BigData Analytics in Materials Science · BigData Analytics in Materials Science Luca M. Ghiringhelli Fritz Haber Institute Handson workshop and HumboldtKolleg: DensityFunctional

BigData Analytics in

Materials Science

Luca M. GhiringhelliFritz Haber Institute

Handson workshop and HumboldtKolleg:DensityFunctional Theory and Beyond Basic Principles and Modern Insights

Isfahan University of Technology, Isfahan, Iran, May 2 to 13, 2016

Data, data, data: big data

Bigdata challenge, fourV:Volume (amount of data)Variety (heterogeneity, of form and meaning of data)Veracity (uncertainty of data quality)Velocity ?

Highthroughput screening: query and read out what was stored

Shouldn't we do more?

Analysis

Identify (so far) hidden correlations Identify which materials should be studied next as most promising candidates Identify anomalies

We have a dream

From the periodic table of the elements to a chart of materials

Mendeleev's 1871 periodic table

We have a dream

From the periodic table of the elements to a chart of materials

Mendeleev's 1871 periodic table

Ga=69.7 Ge=72.6

We have a dream

From the periodic table of the elements to a chart of materials:Organize materials according to their properties and functions, e.g.

figure of merit of thermoelectrics (as function of T )

turnover frequency of catalytic materials (as function of T and p)

efficiency of photovoltaic systems

Training setCalculate properties and

functions P, for many materials, iDensityFunctional Theory

Fast PredictionCalculate properties

and functions for new values of d (new materials)

Big Data Analysis

DescriptorFind the appropriate

descriptor di, build a table: | i | di | Pi |

LearningFind the function PSL(d) for the table;

do cross validation.Statistical learning

(Orbital period)² = C (orbit's major axis)³

Learning Discovery→

Suppose to know the trajectories of all planets in the solar system, from accurate observations (experiment)orby numerically integrating general relativity equations (calculations at the highest level of theory)

Data (collected by Tycho Brahe)

Statistical learning(performed by

Johannes Kepler)

Physical law(assessed by

Isaac Newton)

Databases, platforms

“Just Databases”ICSD Inorganic Crystal Structure DB http://icsd.fizkarlsruhe.deCOD Crystallography Open DB http://www.crystallography.net/ESP Electronic Structure Project http://gurka.fysik.uu.se/ESP/CCCBDB Comp. Chemistry Comparison and Benchmark DB

http://cccbdb.nist.gov/Databases + analytic toolsMaterials project http://www.materialsproject.orgAFLOW Atomatic Flow for Materials Discovery http://aflowlib.orgAiiDA Automated interactive infrastructure and DB for Atomistic Simulations

http://www.aiida.netOQMD Open Quantum Materials DB http://oqmd.org/

http://nomadrepository.euhttp://nomadcoe.eu

Codedependent raw data conversion layer codeindependent representation→ →

Outline

Descriptors and fingerprints

A (personaltaste compiled) zoo of machine learning / data mining techniques

Regularised regression

Linear and nonlinear dimensionality reduction

Feature selection

Some words on causal descriptorproperty relationship

Descriptors

Can we predict an optimal material for a complex process (e.g. heterogenous catalysis)

by looking to a simple (set of) descriptor(s) ?

A simple but insightful descriptor

(Geneticlike) fingerprint: 1D polymers “eugenetics”

Data: 175 linear 4blocks periodic polymers. 7 blocks: CH2, SiF2, SiCl2, GeF2, GeCl2, SnF2, SnCl2,

Descriptor: 20 dimensions [# building blocks of type i, of ii pairs, of iii triplets]

Pilania, Wang, …, and Ramprasad, Scientific Reports 3, 2810 (2013). DOI: 10.1038/srep02810

Isayev, …, and Curtarolo, Chemistry of Materials 27, 735 (2015)

(Geneticlike) fingerprint

Isayev, …, and Curtarolo, Chemistry of Materials 27, 735 (2015)

(Geneticlike) fingerprint

Supervised learning

d → P mapping Support vector machinesNeural networksDecision treesGenetic programming (symbolic regression)Kernel ridge regressionCompressed sensing

Unsupervised learning

d → d'Find patterns / trends

Principalcomponents analysisNonlinear dim. reduction Sketch mapClusteringLocal pattern discovery

Machine learning / data mining : a classification

Focus on “learning”: the algorithm has to improve with data size (“learning by experience”)

Supervised learning

d → P mapping Kernel ridge regressionCompressed sensing (+ symbolic regression)



Principalcomponents analysisNonlinear dim. reduction Sketch map



Figure of merit to be optimized:

Regularization (prefer “lower complexity” in the solution)

(Linear) ridge regression

Explicit solver:

Alternative view, via Hilbert space representation theorem:

Sum over data points!

