge, MA Dane Morgan University of Wisconsin, Madison, WI

122, 084119 (2005) O(N3) Moore, G. ISSCC 2003 slides (http://www.

2 Motivation: materials design through calculation computing power: exponential scaling with time Skylaris, C. et. al. J. Phys. Chem. 122, (2005) O(N3) Moore, G. ISSCC 2003 slides ( Run-time: polynomial scaling with number of atoms O(N)

3 DFT as a predictive tool Burkett, T. et. al. Phys. Rev. Lett. 93 (2004) Norskov, J. et. al. MRS Bulletin 31 (2006) Marzari, N. MRS Bulletin 31 (2006) courtesy of M. Lazzeri, Paris VI Jussieu Marzari, N. MRS Bulletin 31 (2006) courtesy of D. Scherlis, MIT

4 computational materials design strategies Calculating properties of realistic nanostructures ab initio Lee, Y. S. et al. PRL (2005) Galli, G. University of California, Davis

5 computational materials design strategies Which combinations yield the optimal material?

6 Outline Machine learning in Computational Materials Design Searching for Structure: combining historical information with Density Functional Theory Data Mining the High-Throughput engine wrap-up

7 computational materials design strategies Which combinations yield the optimal material?

8 Motivation: searching for new materials for i in (relevant chemistries) { getstablephases(i); calculateproperty(i); i = nextchemistry(); } Depends on which phases are stable and their structure

9 Motivation: materials by design for i in (relevant chemistries) { getstablephases(i); calculateproperty(i); i = nextchemistry(); Machine Learning Depends which needed on here!! phases are stable and their structure }

10 The need for machine learning DFT Code Doesn't know what to calculate next Material Property Predictions

11 The need for machine learning DFT Code Machine Learning Framework Material Property Predictions Database of Computed and Experimental results

12 Computational Materials Design poised for impact 'Commodity' computational resources Open source electronic structure software ~$ k capital investment Computing budget ~50k compounds/year

13 Computational Materials Design poised for impact ICSD: World's Largest database of inorganic crystal structures First Entry: 1913 # of entries: 100,243 # usable compounds: 29,962 Computing budget ~50k compounds/year

14 The structure search problem for i in (relevant chemistries) { getstablephases(i); Where do we put the atoms calculateproperty(i); if no experimental structure i = nextchemistry(); is known?? } Depends on which phases are stable and their structure

15 Strategies to search for structure Coordinate Search: Optimize energy (or free energy) directly in the space of atomic coordinates Heuristic Rules or Chemical Intuition

16 Methods to search for structure Coordinate Search: Optimize energy (or free energy) directly in the space of atomic coordinates GroundState arg min r 1, r 2,, rn E r 1, r 2,, r N # of dimensions = 3N 3 + dim(a,b,c,,, ) complex energy landscape Doye, J. PRL, 88, , (2002)

17 Methods to search for structure Coordinate Search: GroundState arg min r Optimize energy (or free energy) directly in the space of atomic coordinates 1, r 2,, rn E r 1, r 2,, r N # of dimensions = 3N 3 + dim(a,b,c,,, ) Proposed Solutions Calculate energy of a finite set of structure prototypes Doye, J. PRL, 88, , (2002)

18 Methods to search for structure Coordinate Search: GroundState arg min r Optimize energy (or free energy) directly in the space of atomic coordinates 1, r 2,, rn E r 1, r 2,, r N # of dimensions = 3N 3 + dim(a,b,c,,, ) Proposed Solutions Calculate energy of a finite set of structure prototypes Use a stochastic optimization procedure (hop from basin to basin) e.g., Simulated Annealing Genetic Algorithms Doye, J. PRL, 88, , (2002)

r 2,, rn E r 1, r 2,, r N # of dimensions = 3N 3 + dim(a,b,c,,, ) Proposed

not transferred across chemistries Use a stochastic optimization procedure (hop

19 Methods to search for structure Coordinate Search: GroundState arg min r Optimize energy (or free energy) directly in the space of atomic coordinates 1, r 2,, rn E r 1, r 2,, r N # of dimensions = 3N 3 + dim(a,b,c,,, ) Proposed Solutions Calculate energy of a finite set of structure prototypes Knowledge is not transferred across chemistries Use a stochastic optimization procedure (hop from basin to basin) e.g., Simulated Annealing Genetic Algorithms Doye, J. PRL, 88, , (2002)

20 Methods to search for structure Heuristic Rules Use previous experiments to suggest what to calculate How? Identify a set of simple parameters based on alloy constituents 1932: Pauling electronegativity 1935: Laves & Witte 1926,1936-7: Hume-Rothery, Mott & Jones 1976: Miedema r A,B n ate e ws n

21 Methods to search for structure Heuristic Rules Plot stable structures in space of parameters 1986: Pettifor 1983: Villars r A,B n ate

Methods to search for structure Heuristic Rules Plot stable structures in space of parameters 1986: Pettifor 1983: Villars r A,B n ate Heuristic

22 Methods to search for structure Heuristic Rules Plot stable structures in space of parameters 1986: Pettifor 1983: Villars r A,B n ate Heuristic rules efficiently code historical knowledge provide transfer of knowledge Can we leverage historical knowledge to intelligently search for structure?