Ridge Regression: Mathematical formulation

norm

Linear kernel

Kernel Ridge Regression: Mathematical formulation

Nonlinear kernel


Nonlinear kernel

Linear kernel

Gaussian (radial basis function) kernel

Laplacian kernel

Polynomial kernel


Nonlinear kernel

In all cases, a kernel introduces a similarity measure

KRR success stories: Gaussian Approximation Potentials

Translation, rotational, permutational invariant, unique, smooth localenvironment descriptor.(Sphericallyaveraged spherical harmonic expansion of Gaussian densities centered on nuclei)

KRR success stories: Molecular properties

Pilania, Wang, …, and Ramprasad, Scientific Reports 3, 2810 (2013). DOI: 10.1038/srep02810

KRR success stories: 1D polymers “eugenetics”

Data: 175 linear 4blocks periodic polymers. 7 blocks: CH2, SiF2, SiCl2, GeF2, GeCl2, SnF2, SnCl2,

Descriptor: 20 dimensions [# building blocks of type i, of ii pairs, of iii triplets]

Regularized regression in practice: beware of overfitting

Regularized regression in practice: beware of overfitting

Regularized regression in practice: do validation

Supervised learning







Showcase: classification octet binaries crystal structures

The chemical space

Ansatz: atomic features

● Valence number Zv● Energy of valence s orbital Es● Energy of valence p orbital Ep● Radius of valence s orbital rs● Radius of valence p orbital rp



KS

leve

ls [

eV]

Valence p

Valence sRadial probability densities

[Å]

Primary (atomic) features

Radius @ maxAverage radiusTurning point

example: Sn (Tin)

Valence p (HOMO)

Valence s

KS lev els [eV

]

LUMO

(Linear) dimensionality reduction: principal components

Principal component analysis

Pearson, K. "On Lines and Planes of Closest Fit to Systems of Points in Space". Philosophical Magazine 2, 559 (1901)

Orthonormal transformation of coordinates, converting a set of (possibly) linearly correlated coordinates into a new set of linearly uncorrelated (called principal or normal) components, such that the first component has the largest variance and each subsequent has the largest variance constrained to being orthogonal to all the preceding components

Principal component analysis

Pearson, K. "On Lines and Planes of Closest Fit to Systems of Points in Space". Philosophical Magazine 2, 559 (1901)

(Linear) dimensionality reduction: principal components

Saad, …, Chelikowsky, and Andreoni, PRB 85, 104104 (2012)1 2 3A

rb. (

linea

r) s

cale

Components



rs, rp, Es/Zv, Ep/zv,

for A and B atoms



rs, rp, Es/Zv, Ep/zv,

for A and B atoms

What's on the axes?

Linear combination of (possibly all) the initial dimensions

Supervised learning







(Nonlinear) dimensionality reduction



Proximity matchingProximity matching

Sketchmap algorithm

Minimization of the stress function (for a set of landmarks points)

Sketchmap algorithm

From clusters to defects in bulk

The high dimensional representation is still an important choice

From clusters to defects in bulk

What's on the axes?

Supervised learning







What about having a dimensionality reduction, or call it feature selection,

i.e., such that the (best) lowdimensional representation is selected

among (many many) given candidates?

It is time for: compressed sensing

Reference:LMG, J. Vybiral, S. V. Levchenko, C. Draxl, and M. Scheffler,

Phys. Rev. Lett. 114, 105503 (2015)Don't overlook the Supplementary Information!

82 octet AB binary compounds

We have a dreamProof of Concept: Descriptor for the Classification “Zincblende/Wurtzite or Rocksalt?”

Rocksalt

ZincblendeRocksalt/Zincblende


d1

d2 RS

J. A. van Vechten, Phys. Rev. 182, 891 (1969).J. C. Phillips, Rev. Mod. Phys. 42, 317 (1970).J. St. John and A.N. Bloch, Phys. Rev. Lett. 33, 1095 (1974)A. Zunger, Phys. Rev. B 22, 5839 (1980).D. G. Pettifor, Solid State Commun. 51, 31 (1984).Y. Saad, D. Gao, T. Ngo, S. Bobbitt, J. R. Chelikowsky, and W. Andreoni, Phys. Rev. B 85, 104104 (2012).

?


Rocksalt

ZincblendeRS/ZB


d1

d2 RS



● HOMO● LUMO● Ionization Potential● Electron Affinity● Radius of valence s orbital● Radius of valence p orbital● Radius of valence d orbital● … ?



E(Rocksalt) – E(Zinkblende)E(Rocksalt) – E(Zinkblende)

Rocksalt

ZincblendeRS/ZB

Figure of merit to be optimized:

Regularization (prefer “lower complexity” in the solution)

A more complex regularization:

(Linear) ridge regression

Mathematical formulation of the problem

NP – hard !!!

Mathematical formulation of the problem: sparsity

LASSO: convex problem, equivalent to the NP-hard if features (columns of D) are uncorrelated

LASSO, compressed/ive sensing in Materials Science

Find a descriptor AND an accurate evaluation for the difference in energy between RS and ZB crystal structures for all (82) AB octet semiconductors.ΔE = ΔE ( d )

Possibly identify a 2D descriptor which gives a “nice” representation of the materials in a plane

The task






KS level 1 KS level 2

+

Radius 1 Radius 2

/

| x y |

Systematic construction of the feature space

+

Radius 1 Radius 2 KS level 1 KS level 2

| x y |

exp(x)

(x)^n

In practice: formalism borrowed form symbolic regression

Systematic construction of the feature space: EUREQA

Descriptor (candidates: 242)a The largest distance between a H atom and its nearest Si neighborb The shortest distance between a Si atom and its sixthnearest Si neighborc The maximum bond valence sum on a Si atomd The smallest value for the fifthsmallest relative bond length around a Si atome The fourthshortest distance between a Si atom and its eighthnearest neighborf The secondshortest distance between a Si atom and its fifthnearest neighborg The thirdshortest distance between a Si atom and its sixthnearest neighborh The HSi nearestneighbor distance for the hydrogen atom with the fourthsmallest difference between the distances to the two Si atoms nearest to a H atom

T. Müller et al. PRB 89 115202 (2014):Data: ~1000 amorphous structures of 216 Si atoms (saturated)

Property: hole trap depth

EUREQA: genetic programming software. Global optimization (genetic algorithm).Schmidt M., Lipson H., Science, Vol. 324, No. 5923, (2009)

Find a descriptor AND an accurate evaluation for the difference in energy between RS and ZB crystal structures for all (82) AB octet semiconductors.ΔE = ΔE ( d )

Possibly identify a 2D descriptor which gives a “nice” representation of the materials in a plane

The task


● HOMO H● LUMO L● Ionization Potential IP● Electron Affinity EA● Radius of valence s orbital rs● Radius of valence p orbital rp● Radius of valence d orbital rd● Thousands of nonlinear

functions of the above


● HOMO H● LUMO L● Ionization Potential IP● Electron Affinity EA● Radius of valence s orbital rs● Radius of valence p orbital rp● Radius of valence d orbital rd● Thousands of nonlinear

functions of the above


1D

2D

3D

“Extended” LASSO : features are correlated, so the first 25-30 features selected by lasso when scanning from large to low λ are selected and all single features, all pairs, all triplets... are separately tested via linear regression (the NP-hard problem, but only with 25-30 features)

1D 2D 3D

Finding the descriptor

Twodimensional descriptor

0 0.2 eV 0.45 eV 1.0 eV-0.2 eV

A good model must be predictive within the data domain (interpolation):

cross validation

Performance of the descriptors: accuracy, validation

ε

!

“Complexity”

Erro

r

Training err.

Validation err.

Leave 10% out cross validation

Errors are energies, in eV

Max Absolute Error

Convergence with dimensionality of the descriptor

Regularized regression in practice: do validation


cross validation

A better model should be causal:stability analysis

Few words on causality

There are four possibilities (types of causality relationship) behind P(d):

1. d → P : P “listens” to d

2. P → d : d “listens” to P

3. A → d and A → P : There is no direct connection between d and P, but d and P both “listen” to a third “actuator”

4. There is no direct connection between d and P, but they have a common effect (Berkson paradox)

...that listens to both and screams: “I occurred” [Judea Pearl]

[If the admission criteria to a certain graduate school call for either high grades as an undergraduate or special musical talents, then these two attributes will be found to be correlated (negatively) in the student population of that school, even if these attributes are uncorrelated in the population at large (selection bias). Indeed, students with low grades are likely to be exceptionally gifted in music, which explains their admission to graduate school.]

Few words on causality

We are not able to write down a scientific law that connects the descriptor

directly with the total-energy difference between RS and ZB structures.However, ZA, ZB determine these descriptors, and ZA, ZB determine the many-body Hamiltonians and the total-energy difference.

ML has diligently took over “Kepler's work”, but no Newton, yetQuestion: is the latter step always necessary?

Quantitative analysis: effect of noise

The same 2D descriptor is found:


cross validation

A better model should be causal:stability analysis

An ideal model should be predictive outside the data domain (extrapolation)

When both carbon diamond and BN are excluded from training:

If all C containing binaries (C, SiC, GeC, and SnC) are excluded from training, i.e. no explicit information on C is given to the model:

Hadn't we known about diamond … we'd have predicted it!

Hadn't we known about any carboncontaining binary …we'd have predicted carbon chemistry (from atomic features)

E(LDA) E(predicted)

C -2.64 eV -1.37 eV

SiC -0.67 eV -0.48 eV

GeC -0.81 eV -0.46 eV

SnC -0.45 eV -0.23 eV

E(LDA) E(predicted)

C -2.64 eV -1.44 eV

BN -1.71 eV -1.37 eV

Bigdata for Materials Science: Infrastructures

Descriptors and fingerprints

(Selected) machinelearning / datamining methods:Kernel ridge regressionAutomatic descriptor search: dimensionality reduction

Principal component analysisSketch map

Automatic descriptor search: Feature selectionLASSO (compressed sensing) + symbolic regression

Application to a model materialsscience problemApplication of compressed sensing to basisset construction

Some words on causal descriptorproperty relationshipCrossvalidation and Stability analysis

Summary

Documents

BigData Analytics in Materials Science · BigData Analytics in Materials Science Luca M. Ghiringhelli Fritz Haber Institute Handson workshop and HumboldtKolleg: DensityFunctional