23 description of knowledge base Knowledge Base Experimental Data Pauling File binaries edition (Villars, P. et. al. J. of Alloys and Compounds, (2004)) 1335 binary alloys 3975 non-unique compounds 4263 compounds total alloys not containing elements: He, B, C, N, O, F, Ne, Si, P, S, Cl, Ar, As, Se, Br, Kr, Te, I, Xe, At, Rn

24 Machine learning framework: concepts x= x A, x0,, x 1,, xb 2 Data {x 1,, x N } Low temperature state of alloy database of N binary alloys

25 Machine learning framework: concepts x= x A, x0,, x 1,, xb 2 Low temperature state of alloy database of N binary alloys Data {x 1,, xn } p x p x e Probability of low temperature state (fitted to data) Probability of low temperature state conditioned on evidence 'e'

26 how to use the machine learning framework DFT Code Set of likely structure candidates Machine Learning Framework p x e Material Property Predictions Database of Computed and Experimental results

27 Preliminaries and open questions Are probabilities consistent with physical intuition? Do probabilities encode the physics of structure stability?

28 quantifying correlation in probabilistic framework probability that both structures occur in same system estimated from database g(2)(xi,xj) correlated Pair Cumulant 1 uncorrelated anti-correlated p xi, x j gij x i, x j = p x i p x j probability that only xi occurs 0

29 how probabilities represent physics of mixing Do probabilities embody real physical effects? Compounds stabilized by size effect: p xi, x j gij x i, x j = p x i p x j Fe3C 8.48 MgCu cb Data from Pauling File, Binaries Edition 1

Compounds stabilized by size effect: p xi, x j gij x i, x j = p x i p x j

30 how probabilities represent physics of mixing Do probabilities embody real physical effects? Compounds stabilized by size effect: p xi, x j gij x i, x j = p x i p x j Fe3C 8.48 ~0 MgCu cb Data from Pauling File, Binaries Edition 1 Places 'small' atoms on 'large' atom sites G. Ceder

31 how probabilities represent physics of mixing: more interesting correlations Gd2Co7 AABAAB... stacking gij x i, x j =54 PuNi3 ABAB... stacking A B A A A B B A A B Both structures share the same local environments

32 Structure correlation observations Correlation factors are probabilistic analogue of heuristic rules No explicit reference to physics. Physics is embedded in experimental data

33 Information theory for structure stability Suppose I know Fe3C c = ¾, how does this change c = ½? How much information is carried by knowledge of structure? Mutual Information Ii, j= xi, x j p xi, x j p x i, x j log p xi p x j Ii, j= log [ g ij xi, x j ]

34 Information theory for structure stability Ii, j= xi, x j p x i, x j p x i, x j log p xi p x j e.g., Xi= AB prototype and Xj= A2B prototype degree of correlation Each element of matrix is correlation between Xi and Xj

35 Prediction and validation in Li-Pt

36 Predicting structures in Li-Pt?? AlB2 LiRh MgCu2 CuPt7 a.k.a. MgPt7 Use these as conditioning evidence for: p x e

37 Predicting structures in Li-Pt Known phases Suggested phases

38 cross validation to evaluate performance Success of method DFT Code on how depends short this list is Set of likely structure candidates Machine Learning Framework p x e Material Property Predictions Database of Computed and Experimental results

39 Length of List = average 'loss' Cross validation results ~28 candidates req'd for freq. 10 candidates --> 95% chance of seeing GS!! Independent Variables Including structure correlation Nature Materials, 6, , 2006

40 Some open questions ICSD: World's Largest database of inorganic crystal structures What is the information content in a chemical database? How many 'independent' crystal structures exist in nature? First Entry: 1913 # of entries: 100,243 # usable compounds: 29,962 # structure prototypes: 2,485

41 Structure prediction: wrap-up for i in (relevant chemistries) { Much more needed here getstablephases(i); calculateproperty(i); i = nextchemistry(); Now have efficient tool for this }

42 Directions for future work/collaboration DFT Code Database of Computed results Material Property Predictions Machine Learning Framework

43 Directions for future work/collaboration Set of features Charge Density Total energy Bulk moduli Coordination Bond strength Bond character Magnetic moments Polarization... DFT Code

44 Directions for future collaboration DFT Code Database of Computed results Material Properties -catalytic activity -conductivity -plasticity -voltage/energy density Machine Learning Framework (functional mapping)

45 The End Data from High Throughput alloy study Online structure predictor ITR grant (DMR )

46 DELETE ME!!! introduce CMS, what is it being applied to? Data mining and materials design make some outline slide? introduce structure prediction problem, present our solution discuss higher order property prediction. data management, dissemination

47 DATASET NOTES 1335 alloys 3975 non-unique compounds 4263 compounds total alloys not containing elements: He, B, C, N, O, F, Ne, Si, P, S, Cl, Ar, As, Se, Br, Kr, Te, I, Xe, At, Rn

(S1) (S2) = =8.16

(S1) (S2) = =8.16 Formulae for Attributes As described in the manuscript, the first step of our method to create a predictive model of material properties is to compute attributes based on the composition of materials.